Mathematics

Mathematical Symbols - Calculus and Analysis

2020-07-31T23:47:00.003+05:30

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss the eighth topic of Mathematical symbols - Calculus and Analysis on this post.

Mathematics symbols

These are very important Basic Tools frequently used by us not only in Calculus and Analysis but also in Physics, Chemistry, Economics as well as in other branches of applied mathematical sciences, humanities, and engineering.

Table of important Mathematical Symbol of Calculus and Analysis

*Symbol*	*Name of Symbol*	*Definition / Meaning*	*Illustration*
(a, b)	open interval	(a, b) = {x: a < x < b} or (a, b) = {x / a < x < b}	Terminal / End points are not included (Excluded).
[a, b]	closed interval	[a, b] = {x: a ≤ x ≤ b} or [a, b] = {x / a ≤ x ≤ b}	Terminal / End points are also included (Not Excluded).
	Limit	Limit value of any function f	If value of function is not determined then calculated like, Evaluation of f(2), when f(x) = 1/(x-2)
	LHL	Left Hand Limit
	RHL	Left Hand Limit
f(a) or g(x)		Value of function	Value of function f(x) or g(x) when x = a.
	Epsilon	It represents very small number, near to zero.
e	Euler’s Number	It lies between 2 and 3, e = 2.718281828…
dy/dx	Derivative	Derivative- Leibnitz’s Symbol	d(x⁴)/dx= 4x³
D_xy or D	Derivative	Derivative- Euler’s General Notation/Symbol	D(x⁴) = 4x³
D_x²y or D	Derivative	Derivative of Derivative- Euler’s Notation/Symbol	D²(x⁴) = 4.3x²= 12x²
D_x³y or D	Derivative	Three times Derivative - Euler’s Notation/Symbol	D³(x⁴) = 4.3.2x= 24x
D_xⁿy or D	Derivative	n times Derivative - Euler’s Notation/Symbol	D^’n(x⁴) = 0
Y’	Derivative	Derivative- Lagrange’s Symbol	(x⁴)’ = 4x³
Y’’	Second Derivative	Derivative of Derivative Lagrange’s Symbol	(x⁴)’’ = 4.3x²= 12x²
Y’’’	Third Derivative	Three times Derivative- Lagrange’s Symbol	(x⁴)’’’ = 4.3.2x= 24x
Y⁽ⁿ⁾	n^thDerivative	n times Derivative- Lagrange’s Symbol	(x⁴)’ⁿ = 0
∫	integral	Inverse process of differentiation	∫1. dx= x
∫∫	Double integral	Integration of two variable functions	∫∫f (x, y) dxdy Or ∫∫f (x, y) dydx
∫∫∫	Triple integral	Integration of three variable functions	∫∫∫f (x, y, z) dxdydz
∮		Closed contour integral / line integral	∮f(x)dx or ∮f(y)dy
∯		Closed surface integral	∯ f (x, y) dxdy Or ∯ f (x, y) dydx
∰		Closed volume integral	∰ f (x, y, z) dxdydz
z	Complex number	A number having real or imaginary parts	Z = a + i b, where a & b both are Real Numbers
i	iota	Imaginary unit	i ≡ √-1 or i² = -1 or i³ = -i or i⁴ = 1
R(z) or Re(z)	Real part of complex number	If z = a + i b then R(z) = a	If z = 3 +i4 then R(z) = 3
I(z) or Im(z)	Imaginary part of complex number	If z = a +ib then I(z) = a	If z = 3 +i4 then I(z) = 4
Z^*	Complex conjugate	*If z = a +ib then Z = a - ib**	If z = 3 +i4 then *Z =3- i4**
Z bar	Complex conjugate	If z = a +ib then Z(bar) = a - ib	If z = 3 +i4 then *Z =3- i4**
\|z\|	Magnitude of a complex number	If z = a +ib then \|Z\|= √(a²+b²)	If z = 3 +i4 then \|3 +4i\| = √25 = 5
\|z\|	Absolute value of a complex number	If z = a +ib then \|Z\|= √(a²+b²)	If z = 3 +i4 then \|3 +4i\| = √25 = 5
Arg(z)	Argument value of a complex number	The angle made by the radius in the Argand or complex plane.	If z= 1 + i then arg(z)= pie /4

Mathematical Symbols - Logical and General Reasoning

2020-07-31T23:44:00.006+05:30

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical and Reasoning

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss ninth topic Mathematics symbols - Logical and Reasoning on this post.

Mathematical symbols - Logical and Reasoning

These are very important Basic Tools frequently used by us not only in computer, Set theory, statistics and Probability but also in Operation Research, Management as well as in other branches of pure & applied mathematical sciences and engineering.

Table of important Mathematical Symbol of Logical and Reasoning

Mathematics Symbol – Logical & Reasoning

S.N.	Symbol	Name of Symbol	Definition / Meaning	Illustration
1	∴	Therefore
2	∵	Since / Because
3	∃	Their Exist
4	∄	There does not exist
5	∀ or ( )	For all or Universal Quantification	For any, for each, etc	∀ t belongs to W t≤ t²
6	⇒	Material implication	Implies or if … then	a = 3 ⇒ a²= 9 is true but vice versa is not true since a²= 9 ⇒ a = 3 or -3.
7	⇔	Material Equivalence iff (if and only if)	a ⇔ b it means both ways are correct it is valid if a & b both are true or both are false.	a +10 =b + 100 ⇔ a = b + 90
8	↔	Material Equivalence iff (if and only if)	a ⇔ b it means both ways are correct it is valid if a & b both are true or both are false.	a +10 =b + 100 ⇔ a = b + 90
9	⋅	And	And	a. b
10	^	Circumflex or Caret	And	a ^ b
11	&	Ampersand	And	a & b
12	+	Plus	Or	a + b
13	\|	Vertical Line	Or	a \| b
14	V	Reversed Caret	Or	a V b
15	x'	Single Quote	Not - negation	x'
16	!	Exclamation Mark	Not - negation	!x or !a
17	X bar	bar	Not - negation
18	¬	Not	Not - negation	¬ (¬a) = a
19	~	tilde	negation	~ (~a) = a
20	⊕	Oplus or circled plus	Exclusive or xor	a⊕ b

Mathematical Symbols - Roman Symbols (One to Ten Thousand)

2020-07-27T23:38:00.046+05:30

Mathematical Roman Symbols are very important to learn the number system in mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical and Reasoning

Mathematics symbols - Roman Symbols

Mathematics symbols - Greek Symbols

Mathematics symbols - Miscellaneous

Now we shall learn /discuss the tenth topic of Mathematical symbols - Roman Symbols on this post.

Mathematics symbols - Roman Symbols

These are very important Basic Tools frequently used by us not only in Airthematic but also in Management as well as in other branches of applied mathematical sciences and engineering.

Thousands a year ago Romans had a unique system of numbers which had only SEVEN symbols, Each having different values are such as:-

Roman Numerals

Value

I

1

V

5

X

10

L

50

C

100

D

500

M

1000

Note- For 0 (Zero) there are no symbols in Roman Numerals.

Rules for writing Roman symbols

Rule 1- I, X, and C Roman digits are repeated up to three times in succession to form the numbers that is cannot be repeated up to four times.

For illustrations we consider the following examples:

Example 1:

I = 1

II = 1 + 1 = 2

III = 1 + 1 + 1 = 3 (It is valid since used up to three times only.)

IIII = 1 + 1 + 1 + 1 = 4 is wrong

Example 2:

X = 10

XX = 10 + 10 = 20

XXX = 10 + 10 + 10 = 30 (It is valid since used up to three times only.)

XXXX = 10 + 10 + 10 + 10 = 40 is wrong

Example 3:

C = 100

CC = 100 + 100 = 200

CCC = 100 + 100 + 100 = 300 (It is valid since used up to three times only.)

CCCC = 100 + 100 + 100 + 100 = 400 is wrong since repeated more than three times.

Rule 2- V, L, and D Roman digits are not repeated in succession to form the numbers that are the repetition of V, L, and D are not allowed.

so instead of using X for 10, VV is not correct. (VV = 10 is wrong)

Rule 3- If the digit of lower value is written to the right or after / later of the digit of higher value, all the digits of numbers are added.

Illustrations:-

VI = 5 + 1 = 6

VII = 5 + 2 =7

VIII = 5 + 3 =8

XI = 10 + 1 =11

Rule 4- If the digit of lower value is written to the left or ago / before the digit of higher value, then the lower numerals or digits are subtracted from higher values.

Illustrations:-

IV = 5 - 1 = 4

IX = 10 - 1 = 9

1XX = 20 - 1 = 19

XC = 100 - 10 =90

Rule 5 - Multiplication rule

If a vinculum or horizontal line is drawn over the digit of Roman numerals or symbols, then the value of the numerals becomes one thousand ( 1000) times.

Illustrations:-

IV = 4 then   = 4000
V = 5 then ⊽  = 50000
VI = 6 then   = 6000
VII = 7 then    =  7000
VIII = 8 then   = 8000
IX = 9 then   = 9000
X = 10 then    =  10000

Some important Roman Symbols up to 10000 are listed below in the table on behalf of this we can write all numbers up to 10000 with the help of illustrations below the table.

Table of important Mathematical Symbol of Roman Symbols

Number
Roman Numeral

Justification

0

NOT DEFINED

1

I

1+1

2

II

1+1+1

3

III

1+1+1

4

IV

5 -1

(IIII is wrong since I used four times.)

5

V

5

6

VI

5 + 1

7

VII

5 + 1 + 1

8

VIII

5 + 1 + 1

9

IX

10 - 1

(VIIII is wrong since I used four times.)

10

X

10

11

XI

10+1

12

XII

10+1+1

13

XIII

10+1+1+1

14

XIV

10+(5-1)

15

XV

10+5

16

XVI

10+5+1

17

XVII

10+5+1+1

18

XVIII

10+5+1+1+1

19

XIX

10+(10-1)

20

XX

10+10

21

XXI

10+10+1

22

XXII

10+10+1+1

23

XXIII

10+10+1+1+1

24

XXIV

10+10+(5-1)

25

XXV

10+10+5

26

XXVI

10+10+5+1

27

XXVII

10+10+5+1+1

28

XXVIII

10+10+5+1+1+1

29

XXIX

10+10+(10-1)

30

XXX

10+10+10

31

XXXI

10+10+10+1

32

XXXII

10+10+10+1+1

33

XXXIII

10+10+10+1+1+1

34

XXXIV

10+10+10+(5-1)

35

XXXV

10+10+10+5

36

XXXVI

10+10+10+5+1

37

XXXVII

10+10+10+5+1+1

38

XXXVIII

10+10+10+5+1+1+1

39

XXXIX

10+10+10+(10-1)

40

XL

50-10

(XXXX is wrong since X is used four times.)

41

XLI

(50-10)+1

42

XLII
(50-10)+1+1

43

XLIII

(50-10)+1+1

44

XLIV

(50-10)+(5-1)

45

XLV

(50-10)+5

46

XLVI

(50-10)+5+1

47

XLVII

(50-10)+5+1+1

48

XLVIII

(50-10)+5+1+1+1

49

XLIX

(50-10)+(10-1)

50

L

50

51

LI

50+1

52

LII

50+1+1

53

LIII

50+1+1+1

54

LIV

50+(5-1)

55

LV

50+5

56

LVI

50+5+1

57

LVII

50+5+1+1

58

LVIII

50+5+1+1+1

59

LIX

50+(10-1)

60

LX

50+10

61

LXI

50+10+1

62

LXII

50+10+1+1

63

LXIII

50+10+1+1+1

64

LXIV

50+10+(5-1)

65

LXV

50+10+5

66

LXVI

50+10+5+1

67

LXVII

50+10+5+1+1

68

LXVIII

50+10+5+1+1+1

69

LXIX

50+10+(10-1)

70

LXX

50+10+10

71

LXI

50+10+10+1

72

LXXII

50+10+10+1+1

73

LXXIII

50+10+10+1+1+1

74

LXXIV

50+10+10+(5-1)

75

LXXV

50+10+10+5

76

LXXVI

50+10+10+5+1

77

LXXVII

50+10+10+5+1+1

78

LXXVIII

50+10+10+5+1+1+1

79

LXXIX

50+10+10+(10-1)

80

LXXX

50+10+10+10

81

LXXXI

50+10+10+10+1

82

LXXXII

50+10+10+10+1+1

83

LXXXIII

50+10+10+10+1+1+1

84

LXXXIV

50+10+10+10+(5-1)

85

LXXXV

50+10+10+10+5

86

LXXXVI

50+10+10+10+5+1

87

LXXXVII

50+10+10+10+5+1+1

88

LXXXVIII

50+10+10+10+5+1+1+1

89

LXXXIX

50+10+10+10+(10-1)

90

XC

100-10

(LXXXX is wrong since X is used four times.)

91

XCI

(100-10)+1

92

XCII

(100-10)+1+1

93

CXIII

(100-10)+1+1+1

94

CXIV

(100-10)+(5-1)

95

CXV

(100-10)+5

96

XCVI

(100-10)+5+1

97

XCVII

(100-10)+5+1+1

98

XCVIII

(100-10)+5+1+1+1

99

XCIX

(100-10)+(10-1)

100

C

100

200

CC

100+100

300

CCC

100+100+100

400

CD

500-100

(CCCC is wrong since C is used four times.)

500

D

500

600

DC

500+100

700

DCC

500+100+100

800

DCCC

500+100+100+100

900

CM

1000-100

1000

M

1000

1100

MC

1000+100

1200

MCC

1000+100+100

1300

MCCC

1000+100+100+100

1400

MCD

1000+(500-100)

1500

MD

1000+500

1600

MDC

1000+500+100

1700

MDCC

1000+500+100+100

1800

MDCCC

1000+500+100+100+100

1900

MCM

1000+(1000-100)

2000

MM

1000+1000

2100

MMC

1000+1000+100

2200

MMCC

1000+1000+100+100

2300

MMCCC

1000+1000+100+100+100

2400

MMCD

1000+1000+(500-100)

2500

MMD

1000+1000+500

2600

MMDC

1000+1000+500+100

2700

MMDCC

1000+1000+500+100+100

2800

MMDCCC

1000+1000+500+100+100+100

2900

MMCM

1000+1000+(1000-100)

3000

MMM

1000+1000+1000

Table of important Mathematical Symbol - Roman Symbols Greater than 3000

Number	Roman Numeral	Justification
3100	MMMC	1000+1000+1000+100
3200	MMMCC	1000+1000+1000+100+100
3300	MMMCCC	1000+1000+1000+100+100+100
3400	MMMCD	1000+1000+1000+(500-100)
3500	MMMD	1000+1000+1000+500
3600	MMMDC	1000+1000+1000+500+100
3700	MMMDCC	1000+1000+1000+500+100+100
3800	MMMDCCC	1000+1000+1000+500+100+100
3900	MMMCM	1000+1000+1000+(1000-100)
4000		4*1000=4000 (By Multiplication Rule)
4100	C	4000+100
4200	CC	4000+100+100
4300	CCC	4000+100+100+100
4400	CD	4000+(500-100)
4500	D	4000+500
4600	DC	4000+500+100
4700	DCC	4000+500+100+100
4800	DCCC	4000+500+100+100+100
4900	CM	4000+(1000-100)
5000	⊽	5*1000
5100	⊽C	5000+100
5200	⊽CC	5000+100+100
5300	⊽CCC	5000+100+100+100
5400	⊽CD	5000+(500-100)
5500	⊽D	5000+500
5600	⊽DC	5000+500+100
5700	⊽DCC	5000+500+100+100
5800	⊽DCCC	5000+500+100+100
5900	⊽CM	5000+(1000-100)
6000		6 *1000
6100	C	6000+100
6200	CC	6000+100+100
6300	CCC	6000+100+100+100
6400	CD	6000+(500-100)
6500	D	6000+500
6600	DC	6000+500+100
6700	DCC	6000+500+100+100
6800	DCCC	6000+500+100+100+100
6900	CM	6000+(1000-100)
7000		7*1000
7100	C	7000+100
7200	CC	7000+100+100
7300	CCC	7000+100+100+100
7400	CD	7000+(500-100)
7500	D	7000+500
7600	DC	7000+500+100
7700	DCC	7000+500+100+100
7800	DCCC	7000+500+100+100+100
7900	DCM	7000+(10000-100)
8000		8*1000
8100	C	8000+100
8200	CC	8000+100+100
8300	CCC	8000+100+100+100
8400	CD	8000+(500-100)
8500	D	8000+500
8600	DC	8000+500+100
8700	DC	8000+500+100+100
8800	DCC	8000+500+100+100+100
8900	CM	8000+(1000-100)
9000		9*1000
9100	C	9000+100
9200	CC	9000+100+100
9300	CCC	9000+100+100+100
9400	CD	9000+(500-100)
9500	DC	9000+500
9600	DC	9000+500+100
9700	DCC	9000+500+100+100
9800	DCCC	9000+500+100+100+100
9850	DCCCL	9000+500+100+100+100+50
9900	CM	9000+(1000-100)
9950	CML	9000+(1000-100)+50
10000		10*1000

We can use the above rules to write some numerals/numbers in Roman symbols with the help of following illustrations and vice versa are such as:

Illustrations 1:- Write the Roman Symbols for the following numerals/numbers, with justifications?

(I) 111 = CXI =100 +10+1

(II) 121 = CXXI =100 +10+10+1

(III) 129 = CXXIX =100 +10+10+(10-1)

(IV) 135 = CXXXV =100 +10+10+10+5

(V) 149 = CXLIX =100 +(50-10)+(10-1)

(VI) 151 = CLI =100 +50+1

(VII) 159 = CLIX =100 +50+(10-1)

(VIII) 173 = CLXXIII =100 +50+10+10+1+1+1

(IX) 189 = CLXXXIX =100 +50+10+10+10+(10-1)

(X) 199 = CXCIX =100 +(100-10)+(10-1)

Illustrations 2:- Write the Roman Symbols for the following numerals/numbers, with justifications?

(I) 212 = CCXII =100 +100+10+1+1

(II) 223 = CXXIII =100 +100+10+10+1+1+1

(III) 229 = CCXXIX =100+100 +10+10+(10-1)

(IV) 236 = CCXXXVI =100 +100+10+10+10+5+1

(V) 249 = CCXLIX =100 +100+(50-10)+(10-1)

(VI) 254 = CCLIV =100 +100+50+(5-1)

(VII) 259 = CCLIX =100 +100+50+(10-1)

(VIII) 272 = CCLXXII =100 +100+50+10+10+1+1

(IX) 289 = CCLXXXIX =100+100 +50+10+10+10+(10-1)

(X) 294 = CCXCIV =100+100 +(100-10)+(5-1)

Illustrations 3:- Write the Roman Symbols for the following numerals/numbers, with justifications?

(I) 313 = CCCXIII =100 +100+100+10+1+1+1

(II) 323 = CCCXXIII =100 +100+100+10+10+1+1+1

(III) 329 = CCCXXIX =100+100+100 +10+10+(10-1)

(IV) 337 = CCXXXVII =100+100 +100+10+10+10+5+1+1

(V) 349 = CCCXLIX =100 +100+100+(50-10)+(10-1)

(VI) 354 = CCCLIV =100+100 +100+50+(5-1)

(VII) 359 = CCCLIX =100+100 +100+50+(10-1)

(VIII) 373 = CCCLXXIII =100 +100+100+50+10+10+1+1+1

(IX) 389 = CCCLXXXIX =100+100+100 +50+10+10+10+(10-1)

(X) 398 = CCCXCVIII =100+100 +100+(100-10)+5+1+1+1

Illustrations 4:- Write the Roman Symbols for the following numerals/numbers, with justifications?

(I) 413 = CDXIII =(500-100)+10+1+1+1

(II) 428 = CDXXVIII =(500 -100)+10+10+5+1+1+1

(III) 529 = DXXIX =500 +10+10+(10-1)

(IV) 537 = DXXXVII =500+10+10+10+5+1+1

(V) 549 = DXLIX =500+(50-10)+(10-1)

(VI) 454 = CDLIV =(500-100)+50+(5-1)

(VII) 369 = CCCLXIX =100+100 +100+50+10+(10-1)

(VIII) 573 = DLXXIII =500+50+10+10+1+1+1

(IX) 489 = CDLXXXIX = (500-100) +50+10+10+10+(10-1)

(X) 598 = DXCVIII =500+(100-10)+5+1+1+1
We can use like above illustrations to write any numerals up to 10000 into Roman Symbols.
__________________________________________________

Mathematical Symbols - Statistics and Probability

2020-06-30T23:06:00.011+05:30

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical & Reasoning

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss seventh topic Mathematics symbols - Statistics and Probability on this post.

Mathematics symbols - Statistics and Probability

These are very important Basic Tools frequently used by us not only in statistics and Probability but also in Operation Research, Management as well as in other branches of applied mathematical sciences and engineering.

Table of important Mathematical Symbol of Probability and Statistics

S.N.

Symbol

Name of Symbol

Definition

Illustration

1

P(A)

Probability Function

Probability of any event A

P(A) = 0.2 or any number between 0 to 1

0 ≤ P (A ≤ 1

2

P(A) = 0

Complete Uncertainty

The event cannot happen

On tossing a coin we shall get prime number.

3

P(A) = 1

Complete certainty

The event will happen definitely

On tossing a die we shall get a number less than 7.

4

P (A ∩ B)

Probability of intersection of events

Probability of Event A and B

P (A ∩ B) = 0.5

5

P (A ⋃ B)

Probability of union of events

Probability of Event A or B

P (A⋃B) = 0.5

6

P (A / B)

Conditional Probability Function

Probability of Event A when B already occurred.

P (A / B) = 0.2 or any number between 0 to 1

7

P (B / A)

Conditional Probability Function

Probability of Event B when A already occurred.

P (B / A) = 0.6 or any number between 0 to 1

8

μ

Mean or population mean

Mean of population values

Mue = 10

9

f(t) or f(x)

pdf

Probability Density function

P (a ≤ t ≤ b) = ∫ f (t) dt, where t is any variable can be replaced by x or y etc.

10

F (t)

cdf

Cumulative distribution function

F(t) = P (T≤ t)

11

E (X)

Expectation Value

Expected value of any random variable x

E (X) = 10

12

E (X / Y)

Conditional Expectation Value

Conditional Expected value of any random variable X given Y

E (X / Y) = 3

13

E (Y/ X)

Conditional Expectation Value

Conditional Expected value of any random variable Y given X

E (Y / X) = 2

14

Sigma 2 (σ 2)

Var(X)

Variance of random variable x

Var(x) = 16

15

∑

Sigma (Capital)

Summation

Sum of all values in the range.

16

∑∑

Sigma (Capital) Sigma (Capital)

Double Summation

Sum of all values in the range.

17

Range

Range

Difference between maximum and minimum value

For series 2, 4, 6, 10, 10, 10, 20,40

Range= 40-2 = 38

18

MR

Mid-range

½(Maximum + Minimum )

For series 2, 4, 6, 10, 10, 10, 20,40

MR= ½(2 + 40) =21

19

Mo

Mode

Value that frequently occurs in data / population / series. Or

Maximum number of repetitions

For series 2, 4, 6, 10, 10, 10, 20,40

Mode = 10

20

X bar

Sample Mean / average / Arithmetic Mean

Sum of all observations / No. of observations

For series 2, 4, 6, 10, 10, 10, 20,40

Mean = 1/8 (2+4+6+10+10 +10+20+40)

21

Std(X) or

σ(x)

Standard deviation

Standard deviation of random variable x = square root of variance

For any observation, if Var(x) = 16 then

Std(X) = 4

22

Median

Median

Mid value of x

For series 2, 4, 6, 10, 10, 10, 20,40

Median = 10

23

cov (x, y)

Covariance

Covariance of any random variable x and y

cov(x, y) = 4

24

corr (x, y)

Correlation

Correlation of any random variable x and y

Corr (x, y) = 4.5

25

ρ(x, y)

Correlation

Correlation of any random variable x and y

ρ (x, y) = 4.5

26

Q

Quartile

% population are below this average on the basis of Q1, Q2, or Q3.

27

Q1

First Quartile / lower

25 % population are below this average or first quartile

28

Q2

Second Quartile / Median

50 % population are below this average



29

Q3

Third Quartile / Upper

25 % population are above this average or

75 % population are below this average

On having knowledge of the above Mathematical symbols of probability and statistics, we can easily enhance our subject ideas.

___________________________________________________________

Mathematical Symbols - Set Theory

2020-06-30T01:04:00.020+05:30

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss Sixth topic Mathematics symbols - Set Theory on this post.

Mathematical symbols - Set Theory

These are very important Basic Tools frequently used by us in Particular topics of algebra like Sets and relations, Pure Algebra, Mathematical Logics as well as in other branches of applied mathematics and sciences.

Set can be written by capital alphabets like A, B, C, P, Q, X, Y, etc and it can be written in Curley Brackets.

S.N.

Symbol

Name of Symbol

Definition

Illustration

1

{ }

Set

A well-defined collection of objects / elements are known as a set.

A = {1,17,19,24},

B = {3, 7,19,24}, OR

C= Set of word MATHEMATICS

= {M, A, T, H, E, I, C, S}

2

A = B

equality

both sets have the same members (orders may be different)

A = {1,2,3,4}

and

B = {1,2,3,4}

3

A ∪ B

union

Those objects which belong to set A or set B.

If A = {1,17,19,24}, and

B = {3, 7,19,24}, then A ∪ B =

{1, 3, 7, 17, 19, 24}

4

A ∩ B

intersection

Those objects that belong to set A and set B both.

If A = {1,17,19,24}, and

B = {3, 7,19,24}, then A ∩ B = {19,24}

5

A ⊆ B

subset

A is a subset of B or set A is included in set B.

{1,2,3,4} ⊆ {1,2,3,4}

6

P ⊂ Q

proper subset or a strict subset

“P is a subset of Q, but Pis not equal to Q”.

{1,2, 4} ⊂  {1,2,3,4}

7

P ⊄ Q

not a subset

"The first (P) set is not a subset of another or second (Q ) set "

{1,2, 4} ⊄ {1,22, 44}

8

P ⊇ Q

superset

P is a superset of Q or set P includes set Q

{1,14,28} ⊇ {1,14}

9

P ⊅ Q

not a superset

set P is not a superset of set Q

{19,14,208} ⊅ {19,23}

10

power set

All subsets of any set like A

11

P (A)

power set

All subsets of any set like A

12

complement

All the objects that do not belong to given set A

13

complement

All the objects that do not belong to given set A

14

A – B

Set A-minus Set B

Or

relative complement

Objects that belong to set A and but not to set B

If A = {1,2,3,4} and

B = {1, 2, 5, 7}

then

A-B = {3, 4}

15

A / B

Set A minus Set B

or

relative complement

Objects that belong to set A and but not to set B

If A = {1,2,3,4} and

B = {1, 2, 5, 7}

then

A-B = {3, 4}

16

B – A

Set B minus Set A

Or

relative complement

Objects that belong to set B and but not to set A

If A = {1,2,3,4} and

B = {1, 2, 5, 7}

then

B- A = {5, 7}

17

B / A

Set B minus Set A

or

relative complement

Objects that belong to set B and but not to set A

If A = {1,2,3,4} and

B = {1, 2, 5, 7}

then

B- A = {5, 7}

18

a ∈ A

belongs to

OR

elements of

Element is member of a given set

If A = {1,2,3,4}

then

1 ∈ A, 2 ∈ A,

3 ∈ A, 4 ∈ A

19

a ∉ A

Not belongs to OR

not element of

Element is NOT a member of given set

If A = {1,2,3,4}

then

5 ∉ A

20

A ∆ B

Symmetric difference of sets

Set of all those elements that belong to A or B but not to their intersection.

(A ∪ B) – (A ∩ B)

OR

(A - B) ∪ (B - A)

If A = {1,2,3,4}

and

B = {1, 2, 5, 7}

then

A ∆ B = {3,4,5,7}

21

A ⊖ B

Symmetric difference of sets

Set of all those elements that belong to A or B but not to their intersection.

(A ∪ B) – (A ∩ B)

OR

(A - B) ∪ (B - A)

If A = {1,2,3,4}

and

B = {1, 2, 5, 7}

then

A ⊖ B = {3,4,5,7}



22

(x, y)

Ordered Pair of x and y

Set of two elements whose order is fixed.

23

A × B

Cartesian Product

it is a set of all possible ordered pairs from A and B

24

B × A

Cartesian Product

Set of all ordered pairs from B and A

25

n (A)

The cardinality of set A

The total number of elements of the set A

A = {1, 2,3,9,14},

n(A) =5

26

|A|

The cardinality of set A

The total number of elements of the set A

A = {1, 2,3,9,14}, |A|=5

27

(Phi)
Empty Set

The set having No elements.

28

(Zeta)
Universal Set

Superset of all sets

On having knowledge of the above Mathematics symbols - Set Theory we can easily enhance our subject ideas.

___________________________________________________________

Mathematics Formula Algebra - Exponents And Powers

2020-06-28T01:10:00.008+05:30

Mathematics Formula Algebra

In algebra, we have various topics having formulas and identities, Some are listed below:-

Algebraic Identity
Exponents and Powers
Least Common Multiple (LCM) & Highest Common Factor(HCF)

Exponents and Powers

if we factorize 4,9,16 or 25 then can be written as

4 = 2 x 2 x 1

9 = 3 x 3 x 1

16 = 2 x 2 x 2 x 2 x 2 x 1

25 = 5 x 5 x 1

The above expression can be also written in a shorter way using exponents in this way.

4 = 2 x 2 = 2²

9 = 3 x 3 = = 3²

16 = 2 x 2 x 2 x 2 x 2 = 2⁴

25 = 5 x 5 = 5²

^{In the above expression that represents repeated multiplication of the same factor is called power.}

The number 2 is called the Base and 4 is known as Power or Exponents or Index.

The conventions for reading are such as

The above short notation stands for the product 2 x 2 x 2 x 2. Here ' 2' is called the base and'4' the exponent . The above number is read as " 2 raised to the power of 4" or simply as the "fourth power of 2" . It is also called the exponential form of 16.

Example 1:- Express 64 as the power of 2?

Solution:- By prime factorisation method we have

64 = 2 x 2 x 2 x 2 x 2 x 2 x 1

So we can say 64 = 26

Mathematics Symbols - Trigonometry

2020-06-21T20:29:00.006+05:30

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical & Reasoning

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss Fifth topic Mathematics symbols - Trigonometric on this post.

Mathematics symbols - Trigonometric

These are very important Basic Tools frequently used by us in Geometry as well as in other branches of mathematics, physics,
sciences, and Engineering.

We know that Trigonometry is a branch of mathematics that deals with sides and angles of triangles.

Trigonometry is frequently used in Physics and other sciences so we can say without trigonometry imagination of physics and engineering sciences is not possible.

Note:- Degree, Gradians and Radians are units of measurements of angles in various systems.
On having knowledge of the above Mathematics symbols - Trigonometric we can easily enhance our subject ideas also using standard relations and Trigonometric Identities.

Mathematics Formula Algebra - Identities

2020-05-30T01:05:00.002+05:30

What is a formula? We can say 'Any conventional or fixed methods based on experiences for doing something' or literally 'a set form of words, as for stating or declaring something definitely or authoritatively, for indicating procedure to be followed, or for prescribed use on some ceremonial occasion' is the general meaning of the formula. Mathematically we can express formula as 'A principle or rule which is frequently expressed in the language of Mathematically symbols'.

Similar to other physical science, computational, and engineering subjects mathematics is also has varied and the vast list of formulas. Classification of formulas can be done according to standard (Class) or Branches of Mathematics. In my opinion, it is beneficial to classify on the basis of Branches of Mathematics is more useful so we can classify such as

Mathematics Formula Algebra
Mathematics Formula Geometry
Mathematics Formula Commercial mathematics
Mathematics Formula Mensuration
Mathematics Formula Trigonometry
Mathematics Formula Coordinate Geometry
Mathematics Formula Statistics & Probability
Mathematics Formula Calculus
Mathematics Formula Miscellaneous

Mathematics Formula Algebra

In algebra, we have various topics having formulas and identities, Some are listed below:-

Algebraic Identity
Exponents and Powers
Least Common Multiple (LCM) & Highest Common Factor(HCF)

Now we shall write about algebraic identities.

Algebraic identities are algebraic equations that are true for all the variables involving in it. Algebraic identities are also an integral part of Formula in Algebra. These identities contain various constants and variables on both sides of the equation. When we factorize a polynomial, we often use these algebraic identities. On the basis of terms of identities, it can be divided into two parts.

Binomial Algebraic Identity
Trinomial(Polynomial) Algebraic Identity

Binomial Algebraic Identity

We know that the algebraic expression having two terms is known as Binomial. The few basic important algebraic identities frequently used in algebra as well as other branches of mathematics are such as( Instead of a and b we can use any alphabets of many languages).

Trinomial(Polynomial) Algebraic Identity

The algebraic expression having three terms is known as Trinomial. The few basic important algebraic identities frequently used in algebra as well as other branches of mathematics are such as( Instead of a and b we can use any alphabets of many languages).

The First Four Binomial Algebraic Identities are in the syllabus of Middle Standard (Class 6 to 8) While all remaining along with Trinomial(Polynomial) Algebraic Identityare used in secondary Standards in mathematics.

We can use the above identities to solve problems, Few illustrations are such as

Similarly, we can use the above-mentioned identities to factorize algebraic polynomials and evaluate values like examples 3 and 4.

From the above illustration 8, we can conclude that identity second is obtained by first with substitution of b by (-b). Similar results can be obtained for five and six, seven & eight.

Note:- The number of identities is not fixed they are written only for convenience.

Aims for mathematics education at the elementary stage & Secondary stage

2020-05-25T21:19:00.001+05:30

Aims or the main goal for mathematics education in schools is the mathematisation of the child’s thinking using various strategies/techniques.
if we consider a real situation that is similar to most of us (Maybe up to 99.99 % in my opinion) that, Among all the school subjects, maximum emphasis is given by any chid to learning mathematics. We must all have experienced, during our school days and also being a teacher(Now), the extent of pressure exerted on children to perform at a higher level in mathematics in comparison with other subjects like Languages, Sciences, Social Sciences, Music & art, etc. Although the parents, irrespective of their educational status (including the first-generation Learner), suppress their children very hard to perform well in mathematics and perhaps maximum study time at home is spent on doing homework and creative activity (Learning of Tables is common phenomena for Parents up to 5 /10 /15 or 30 for brilliant students) in mathematics than in any other subject.
Aims or the main goal for mathematics education in schools may differ according to the opinion of persons but rigorously may be similar like

At Elementary Stage

At the most basic level, mathematics involves counting objects, recognizing numbers, and performing simple operations like addition, subtraction, multiplication, and division.
In the primary grades, students should be master these skills including general concepts of maths like factors, multiples, and classification of numbers and we have to provide numerous opportunities instead of rote learning methods.
Upon exiting at elementary stage (primary school), children should be comfortable with writing and identifying numbers, rote counting forward and backward & comparing numbers and quantities.
They should also be able to add, subtract, multiply, and divide numbers, and knowing about vice versa of basic operations.
Clarity of thought and pursuing assumptions to logical conclusions is the main aim of learning mathematics.
There are various ways of thinking, and the kind of thinking one learns in mathematics is an ability to handle to approach about problem-solving.
The school mathematics must be activity-oriented for the learner.

At Secondary Stage

To develop the powers of logical thinking and reasoning in the learner.
To Shifting the focus of mathematics education from achieving ‘narrow’ goals to ‘higher’ goals to the learner.
To solve mathematical problems of daily life by mathematisation of the learner.
To understand and acquainted with the environment and culture with mathematisation.
To prepare the learner for various technical and auspicious future professions.
To prepare the learner for higher study.
To develop in the learner the power for invention

The general aim of Mathematics Education

To engage every student with a sense of success, while at the same time offering conceptual challenges to the emerging mathematician.
To Changing modes of assessment to examine students mathematisation abilities rather than procedural knowledge continued from ancient times.
Enriching teachers with a variety of mathematical resources from time to time.

As per position paper on 'Teaching of mathematics' by national focus group(NCF ) In India by NCERT are such as

The mathematisation of the child’s thought processes.
Curriculum may be ambitious, coherent, and teaches important mathematics.
School mathematics must be activity-oriented.
Children learn to enjoy mathematics.
Children learn important mathematics.
Children see the importance of mathematics in various ways as something to talk about, to communicate, to discuss among themselves, to work together on.
Children pose and solve meaningful problems.
Children understand the basic structure of mathematics.
Teachers expect to engage every child in the class.

Therefor to know about the aim of mathematics Education is very important before learning and inculcating among children or Mathematics lovers fraternity.

Mathematics symbols - Geometric

2020-05-25T14:54:00.010+05:30

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss Fourth topic Mathematics symbols - Geometric on this post.

Mathematics symbols - Geometric

These are very important Basic Tools frequently used by us in algebra as well as in other branches of mathematics and sciences.

Symbol

NAME OF SYMBOL

DEFINITION / MEANING

ILLUSTRATION

∠

Angle

An angle is formed by two rays

∠ABC = 60°

Measured angle

ABC = 30°

Zero Angle
An angle of measure zero degree ∠ABC = 0°

∟

Right angle

An angle of measure ninety degrees = 90°

α = 90°

°

Degree

1 turn or round = 360°

α = 60°

deg

Degree

1 turn or round = 360 deg

α = 60 deg

′

Prime ( Read as different from prime numbers)

Arcminute, 1° = 60′

α = 60°59′

″

Double prime

Arcsecond, 1′ = 60″

α = 60°59′59″

Line

Infinite line

Drawn from any point

AB

Line segment

The lines from point A to point B

The lines from point B to point A is written as BA

Ray

The lines that start from any point like A

Arc

an arc from point A to point B

Part of the circumference of a circle of any measure = 60°

⊥

Perpendicular

Those lines intersect at right angles are perpendicular lines (90° angle).

AC ⊥ BC

∥

Parallel

Those lines do not intersect each other are parallel lines.

AB ∥ CD

≅

Congruent to

Those geometric figures having same shapes and sizes.

∆ABC≅ ∆XYZ

~

Similarity

Those geometric figures having the same shapes but sizes may be different.

∆ABC~ ∆XYZ

Δ

Triangle

Triangle shape

ΔABC≅ ΔBCD

|x-y|

Distance

Distance between any two points x and y

| x-y | = 2 or 3 or any number

π

Pi constant

π = 3.141592654...

is the ratio between the circumference and diameter of a circle and it is an irrational numbers

c = π⋅d = 2⋅π⋅r where r is the radius of circle and d = 2r

rad

Radians

Radians angle unit

360° = 2π rad or 2π ^c

^c

Radians

Radians angle unit

360° = 2π ^{c or}2π rad

grad

Gradians

Grads angle unit

360° = 400 grad or g

^g

Gradians

Grads angle unit

360° = 400 ^{g or}grad

Note:- Degree, Gradians and Radians are units of measurements of angles in various systems.
On having knowledge of the above Mathematics symbols - Geometric we can easily enhance our subject ideas.
________________________________________________

Symbol	*NAME OF SYMBOL*	*DEFINITION / MEANING*	*ILLUSTRATION*
∠	Angle	An angle is formed by two rays	∠ABC = 60°
	Measured angle		ABC = 30°
	Zero Angle	An angle of measure zero degree	∠ABC = 0°
∟	Right angle	An angle of measure ninety degrees = 90°	α = 90°
°	Degree	1 turn or round = 360°	α = 60°
deg	Degree	1 turn or round = 360 deg	α = 60 deg
′	Prime ( Read as different from prime numbers)	Arcminute, 1° = 60′	α = 60°59′
″	Double prime	Arcsecond, 1′ = 60″	α = 60°59′59″
	Line	Infinite line	Drawn from any point
AB	Line segment	The lines from point A to point B	The lines from point B to point A is written as BA
	Ray	The lines that start from any point like A
	Arc	an arc from point A to point B	Part of the circumference of a circle of any measure = 60°
⊥	Perpendicular	Those lines intersect at right angles are perpendicular lines (90° angle).	AC ⊥ BC
∥	Parallel	Those lines do not intersect each other are parallel lines.	AB ∥ CD
≅	Congruent to	Those geometric figures having same shapes and sizes.	∆ABC≅ ∆XYZ
~	Similarity	Those geometric figures having the same shapes but sizes may be different.	∆ABC~ ∆XYZ
Δ	Triangle	Triangle shape	ΔABC≅ ΔBCD
\|x-y\|	Distance	Distance between any two points x and y	\| x-y \| = 2 or 3 or any number
π	Pi constant	π = 3.141592654... is the ratio between the circumference and diameter of a circle and it is an irrational numbers	c = π⋅d = 2⋅π⋅r where r is the radius of circle and d = 2r
rad	Radians	Radians angle unit	360° = 2π rad or 2π ^c
^c	Radians	Radians angle unit	360° = 2π ^{c or}2π rad
grad	Gradians	Grads angle unit	360° = 400 grad or g
^g	Gradians	Grads angle unit	360° = 400 ^{g or}grad

Mathematics symbols - Algebraic

2020-05-25T12:31:00.008+05:30

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss Third topic Mathematics symbols - Algebraic on this post.

Mathematics symbols - Algebraic

These are very important Basic Tools frequently used by us in algebra as well as in other branches of mathematics and sciences.

Symbol

Name of Symbol

Definition / Meaning

Illustration

Any alphabets like x or y or z

Variable x or variable y or etc...

The unknown value we have to find

If 4x = 4, then x = 1 or
If -12z = 48, then z = -4

≡

Equivalence (Used in Relation)

Identical to

a ≡ b, a is identical to b.

≜

Equal by definition

Equal by definition (that is given)

This symbol is also used for the same:=

:=

Equal by definition

Equal by definition (that is given)

This symbol is also used for the same ≜

~

Approximately equal

Weak approximation

11.0001 ~ 10.9999

≈

Approximately equal

Approximation (more near in comparison to ~ )

tan x≈ x if x is very small
tan(0.001) ≈ 0.001 or
sin x≈ x if x is very small

∝

Proportional to

Proportional to ( If number varies in definite proportion)

y ∝ x when y = k x, where k is a proportionality constant.

∞
Lemniscate
Symbol of Infinity

it can be also used in both sense + ∞ and - ∞.

≪

Much less than

Much less than (Very)

1 ≪ 1000000

≫

Much greater than

Much greater than (Very)

1000000 ≫ 1

( )

parentheses

calculate expression inside first

-2 × (3+5) = -16

[ ]

brackets

calculate expression inside first

[(1+3)×(1+5)] = 24

{ }

Braces

Used for a set or known as Curley Bracket also

A = {a, e, i, o, u}

⌊x⌋

floor brackets

Rounds number to lower integer

⌊5.1⌋ = 5 =⌊5.9⌋ = ⌊5.7⌋ =⌊5.3⌋

⌈x⌉

Ceiling brackets

rounds number to upper integer

⌈4.3⌉ = 5

n!

Factorial / Exclamation mark

In mathematics used for Factorial Notation

1! =1
2! = 12= 2
3! = 123 = 6
4! = 1234 = 24 ...
n! = 123*4 ... continued products upto n

| x |

Vertical bars

Absolute value in Real numbers and Modulas in Complex Numbers

| -4 | = 4 or | -3.2 | = 3.2 or | -101 | = 101.

f (x) or g(x)

function of x

Mappings values of x to f(x)

f (y) = 4y-3

(f ∘ g)

Composition of functions

(f ∘ g) (x) = f (g(x))

f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)

(a,b)

Interval Open

(a,b) = {y | a < y < b} Initial and final values are not incliuded

y ∈ (0,6)

[a,b]

Interval Closed

[a,b] = {y| a ≤ y ≤ b} Initial and final values are also incliuded

y ∈ [-4,4]

∆

Delta

Difference ( Used in Numerical Analysis and higher studies)

∆t = t₁- t₀

∆

Discriminant

Δ = b² - 4ac in quadratic equation where
a = coefficient of square of x
b = coefficient of x
c = constant term

Used for finding Nature of Roots - Quadratic Equations

∑

Sigma used for Summation

Used for summation - the sum of all values in the range of series

∑ x_i= x₁+x₂+...+x_n

∑∑

sigma

double summation

∏

capital pi

Used for Product - Product of all values in range of series

∏ x_i=x₁∙x₂∙...∙x_n

e

Euler's number

e = 2.718281828...

e = lim (1+1/y)^y , Where y→∞

e constant For logarithms
e = lim (1+1/y)^y , Where y→∞

*Symbol*	*Name of Symbol*	*Definition / Meaning*	*Illustration*
Any alphabets like x or y or z	Variable x or variable y or etc...	The unknown value we have to find	If 4x = 4, then x = 1 or If -12z = 48, then z = -4
≡	Equivalence (Used in Relation)	Identical to	a ≡ b, a is identical to b.
≜	Equal by definition	Equal by definition (that is given)	This symbol is also used for the same:=
:=	Equal by definition	Equal by definition (that is given)	This symbol is also used for the same ≜
~	Approximately equal	Weak approximation	11.0001 ~ 10.9999
≈	Approximately equal	Approximation (more near in comparison to ~ )	tan x≈ x if x is very small tan(0.001) ≈ 0.001 or sin x≈ x if x is very small
∝	Proportional to	Proportional to ( If number varies in definite proportion)	y ∝ x when y = k x, where k is a proportionality constant.
∞	Lemniscate	Symbol of Infinity	it can be also used in both sense + ∞ and - ∞.
≪	Much less than	Much less than (Very)	1 ≪ 1000000
≫	Much greater than	Much greater than (Very)	1000000 ≫ 1
( )	parentheses	calculate expression inside first	-2 × (3+5) = -16
[ ]	brackets	calculate expression inside first	[(1+3)×(1+5)] = 24
{ }	Braces	Used for a set or known as Curley Bracket also	A = {a, e, i, o, u}
⌊x⌋	floor brackets	Rounds number to lower integer	⌊5.1⌋ = 5 =⌊5.9⌋ = ⌊5.7⌋ =⌊5.3⌋
⌈x⌉	Ceiling brackets	rounds number to upper integer	⌈4.3⌉ = 5
n!	Factorial / Exclamation mark	In mathematics used for Factorial Notation	1! =1 2! = 12= 2 3! = 123 = 6 4! = 1234 = 24 ... n! = 123*4 ... continued products upto n
\| x \|	Vertical bars	Absolute value in Real numbers and Modulas in Complex Numbers	\| -4 \| = 4 or \| -3.2 \| = 3.2 or \| -101 \| = 101.
f (x) or g(x)	function of x	Mappings values of x to f(x)	f (y) = 4y-3
(f ∘ g)	Composition of functions	(f ∘ g) (x) = f (g(x))	f (x)=3x,g(x)=x-1 ⇒(f ∘ g)(x)=3(x-1)
(a,b)	Interval Open	(a,b) = {y \| a < y < b} Initial and final values are not incliuded	y ∈ (0,6)
[a,b]	Interval Closed	[a,b] = {y\| a ≤ y ≤ b} Initial and final values are also incliuded	y ∈ [-4,4]
∆	Delta	Difference ( Used in Numerical Analysis and higher studies)	∆t = t₁- t₀
∆	Discriminant	Δ = b² - 4ac in quadratic equation where a = coefficient of square of x b = coefficient of x c = constant term	Used for finding Nature of Roots - Quadratic Equations
∑	Sigma used for Summation	Used for summation - the sum of all values in the range of series	∑ x_i= x₁+x₂+...+x_n
∑∑	sigma	double summation
∏	capital pi	Used for Product - Product of all values in range of series	∏ x_i=x₁∙x₂∙...∙x_n
e	Euler's number	e = 2.718281828...	e = lim (1+1/y)^y , Where y→∞
	e constant	For logarithms	e = lim (1+1/y)^y , Where y→∞

Mathematics symbols - Basic

2020-05-24T17:12:00.005+05:30

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

Now we shall learn /discuss second topic Mathematics symbols - Basic on this post

These are very important Basic Tools frequently used by us

Symbol

Name of Symbol

Definition / Meaning

Illustration

=

Sign of Equality / Equal sign

equality

1+3 =4
1+3  is equal to 4

≠

Not equal sign

inequality

-4 ≠ 4
-4 is not equal to 4

≈

Approximately equal

approximation

sin(0.0001) ≈ 0.0001,
a ≈ b means a is approximately equals to b

>

Strict inequality

greater than

9 > 4
9 is greater than 4

<

Strict inequality

less than

-3 < -2
-3 is less than -2

≥

Inequality

greater than or equals to

15 ≥ 14,
a ≥ b means a is greater than or equals to b

≤

Inequality

less than or equals to

-1 ≤ 5,
a≤ b means a is less than or equals to b

( )

Parentheses

calculate expression inside first

-2 × (3+5) = -16

[ ]

Brackets

calculate expression inside first

[(1+3)×(1+5)] = 24

+

Plus sign

addition

0 + 1 = 1 or 1 +1 = 2 etc

−

Minus sign

subtraction

5-2 = 3 or -2 − 1 = -3 etc

±

Plus - Minus

both plus and minus operations

4 ± 5 = 9 or -1

∓

Minus - Plus

both minus and plus operations

4 ∓ 5 = -1 or 9

*

Asterisk

multiplication

2 * 4 = 8 or 11 = 1 etc

×

Times sign

multiplication

2 × 4 = 8 or -2 x -3 = 6 etc

⋅

Dot sign

multiplication

2 ⋅ 4 = 8 or -2 .3 = -6 etc

÷

Sign of Division

division

10÷ 2 = 5 Division by 0 is not possible

/

Slash sign for Division

division

10 / 2 = 5

- Horizontal line Horizontal line - generally used for fractions

mod

Modulo

remainder calculation

3 mod 2 = 1

.

Period /Point sign

decimal point, decimal separator

3.56 = 3+56/100

a^b

Power / Index

exponent

4³= 4x4x4= 64

a^b*

Caret

exponent

(-3) ^ 3= (-3) x (-3) x (-3) = -27 etc

√a

Square root of positive number

√p ⋅ √p = p,   where p is any number

√25 = ±5
√100 = ±10
√625 = ±25 etc

³√a

Cube root

³√p ⋅ ³√p  ⋅ ³√p =p,

where p is any number

³√1 = 1
³√27 = 3
³√125 = 5
³√1000 = 10 etc

⁴√a

Fourth root

⁴√p ⋅ ⁴√p ⋅ ⁴√p  ⋅ ⁴√p = p where p is any number

⁴√1 = ±1
⁴√16 = ±4
⁴√81 = ±3
⁴√10000 = ±10 etc

ⁿ√a

The n-th root of any number

Radical

for n=5, ⁿ√32 = 2

%

Percent

1% = 1/100

10% × 50 = 5

‰

Per-mile

1‰ = 1/1000 = 0.1%

10‰ × 50 = 0.5

ppm

Part per million

1ppm = 1/1000000

10ppm × 50 = 0.0005

ppb

Part per-billion

1ppb = 1/1000000000

10ppb × 50 = 5×10^-7

ppt

Part per-trillion

1ppt = 10^-12

10ppt × 50 = 5×10^-10

*Symbol*	*Name of Symbol*	*Definition / Meaning*	*Illustration*
=	Sign of Equality / Equal sign	equality	1+3 =4 1+3 is equal to 4
≠	Not equal sign	inequality	-4 ≠ 4 -4 is not equal to 4
≈	Approximately equal	approximation	sin(0.0001) ≈ 0.0001, a ≈ b means a is approximately equals to b
>	Strict inequality	greater than	9 > 4 9 is greater than 4
<	Strict inequality	less than	-3 < -2 -3 is less than -2
≥	Inequality	greater than or equals to	15 ≥ 14, a ≥ b means a is greater than or equals to b
≤	Inequality	less than or equals to	-1 ≤ 5, a≤ b means a is less than or equals to b
( )	Parentheses	calculate expression inside first	-2 × (3+5) = -16
[ ]	Brackets	calculate expression inside first	[(1+3)×(1+5)] = 24
+	Plus sign	addition	0 + 1 = 1 or 1 +1 = 2 etc
−	Minus sign	subtraction	5-2 = 3 or -2 − 1 = -3 etc
±	Plus - Minus	both plus and minus operations	4 ± 5 = 9 or -1
∓	Minus - Plus	both minus and plus operations	4 ∓ 5 = -1 or 9
*	Asterisk	multiplication	2 * 4 = 8 or 1*1 = 1 etc
×	Times sign	multiplication	2 × 4 = 8 or -2 x -3 = 6 etc
⋅	Dot sign	multiplication	2 ⋅ 4 = 8 or -2 .3 = -6 etc
÷	Sign of Division	division	10÷ 2 = 5 Division by 0 is not possible
/	Slash sign for Division	division	10 / 2 = 5
	- Horizontal line	Horizontal line -	generally used for fractions
mod	Modulo	remainder calculation	3 mod 2 = 1
.	Period /Point sign	decimal point, decimal separator	3.56 = 3+56/100
a^b	Power / Index	exponent	4³= 4x4x4= 64
*a^b*	Caret	exponent	(-3) ^ 3= (-3) x (-3) x (-3) = -27 etc
√a	Square root of positive number	√p ⋅ √p = p, where p is any number	√25 = ±5 √100 = ±10 √625 = ±25 etc
³√a	Cube root	³√p ⋅ ³√p ⋅ ³√p =p, *where p is any number*	³√1 = 1 ³√27 = 3 ³√125 = 5 ³√1000 = 10 etc
⁴√a	Fourth root	⁴√p ⋅ ⁴√p ⋅ ⁴√p ⋅ ⁴√p = p where p is any number	⁴√1 = ±1 ⁴√16 = ±4 ⁴√81 = ±3 ⁴√10000 = ±10 etc
ⁿ√a	The n-th root of any number	Radical	for n=5, ⁿ√32 = 2
%	Percent	1% = 1/100	10% × 50 = 5
‰	Per-mile	1‰ = 1/1000 = 0.1%	10‰ × 50 = 0.5
*ppm*	Part per million	1ppm = 1/1000000	10ppm × 50 = 0.0005
*ppb*	Part per-billion	1ppb = 1/1000000000	10ppb × 50 = 5×10^-7
*ppt*	Part per-trillion	1ppt = 10^-12	10ppt × 50 = 5×10^-10

Mathematics Symbols - Number

2020-05-21T12:20:00.007+05:30

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Number

Mathematics symbols - Basic

Mathematics symbols - Algebraic

Mathematics symbols - Geometric

Mathematics symbols - Trigonometry

Mathematics symbols - Set Theory

Mathematics symbols - Statistics & Probability

Mathematics symbols - Calculus And Analysis

Mathematics symbols - Logical

Mathematics symbols - Roman

Mathematics symbols - Greek

Mathematics symbols - Miscellaneous

In mathematics numbers, we required are generally represented by using combinations of ten (10) number symbols 0 (Zero / Nil), 1(One ), 2 ( Two), 3 ( Three ), 4 ( Four), 5 ( Five), 6 ( Six), 7 ( Seven), 8 ( Eight), and 9 ( Nine ),{These symbols also called as numerals or digits} along with the mathematical symbols/operators "." ( Dot ), "+" ( Plus ), and "–" ( Minus) and " / " Division. Now we are using the following numbers in our day to day life.

Mathematics symbols - Number

These are very important Number Tools frequently used by us in arithmetics, algebra as well as in other branches of mathematics and sciences.

On the using of below concepts, we can easily have answers of such type of questions like

Example 1) Write 20 lakh crore in mathematics language?

Answer:- 20,000,000,000,000

in other ways this can be also written as

20,00,000 Crores = 200000000000 Hundreds

20,00,000 Crores = 20,00,00,00,000 Thousands

20,00,000 Crores = 20,00,00,000 Lakhs

Example 2) Write 130 crore in mathematics form or in mathematical language?

Answer:- 1,30,00,00,000

We can also observe that 130 crore is the same as 1.3 billion.

S. N.	Number	Number Name
1	0	Zero
2	1	One
3	2	Two
4	3	Three
5	4	Four
6	5	Five
7	6	Six
8	7	Seven
9	8	Eight
10	9	Nine
11	10	Ten
12	11	Eleven
13	12	Twelve
14	13	Thirteen
15	14	Fourteen
16	15	Fifteen
17	16	Sixteen
18	17	Seventeen
19	18	Eighteen
20	19	Nineteen
21	20	Twenty
22	30	Thirty
23	40	Forty
24	50	Fifty
25	60	Sixty
26	70	Seventy
27	80	Eighty
28	90	Ninety
29	100	One Hundred
30	1,000	One Thousand
31	10,000	Ten Thousand
32	1.00,000	Hundred Thousand / One Lakh
33	1,000,000	One Million / 10 Lakh
34	10,000,000	Ten Million / One Crore etc...

Classification of Numbers - Natural, Whole, Integers, Rationals, Irrationals, Real,even, Odd...

2020-05-19T19:24:00.009+05:30

In general numbers in Mathematics from Ancient Times can be classified according to how they are represented or according to their properties.

Classification of Numbers - Formulating numbers

In mathematics numbers, we required are generally represented by using combinations of ten (10) number symbols 0 (Zero / Nil), 1(One ), 2 ( Two), 3 ( Three ), 4 ( Four), 5 ( Five), 6 ( Six), 7 ( Seven), 8 ( Eight), and 9 ( Nine ),{These symbols also called as numerals or digits} along with the mathematical symbols/operators "." ( Dot ), "+" ( Plus ), and "–" ( Minus) and " / " Division.
We can write numbers in various types by using Decimal Systems are such as: -

1, 1.1, 1.001, 2.0001, 3.33333333..., 0.777777..., 22 / 7, 2212, 4.01001000100001....., -1, -7, -2.6100, -.555552, - 3 / 5, - 7.777..., - 5.12112111211112..., 2² , 40231² etc

Number System will be introduced to the mathematics Fraternity gradually, and each new type comes with its own uses or due to complete the shortfall of the previous one and its own challenges.
The main types of numbers /Classification of Numbers used in modern mathematics are such as:

Classification of Numbers

Classification of Numbers

On the basis of Properties, Classification of Numbers can be done in two ways according to my opinion -

(A ) BASIC TYPES OF NUMBERS

Natural Numbers (N)

Whole Numbers (W)

Integers Numbers (I or Z)

Rational Numbers (Q)

Irrational Numbers

Real Numbers (R)

Imaginary Numbers, Complex Number, etc

(B) OTHER STANDARD TYPES OF NUMBERS

Even Numbers
Odd Numbers
Prime Numbers
Composite Numbers
Perfect square Numbers
Perfect cubic Numbers
Perfect Numbers
Co-Prime Numbers
Discrete or continuous etc

(A ) Classification of Numbers - BASIC TYPES OF NUMBERS

Natural Numbers (N)

Counting Numbers
OR

Positive integers

A number that generally occurs commonly and obviously in our surroundings or nature. From Ancient Time, first of all, Number is used for counting Fruits, Pebbles, Pet's, animals, or Balls, etc...

The set of natural numbers Generally denoted by boldface “N” (The first letter of Natural Number, It is a general convention adopted in Mathematics).
it's set is written as

N= Set of Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8, …}

More preciously it can be defined as "Those numbers which start from 1 ( One ) and Gradually increased By STEP 1 are known as Natural Numbers or Counting Numbers."

Examples

One Orange

Image source- Google|image by-

food-healthy-orange-white-42059

Two Oranges

+ =Two Oranges...

Three Oranges

= Three Oranges ...

Whole Numbers (W)

OR
Complete Numbers

Those numbers that generally occur commonly and obviously in our surroundings or nature are known as Naturals, From Ancient Time of they are used for counting Fruits, Pebbles, pets, animals, or Ball's, etc...

If these are sufficient then what is the need for more extension in Naturals, Think ...???

Image source- Google|image by-ExplorerBob (pixabay.com)

In the above picture according to the Natural Number concept, we have 2 ( Two ) water bottles (Glass).

Let us assume if all the above Two water bottles (Glass) BROKEN then how many Number of Water bottles, We have left ...........???

In Natural numbers we are not able to give an answer to the above problems, So the extension of Naturals is required.

Mathematically it can be expressed such as:-

if we solve the following equation:-

x + 2 = 2 (for every x belongs to N )

x = ????

In naturals no solution Exists (possible) since ZERO (Null) does not belong to Natural Numbers, Therefore an Extension in Natural Number is required.

Natural numbers extension along with ‘0’ ( Zero)are called Whole Numbers.

The set of Whole Numbers Generally denoted by usual convention boldface 'W'The first letter of the Whole Number

(It is a general convention adopted in Mathematics from ancient times).

W = Set of Whole Numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, …}

More preciously it can be defined as "Those numbers which start from 0 ( Zero ) and Gradually increased By STEP 1 are known as Whole Numbers."

The whole numbers are the set or collection of positive integers and zero.

Integers (I or Z)

If Whole Numbers are sufficient then what is the need for more extension in Wholes, Think ...???

In the above picture according to the Number concept (till Whole Numbers), we have 2 ( Two ) water bottles (Glass).

Let us assume if any Nears & Dears ask to give three water bottles (Glass) to us then how many Number of Water bottles, We have left ...........???

Up to concepts of Whole Numbers we are not able to give the answer to the above problems, So the extension of Whole Numbers is required and essentials.

Mathematically it can be expressed such as:-
if we solve the following equation:-

x + 5 = 2 (for every x belongs to W )

x = ????

In naturals, no solution Exists (possible), since Negative of Naturals (-3) is not belongs to Whole Numbers, Therefore Extension in Whole Numbers is required.

Therefore Whole Numbers along with the negative of Natural numbers are called Integers.
The set of Integers Generally denoted by boldface ' I' The first letter of the Integer, (It is a general convention adopted in Mathematics) but more popular Notation is boldface " Z" ( It is the first Letter of Zahlen. It is a German word which means Numbers).

Z or I = {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, …}

More preciously it can be defined as "Those numbers which start from 0 ( Zero ) and Gradually increased By STEP 1 (one only) on both sides (Positive and Negative) are called Integers."

The integers are the set or collection of all whole numbers plus all negative of the Natural Numbers.

Rational Numbers (Q).

If Integers Numbers are sufficient then what is the need for more extensions in integers, Think ...???

Image source- Google|image by-
Petr Kratochvil

In the given picture of an apple according to the Number concept (till Integers Numbers), we have 1 Apple. Let us assume if we have to divide this apple into our Three friends also. Then how many Parts of Apple each friend got, ...........???Up to concepts of Integer Numbers we are not able to give the answer to the above problems, So the extension of Integers is also required and essentials.

Mathematically it can be expressed in a similar manner such as:-
if we solve the following equation:-

2x+3=12 (for every x belongs to Z)

x =?

In Integers no solution Exists (possible), since 9/2 fractions do not belong to Integers Numbers, Therefore Extension in IntegersNumbers is required.

As per the famous Proverb "Necessity is the mother of Invention", so Rationals Discovered by Mathematicians to Solve Above Problems. Therefore Integers along with non zero fractions. In Mathematics, Rational numbers play an important role in Number System.
The general conventions in the first learner about Rational Numbers are such as:-

Rational Numbers are simply quotients/fractions having Denominators not zero.

Rational Numbers can be obtained by dividing two integers.

The fraction with denominators ( none zero)is a Rational number.

Rational Numbers contains all Natural Numbers, Whole Numbers, Integers.

Rational Numbers are expressed in the "Numerator / Denominator" form.

Although all the above conventions are considered as definition But the definition on the basis of all above assumptions can be given in a more appropriate way as:

Rational Numbers are generally expressed in the form 'Numerator (p) / Denominator (q) from ancient times, where p and q both are integers, and q is not equal to zero.

If we express in Decimal Fractions then Either Terminating Or Non-Terminating But RECURRING / Periodic / Repeating are known as Rational Numbers.

Like all other Numbers, Rational numbers are usually denoted by a

Q

(boldfaced) Where "Q" is taken from quotients.

now a few examples are such as:

2/3, 3/5, 4/7, 11/2, 303/12654, 0/1, 100/1, 51/ 9, -1/2, -2/3, -4/5, -11/2, - 2013/11111,1/2, 3/4, 17/2, 14/13, 41/1 ...
In decimal representation can be written as
3.0, 1.25, 4.33..., 1.232323232323...., 4.1234,14.1234123412341234..., etc
Few important properties of Rational Number are listed below:-

Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, -0/1, 0/2, 0/3, etc.
But it is important to keep in mind that -1/0, -15/0, 1/0, 2/0, 3/0, etc. are not rational.
Thus all the fractions where the top and bottom numbers are integers(Bottom number can not be zero); e.g., 2/3, 3/5, 4/7, 11/2, 303/12654, 0/1, 100/1, 51/ 9, -1/2, -2/3, -4/5, -11/2, - 2013/11111,1/2, 3/4, 17/2, ⁻14/13, 41/1 ... are Rationals.
The denominator of Rationals cannot be 0, but the numerator can be ( Important).
Since the denominator may be equal to 1, so every Natural Number is a rational number. ( 1/1, 2/1,3/1, etc )
Since the denominator may be equal to 1, so every Whole Number is a rational number. ( 0/1,1/1, 2/1,3/1, etc )
Since the denominator may be equal to 1, so every Integer is a rational number.( 1/1, 2/1,3/1, etc )

Irrational numbers

If Rational Number is sufficient then what is the need for more extension in integers, Think ...???

if we solve the following equation:-

x = ????

In Rationals no solution Exists (possible), since the root of three does not belong to Rational Numbers, Therefore Extension in Rational is required.

As per the famous Proverb "Necessity is the mother of Invention", so Irrationals Discovered by Mathematicians to Solve Above Problems. Therefore Rationals along with their positive roots.

Irrational Numbers are generally expressed in the form root of x from ancient times, where x is non-negative and not a perfect square Number.

If we express in Decimal Fractions then NeitherTerminating Or Not RECURRING / Periodic / Repeating are known as Irrational Numbers.
Now a few examples in decimal expansions are such as:

1.31323334..., 0.30313234785...,

0.70700700070000 ( See the Patterns)..., etc

Real numbers ( R)

Union of Rational and Irrational Number is generally known as Real Numbers and denoted by as usual conventions by 'R'.
These numbers are in daily life used for Measurements so also called Measuring Numbers.
Every Number as discussed above Like- Natural Number, Whole Number, Integer, Rationals, Irrationals are all subsets of real numbers.
Every Real number can be represented on Number Lines with a unique representation.

(B) Classification of Numbers - OTHER STANDARD TYPES OF NUMBERS

Even Numbers

A whole number that can be exactly divisible by 2 ( Two) is known as Even Numbers or in other words, we divide by 2 then leaves remainder always zero.

Even numbers always end with the last digits 0, 2, 4, 6, 8.

Some examples are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ...etc

Generally, we can say multiples of 2 are always even numbers.

Odd Numbers

A whole number that can be exactly not divisible by 2 ( Two) is known as Odd Numbers or in other words, we divide by 2 then leaves remainder always one (1).

Odd numbers always end with last digits 1, 3, 5, 7, 9.

Some examples are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 ...etc

Generally, we can say multiples of 2 are cannot be odd numbers.

A number may be either Even or Odd not Both, so a natural number can be again redefined as a "Collection of all even and odd numbers".

Prime Numbers

Any number is said to be a prime number 'if it is a positive integer that has exactly two positive integral factors', 1 and itself.

For example, if we list the factors of 14, we have 1, 2, 7, and 14. That's more than two (four) factors. So we say that 14 is not a prime number.

If we consider another example 19, we only have 1 and 19. That's two factors. So we say that 19 is a prime number.

we can also have a property that prime number is a positive integer that is not the product of two smaller positive integers.

According to the definition of prime number 1 is not a prime number since it has only one factor.

When a number can not be divided up exactly called Prime Numbers. For Example, 5 can not be divided into two equal groups so also Prime numbers.

we can write a few prime numbers are such as:-

2 (Two), 3 (Three), 5 (five), 7 (Seven), 11 (Eleven), 13 ( Thirteen), 17, 19, 23, 29, 31, 37, 43, 41, 47, 53, 59, 61, 67, 73, 71, 79 (seventy Nine), 83, 89, 97, 101, 103, 107, 109, 113, 127, 137, 131, 139, 149, 151 (One hundred Fifty One), 157, 163, 167, 173, 179, 181, 191, 193, 197 ( One hundred Ninety seven), 199, 203, 207, 211, 213... etc.

Composite Numbers

If any number has more than two factors then it is called a composite number.

For example, if we list the factors of 14, we have 1, 2, 7, and 14. That's more than two (four) factors. So we say that 14 is a composite number.

If we consider another example 19, we only have 1 and 19. That's two factors. So we say that 19 is not a composite number.

When a number can be divided up exactly called composite Numbers. For Example, 6 can be divided exactly into two groups of three's (6 = 3 + 3) as well as two's ( 6 = 2+ 2 + 2) equal groups so 6 is composite numbers.

One is neither Prime nor composite Numbers.

We can write the few composite numbers are such as:- 4 (four), 6 (six) , 8 ( Eight), 9 (Nine), 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70 (Seventy), 72, 74, 75, 76, 77, 78, 80, 81, 84, 82, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110 (One hundred Ten), 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 130, 132, 134, 136, 135, 133, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150 ( One Hundred Fifty), 151, 152, 154, 155, 156, 157... etc

Perfect square Numbers

The squares of the whole numbers are called perfect squares numbers.

We can write the few perfect square numbers are such as:- 0(Zero), 1 (One), 4 (four), 9 (Nine),16 (Sixteen), 25, 36, 49, 64, 81, 100, 121, 144 (One Hundred Forty Four), 169, 196, 225, 256, 289, …etc

Perfect cubic Numbers

The cubes of the whole numbers are called perfect cubic numbers.

We can write the few cubic numbers are such as- 0(Zero), 1 (One), 9 (Nine), 27 ( Twenty Seven), 64, 125, 216, 343 (Three Hundred Forty-Three), …etc

Perfect Number

A positive integer is said to be the perfect number that some of the proper divisor (Excluding given Number) equal to the number itself.

Proper divisors of 6 are 1, 2, and 3, Now sum is 1 + 2 + 3 = 6 (Given number 6), So 6 is perfect Numbers.

We can write a few perfect numbers are such as- 6, 28, 496, 8128...

Co-Prime Number

Those numbers having only one common factor 1 (HCF = 1) are known as Co-Prime Numbers or Relatively Prime.

It is not essential that the Co-prime number must be Prime.

if we consider few examples like (1, 2), (1, 10), (1, 100), (5, 2), (3, 4), (5, 6), (19, 20) all are Co-Primes since their HCF = 1, although some of them are not Prime.

Since by the definition of the co-prime number we know that "HCF of co-prime numbers is 1". Therefore LCM of given co-prime numbers is equal to the product of the co-prime numbers.

We have an important property about co-prime numbers that "Co-prime Numbers LCM = Product Of The co-prime Numbers".

Discrete or continuous Numbers

After the above discussion, all types of numbers can be split up into two groups discrete Numbers or continuous numbers.
The first four of the above basis tyes of the number are Natural Numbers (N), Whole Numbers (W), Integers Numbers (I or Z), and Rational Numbers (Q) are considered as discrete numbers.
It means that they are separate and distinct entities. Also, each of these sets is countable (those that can be counted are known as countable).
The set of Irrational numbers (R) is also a discrete number but they can not be counted (Uncountable).
The set of Real numbers (R) is continuous so they can not be counted.

We know that between any two real numbers, there may be infinitely many real numbers that may be very close or dense.
____________________________________________________

Career Opportunities After 12 Standard

2020-05-15T16:51:00.002+05:30

In life scopes are unlimited depends on our interests and utilization of skills. If any students Passed 12 class then in my opinion Career Opportunities After 12 Standard are depends on their subjects / Streams (Science / Commerce / Humanities ) studied in 11 and 12 standards.

On behalf of optional subjects of 12 standards, it can be categorized in the following groups:-

(A ) Career Opportunities After 12 Standard in Science with Mathematics

(B ) Career Opportunities After 12 Standard in Science with Biology.

(C ) Career Opportunities After 12 Standard in Commerce.

(D) Career Opportunities After 12 Standard Arts / Humanities.

(E ) Career Opportunities After 12 Standard General.
(F ) Career Opportunities After GRADUATION LEVEL General.

According to my opinion following choices are available to students according to their categories of the optional subject after passing of 12 standard examinations by CBSE / ICSE/ NIOS or any state boards.

There are the following choices for both mathematics and biology groups.

A) Career Opportunities After 12 Standard in Science with Mathematics

B TECH / B, E. ( By IIT / NIT/ IIIT/ ANY ENGINEERING COLLEGES )
N D A
INTEGRATED B. Sc.B.Ed.
INTEGRATED M SC ( PHYSICS / CHEMISTRY / MATHEMATICS / STATISTICS / or any APPLIED subjects).
SCRA
Pharmacy ( Diploma / Bachelor / Master )
Agriculture ( B Sc ./ M Sc. )

B ) Career Opportunities After 12 Standard in Science with Biology.

MBBS ( By AIIMS / ANY govt / private / foreign college )
B D S
N D A
Ayurvedic BAMS
Homeopathy BHMS
Veterinary science
Pharmacy ( D / B / M )
INTEGRATED B. SC.BED.
INTEGRATED M SC. ( Chemistry / Botany / zoology/ applied Science ).
Agriculture ( B Sc ./ M Sc. )

(C ) Career Opportunities After 12 Standard in Commerce.

There are the following choices after 12 commerce. You may appear in competitive examination or take admission in any of the following courses:-

B Com.

B A

B B A

Integrated M B A

C A

I C WA

C S

Integrated B Com B Ed.

Integrated M Com from the central university.

N D A for Army only.

B C A / integrated M C A.

Bachelor in mass communication.

Integrated LLB by CLAT.

(D) Career Opportunities After 12 Standard Arts / Humanities.

B A

B B A

Integrated B A B Ed.

Integrated M A from the central university.

N D A for Army only.

B C A / integrated M C A.

Bachelor in mass communication.

Integrated LLB by CLAT.

(E ) Career Opportunities After 12 Standard General.

B C A / integrated M C A.
Bachelor in mass communication.
Integrated LLB by CLAT.
STC (Teacher Training)
Diploma or degree in Library Science.
ITI ( Also Eligible After 10)
Polytechnic Diploma in all trades.
Diploma or degree in Physical Science / Education.
Fashion Designing.

(F ) Career Opportunities After GRADUATION LEVEL (General).

Following opportunities after completion of Graduation in any field.

Indian Civil service ( ICS).

Bank P O / clerical grades

Railway

C D S

LIC (AAO)

POST OFFICE

SSC

B.Ed.

SO MANY competitive examinations.

________________________________________________________________________________

Properties of HCF (GCD) & LCM

2020-05-08T17:08:00.000+05:30

To understand the concept of HCF (Highest Common Factor) / GCD (Greatest Common Divisor) & LCM (Lowest Common Multiple) in completely, we have to recall the terms Factors and Multiples.

We may also recall Least Common Multiple ( LCM) and Highest Common Factor (HCF) or Greatest Common Divisor (GCD) before going through their properties.

Least Common Multiple ( LCM)

The least common multiple ( LCM ) is also referred to as the lowest common multiple or smallest common multiple of two integers. For any two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.

For example, The L.C.M of 3 and 4 is 12.

Highest Common Factor (HCF) Or Greatest Common Divisor (GCD / gcd)

The Highest Common Factor ( HCF) or greatest common divisor (GCD / gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
For any two integers a and b, usually denoted by HCF (a, b), is the largest integer that can divide both numbers a and b. (a , b) can be also written as for simplicity.
For example, The H.C.F of 10 and 15 is 5.

Important Properties of HCF and LCM

1) The Highest Common Factor ( HCF) or greatest common divisor (GCD) of two or more integers is always LESS than to the given numbers.

For example, The H.C.F of 10 and 15 is 5.

If we consider another example,The H.C.F of 14 and 28 is 14.

2) The Least Common Multiple LCM) of two or more integers is always GREATER than or EQUALS to given numbers.

For example, The L.C.M of 10 and 20 is 20.

If we consider another example,The L.C.M of 10, 15 and 20 is 60.

3) Relation between LCM and HCF (Formula for finding HCF and LCM)

Given relations is very popular in mathematics to find either LCM (if HCF is known) or HCF (if LCM is known) or both are known along with one number then we can find another unknown number.

'The product of Least Common Multiple( LCM) and Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of any two Numbers/ Polynomials is equaled to their Products'.

in other words, mathematically we can write as:-

LCM × HCF = Product of the Numbers

if a and b are two numbers, then.

HCF (a , b) x LCM (a , b) = a × b.

Above results can be used directly or summarized as in different situations

LCM (a , b) = (a × b) / HCF (a , b).

HCF (a , b) = (a × b) / LCM (a , b).

a ( one number) = LCM (a , b) × HCF (a , b) / b

Illustrative Examples for this third property / Formula / Results.

Example 1:- The LCM and HCF of two numbers are 150 and 4 respectively. If 25 is a number given then find the other number.

Solution: As we know that the product of two numbers is equal to the product of HCF and LCM of two numbers & HCF and LCM are given to us. Also, one of the numbers is given to us. Thus, we need to find the other number so we may use the above relation.

So, HCF x LCM (or LCM x HCF ) = Product of the Numbers

=> 150 × 4 = 25 × x

=> x = 24

Therefore required answer is 24.

We can use direct relation to verify the above results

a ( one number) = LCM (a , b) × HCF (a , b) / b

a = (150 x 4 ) / 25

a = 24.

Therefore required answer is 24.

Example 2:- Prove the above formula

LCM (18 & 12) × HCF (18 & 12) = Product of 18 and 12

Solution:- First of all we have to find the prime factorisation of given Numbers 18 and 12

18 = 2 x3 x 3 = 2 x 3²

12 = 2 x 2 x 3 = 2² x 3

Now we have to calculate their LCM and HCF

LCM of 18 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 18 and 12 = 3 × 2 = 6

LHS (Left Hand Side) =

LCM (18 & 12) × HCF (18 & 12) = 36 × 6 = 216

RHS (Right Hand Side) =

Product of 18 and 12 = 18 × 12 = 216

Now

LHS (Left Hand Side) = RHS (Right Hand Side) = 216

Hence, LCM (18 & 12) × HCF (18 & 12) = 18 × 12 = 216

4) Formula for finding HCF and LCM of Fraction is such as

L.C.M. = L.C.M. of Numerator / H.C.F. of Denominator

H.C.F. = H.C.F. of Numerator / L.C.M. of Denominator

Now we solve some problems for finding LCM and HCF of Fractions:-

CFofnumeratorsLCMofdenominators

Example 3:- Find L.C.M. of

4 / 3, 8 / 81, 64 / 9 ,10 / 27

Solution:

Before using this formula we have to find LCM of Numerator and HCF of Denominator.

L.C.M. =	L.C.M. of Numerator
	H.C.F. of Denominator

L.C.M. of Numerators = 4, 8, 64, 10 ( By Prime Factorisation)

4 = 2²
8 = 2³
64 = 2⁶
10 = 2 × 5
L.C.M of 4, 8, 64, 10 = 2⁶ × 5 = 320

H.C.F. of Denominators = 3, 81, 9, 27 ( By Prime Factorisation)
3 = 3¹
81 = 3⁴
9 = 3²
27 = 3³
H.C.F. of 3, 81, 9, 27 = 3

L.C.M. of

4 / 3, 8 / 81, 64 / 9 ,10 / 27 = 320 / 3

Therefore 320/ 3 is our required Answer.

We will solve one more Examples

Example 4:- Find the HCF of

415, 910, 1835, 2130

Solution: Before using this formula we have to find LCM of Denominator and HCF of Numerator.

H.C.F. =	H.C.F. of Numerator
	L.C.M. of Denominator

H.C.F. of Numerators = 4, 9, 18, 21 ( By Prime Factorisation)

4 = 2 × 2
9 = 3 × 3
18 = 2 × 3 × 3
21 = 3 × 7
HCF (4, 9, 18, 21) = 1
L.C.M. of Denominators = 15,10,35, 30 ( By Prime Factorisation)

15 = 3 × 5
10 = 2 × 5
35 = 5 × 7
30 = 2 × 3 × 5
LCM(15, 10, 35, 30) = 5 × 3× 2 × 7 = 210
The required

HCF = HCF(4, 9, 18, 21)/LCM(4, 9, 18, 21) = 1/210

Therefore 1 / 210 is our required Answer.

Example 5:- Find the HCF of

415, 1681, 89.

Solution: Before using this formula we have to find LCM of Denominator and HCF of Numerator.

H.C.F. =	H.C.F. of Numerator
	L.C.M. of Denominator

OR H.C.F. =	H.C.F. of 4, 16, 8
	L.C.M. of 15, 81, 9

H.C.F. of Numerators = 4, 16, 8 ( By Prime Factorisation)

4= 2²
16 = 2⁴

8 = 2³
Number with least index = 2² = 4
H.C.F. of 4, 16, 8 = 4

L.C.M. of Denominators = 15,81,9 ( By Prime Factorisation)

15= 3 x 5¹
9 = 3²
81 = 3⁴
Number with highest index = 5 x 3⁴ = 5 x 81 =405
L.C.M. of 15, 81, 9 = 405

H.C.F. =	4
	405

Therefore 4 / 405 is our required Answer.

5 ) Since by the definition of the co-prime number we know that "HCF of co-prime numbers is 1". Therefore LCM of given co-prime numbers is equal to the product of the co-prime numbers.

Co-prime Numbers LCM = Product Of The Numbers

Example 6:- Prove the above formula

LCM of Co-prime Numbers 5&7 = Product Of The Numbers 5&7

Solution:-

LHS (Left Hand Side) =
5= 5¹ 7 = 7¹

LCM (5 & 7) = 5 × 7 = 35

RHS (Right Hand Side) =

Product of 5 and 7 = 5× 7 = 35

Now

LHS (Left Hand Side) = RHS (Right Hand Side) = 35.

______________________________________________________

Highest Common Factor or Greatest Common Divisor

2020-05-05T17:10:00.000+05:30

Highest Common Factor (HCF)
or
Greatest Common Divisor ( GCD)

In arithmetic ( mathematics) and number theory, the greatest common divisor (GCD/gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. The greatest number which divides each of the two or more numbers is popularly called HCF or Highest Common Factor (HCF). It is also called the Greatest Common Measure (GCM) and Greatest Common Divisor(GCD).
For any two integers a and b, usually denoted by HCF (a, b), is the largest integer that can divide both numbers a and b. (a , b) can be also written as for simplicity.
The HCF of more than two integers is also well-defined: it is the largest integer that can divide to all of them.

What is the Highest Common Factor (HCF)?
It is a method to find the highest common factor between numbers ( any two or more). Generally according to definitions the common factor is a number which is a factor of two or more numbers therefore HCF is used to determine the common factor.
Example 1:-
If we consider an example, To find HCF of 2 and 3?
The factors/ divisors of 2 = 1, 2.
The factors/ divisors of 3 = 1, 3.
Common factors/ divisors = 1
Therefore the highest among common Factors (HCF) = 1 (where 1 is the highest common factor for numbers 2 and 3 since both numbers have only one factor in common.)
Therefore HCF(2, 3) = gcd (2, 3) = 1 is the required solution of given numbers.

Those numbers having only one common factor 1 (HCF = 1) are known as Co-Prime Numbers or Relatively Prime.
It is not essential that the Co-prime number must be Prime.
if we consider few examples like (1, 2), (1, 10), (1, 100), (5, 2), (3, 4), (5, 6), (19, 20) all are Co-Primes since their HCF = 1, although some of them are not Prime.

Various methods to find HCF of Numbers:-

By Factorization Method,
By Division Method,
By Relation between LCM and HCF ( By Formula)

By Factorization

The method to find the highest common factor of any given numbers is such as:-

First of all write down the Factors or Divisors of individual numbers.
Find the common factors/ divisors between them.
The product of all common factors/ divisors is our HCF or GCD.

Example 2:- Find HCF of 12 & 18?

Solution:-

The factors/ divisors of 12 = 1, 2, 3 , 4, 6, 12.

The factors/ divisors of 18 = 1, 2, 3 , 6, 9, 18.
Common Factors = 1, 2, 3, 6.
Therefore the highest among common Factors (HCF) = 6 (where 6 is the highest common factor for numbers 2 and 3 since both numbers have four factors/ divisors in common).
Therefore HCF(12, 18) = gcd (12, 18) = 6 is the required solution.

Example 3:-To find HCF of 20, 30 & 50 ( Three numbers)?

Solution:-

Factors of 20 = 1, 2, 4, 5, 10 & 20 ( Since 1x20= 4x5=2x 10).

The factors of 30 = 1, 2, 3, 5, 6, 15, 30.
The factors of 50 = 1, 2, 5, 10, 25 & 50 ( Since 1 x50 = 2 x 25 = 5 x 10).
Common Factors = 1, 2, 5, 10.
Therefore the highest among common Factors (HCF) = 10 (where 10 is the highest common factor for numbers 20,30 and 50 since all numbers have four factors in common).
Therefore HCF (20, 30, 50) = gcd (20, 30, 50) = 10 is the required solution.

Example 4:-To find HCF of 18,24 & 54?

Solution :-

The factors/ divisors of 18 = 1, 2, 3, 6, 9, 18.

The factors / divisors of 24 = 1, 2, 3, 4, 6, 8,12, 24.

The factors / divisors of 54 = 1, 2, 3, 6, 9, 18, 27, 54.

Common factors / divisors = 1, 2, 3, 6.
Therefore the highest among common Factors (HCF) = 6 (where 6 is the highest common factor for numbers 18,24 and 54 since all numbers have four factors/divisors in common.)

Therefore HCF (18, 24, 54) = gcd (18, 24, 54) = 6 is the required solution.

Example 5:-To find GCD / gcd / HCF of 6 , 16 & 20 ?

Solution :- There are given three numbers 6, 16, and 20. Now we have to write the prime factors ( so do not write 1) of all three numbers individually.

6 = 2 x 3

16 = 2 x 2 x 2 x 2 x 1

20 = 2 x 2 x 5

The common prime factors divisors of above numbers we get the HCF. Hence, there are ONE pairs of 2 . So the HCF of 6, 16 and 20 will be 2.
HCF (6, 16, 20) = 2

By Division Method

We have to know about the method of finding the highest common factor using prime factorization or division. we have to keep in mind that the division method is nothing but dividing the given numbers simultaneously to get the common factors between them.

ALGORITHMS to solve problems of HCF by are such as.

First of all write the given numbers horizontally, in a sequence, by separating it with commas ( It is for tradition to separate by commas) order is not important.
Find the smallest prime number which can divide the all given number. It should exactly divide the all given numbers and we have to write in the left side.
Now we have to write the quotients carefully.
We have to repeat the process, unless and until we reach the stage, where there is no further division is possible for all.
We will get the common prime factors as the factors in the left-hand side divides all the numbers exactly.
The product of these common prime factors is the HCF of the given numbers is our required answers.

Now Let us solve problems by the above algorithms to find the HCF by division method with the help of these examples "To find HCF of numbers 12, 24, 36".

Example 6:-To find HCF of 12, 24 & 36 ?

Solution:-

For solving this problem we have to follow the above mentioned algorithms.

By Long Division Method.

Algorithms to find the HCF of given two numbers, For more than three processes can be done similarly taking two at a time in order.

For two numbers:-

Identify larger and smaller Numbers,
Divide larger number by smaller number first, such that

Larger Number/Smaller Number

Divide the divisor of the above step by the remainder left.

The divisor of the above step /Remainder

Again divide the divisor of the above step by the remainder similar to the previous step.

The divisor of the above step /Remainder

Repeat the process until the remainder is zero.
The divisor of the last step is the HCF.
Example 6:-To find HCF of 135 & 225?
Solution:- Since 225 is greater than 135, so we have to divide 225 to 135.

Therefore HCF (135, 225) = 45 is our required answer.

For three numbers:-

Repeat the above process taken two at a time in order.

________________________________________________

Least Common Multiple ( LCM )

2020-05-02T20:09:00.003+05:30

In arithmetic, mathematics, and number system, the least common multiple ( LCM ) is also known as the smallest common multiple or lowest common multiple ( LCM ) of two integers. For any two integers p and q, usually denoted by LCM(p, q), is the smallest positive integer that is divisible by both p and q.

The LCM is the Lowest Common Denominator (LCD), that can be used before fractions can be added, subtracted, or compared. The LCM of more than two integers (like for 3 or 4 or 5 any times ) is also well-defined since LCM is the smallest positive integer that is divisible by each of them.

What is the Least Common Multiple(LCM)?
It is a method to find the smallest common multiple between numbers ( any two or more). Generally according to definitions the common multiple is a number which is a multiple of two or more numbers therefore LCM is used to determine the least common factor or multiple ( Generally called as Tables) of an integer. Since the division of integers by zero is undefined or not possible, this definition has meaning only if a and b are both different from zero (so not zero ).
Example 1:-
If we consider an example, To find L.C.M of 2 and 3?
Now multiples of 2 = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 , 24...
Again Multiples of 3 can be written as = 0, 3, 6, 9, 12, 15, 18, 21, 24, 27,30, 33. 36, 39.42. 45, 48. ...
Common Multiples = 0, 6, 9, 18, 24, ....
Therefore least among common Factors ( LCM) = 6 ( where 6 is the smallest common multiple for numbers 2 and 3.)

Algorithm to find LCM of Numbers:-

By Finding the Multiples,
By Prime Factorization
By Multiple Tree Diagram.
By Relation Between LCM and HCF ( By Formula ). 1 By Finding the Multiples

The method to find the least common multiple of any given numbers is first to write down the multiples (Tables) of individual numbers and then find the first common multiple between them.

Example 2:- To find LCM of 3 & 4 ? Solution:- Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24 ... Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, …. Common Multiples = 12, 24, ....

Therefore least among common Factors ( LCM) = 12 ( where 12 is the smallest common multiple for numbers 3 and 4. ) Example 3:-To find LCM of 3 4 & 6 ( Three numbers)?

Solution:-

Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24 , 27,30, 33... Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36,40, 44, 48,…. Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54... Common Multiples = 12, 24,36 ....

Therefore least among common Factors ( LCM) = 12 ( where 12 is the smallest common multiple for numbers 3, 4 and 6.)

2 By Prime Factorization

Most Popular method to find the LCM of the given numbers is by prime factorization.

Example 4:-To find LCM of 6 16 & 20 ( Three numbers)?Solution :- There are given three numbers 6, 16, and 20. Now we have to write the prime factors of all three numbers individually.

6 = 2 x 3 == 2 x 3 x 1 ( Also written as )

16 = 2 x 2 x 2 x 2 = 2 x 2 x 2 x 2 x 1 ( Also written as )

20 = 2 x 2 x 5 = 2 x 2 x 5 x 1 ( Also written as )

On pairing the common prime factors of above numbers we get the LCM. Hence, there are four pairs of 2 and one pair of 3 and 5 each. So the LCM of 6, 16 and 20will be;

LCM (6, 16, 20) = 2 x 2 x 2 x 2 x 3 x 5 = 240 3 By Multiple Tree Diagram.

Example 5:- To find LCM of 6 & 10?

The multiple trees can be formed by using the prime factorization method this is similar to the above method.

Therefor LCM of 6 and 10 = 1 x 2 x 3 x 5 = 30.

this method is known as Tree Diagram.

Use's of LCM:-
LCM can be used before the addition, subtraction, or comparison of fractions can take place, Therefore it is basic fundamental tools for using mathematical operations in Fractions.
The LCM of more than two integers also happens to be well-defined: it is the smallest positive integer whose division can take place by each of them.

________________________________________________

NTSE ( National Talent Search Examination)

2020-05-02T11:24:00.007+05:30

National Talent Search Examination

To identify and nurture talented students across the country along with Indian students studying abroad The "National Talent Search Examination: popularly known as 'NTSE' is a unique National Level scholarship program in India for students studying in 10 class organized By the Government of India by ( NCERT as a Nodal agency)
It recognizes the young talents of India and honors them also helps talented students by providing continuous financial assistance in the form of a monthly scholarship for the entire Academic Career.
NTSE Examination conducted by the National Council of Educational Research and Training (NCERT) for the second Phase and first phase by Boards Or respective nodal agencies of states.

This examination is divided into two papers MAT ( Mental Ability Test ) and SAT (Scholastic Aptitude Test).

For the preparation of NTSE following books may be consulted/suggested (as per my opinions ) according to subject wise are as follows:-

Paper - I
MAT ( Mental Ability Test ) ( IMPORTANT PART )

Any book according to your choice few topics may be followed from R. S. Agarwal
Mental Ability for NTSE stage 1 & 2 by Disha Publication is also concise and useful.

Paper-II Scholastic Aptitude Test (SAT)

Social science:-

NCERT book of 9 and 10 with all parts.
Some latest general knowledge questions ( 1 or 2 questions may or may not be asked).
Stress on maps and facts and figures are given in the books.

Science:-

NCERT books of 9 and 10.
NCERT Exemplars.
Some reference books according to your choices like H. C. Verma for Physics ( 9 and 10 Both classes ) and Lakhmir Sing & Kaur for Biology and Chemistry.

Mathematics: -

NCERT books of 9 and 10.
NCERT Exemplars.
Solve a few advanced problems in geometry and number system by any book like R. D. SHARMA / R, S, AGARWAL ETC.

Solve last 5 Year solved papers of BOTH stages along with few modal papers from any standard reference book ( Arihant publication or TMH or etc ) for time management. Process and pattern of examination along with useful information are such as:-

How to apply:-The students studying in Class X in the country ought to be on the lookout for any advertisement in the newspapers or circulars in the school by the respective Government of their State/UT for the above said examination and act as per the requirement given in the State advertisement/circular in the beginning of schools. Any other information/query about the details of the state level examination maybe had from Liaison Officers of the State/UT agencies may be obtained generally their state birds are nodal agencies. The filled-in application by the student be submitted to the State Liaison Officer {may be changed from time to time (http://www.ncert.nic.in/programmes/talent_exam/pdf_files/LO_list_2019-20.pdf)} duly signed by the Principal of the school in which studying 10 classes before the last/due date as advertised/circulated by their respective State/UT.

For Indian Students Studying Abroad:- Studying in class X can directly apply to NCERT for second stage NTS examination under conditions prescribed by NTSE in advertisement brochure up to 31DECEMBER ( May be changed see on NCERT web regularly).

The medium of Examination:-The medium of the test shall be as announced by their respective State/UT.

Fee :- State may impose any fee for examination and/or for application form while for the second stage no fees.

Examination:- State level examination also has two parts like to second stage (but you may check before appearing in examinations ): Part-I Mental Ability Test (MAT) and Part II Scholastic Aptitude Test (SAT) for nominating the required number of candidates ( already fixed foe each state and union territories) for the second level test to be conducted by the NCERT.

Written Examination:-The written examination shall consist of two paper; Paper-I Mental Ability Test (MAT) and Paper-II Scholastic Aptitude Test (SAT). Both the papers (tests) will be administered on the same date

Mental Ability Test:-There shall be 100 multiple-choice type items, with four alternatives. Each item will carry an equal ( one) mark. Candidates are required to answer the items on a separate OMR ( Optical Mark Recognition) sheet as per instructions given in the test booklet and on the OMR sheet.

Scholastic Aptitude Test:-The scholastic Aptitude Test will consist of 100 multiple-choice problems/questions of one mark each. Each problem/question shall have four alternatives/options, out of which only one will be the correct answer. There shall be 40 problems/questions s from Science, 40 from Social Science and 20 from Mathematics, Candidates are required to answer the items on a separate OMR sheet given at the examination center. For the convenience of students, they are allowed to take away question booklets of both the tests (Mental Ability Test and Scholastic Aptitude Test) along with a copy of the OMR sheet after the examination.

Medium:- The tests will be available in the following languages: Asamiya, Bangla, English, Gujarati, Hindi, Kannada, Marathi, Malayalam, Odia, Punjabi, Tamil, Telugu and Urdu.

Venue, Date, and Time of the Test:-The admission letter shall have all the information about the venue, the roll number, the date and time of the test. Candidates can download their E-Admit cards themselves before 21 days 2 or 3 weeks of examination.

FOR the FIRST STAGE:- The state-level stage first screening examination is conducted in all State/UT’s on the first Sunday of November (You may check )except in Nagaland, Andaman and Nicobar Island, Meghalaya and Mizoram where it will be generally conducted on first Saturday of November every year (You may check )until and unless some special circumstances occur

Syllabus:-There is no prescribed syllabus for the National Talent Search Examination (NTSE). However, the standard of problems/questions shall be conforming to the level of Classes IX and X. Information about test and problems may be obtained from (separate booklet called ‘Learn about the Test’ containing sample items for both the Tests- MAT and SAT is available on the NCERT website ) along with in some standard Guides.

Centre of Examination: Generally the candidate belonging to a particular state shall be allotted the center in their state where studying 10 class. Normally there are one/two/ three / four centers ( According to the number of students and region of sate )in each state for the second stage National Level Examination. The conduct of examination at the centers shall be done under the supervision of the Centre Superintendent (CS) appointed by the NCERT.

Scholarship At present National Talent Search Examination (NTSE) Scholarship Amount are such as:-

Stage	Scholarship Amount
Higher Secondary level (for class 11 & 12)	Rs.1250/-
Graduate and Post Graduate (in all streams like B, Tec / B. E./ M. Tech/ MBBS/MS/ B. Sc./ B. A./B. Com./ BBA/ MBA/ etc.)	Rs.2000/-
For a Ph. D. degree (four years) in all disciplines.	As per UGC norms for JRF Follows.

Selection Criteria:-• Only candidates qualifying in both the papers separately will be considered for merit.

Selection of the awardees will be made on the basis of total Marks scored in MAT & SAT based on merit according to their respective categories.

There will be No Negative Mark

Qualifying marks 32% SC, ST and PWD and 40% for Others (General& OBC) ___________________________________________________________________________

Factors in Mathematics

2020-05-01T16:11:00.001+05:30

If we multiply to get the number is Factors of that numbers. Factors and multiples are different things. But they both involve the general concept of multiplication.

In another way, a factor is a number that divides into another number exactly and without leaving a remainder.
The Factors of any number are always less than or equals to given Numbers and it is different from Multiples.

Example:-Let us find the factors of 6:-

1 x 6 = 6
2 x 3 = 6

Therefore 1, 2, 3, and 6 are factors of 6.
INCOMPLETE ANSWER Why ???

Because multiplying negatives make a positive, −1, −2, −3, and −6 are also factors of 6:

(−1) × (−6) = 6
(−2) × (−3) = 6

So ALL the factors of 6 are:

1, 2, 3, and 6 AND −1, −2, −3, and −6

Example:- Let us find the factors of 8:-

1 x 8 = 8 or 8 x 1 = 8
2 x 4 = 8 or 4 x 2 = 8

So factors of 8 are 1, 2, 4, 8.
INCOMPLETE ANSWER Why ???

Since because multiplying negatives make a positive, −1, −2, −4, and −8 are also factors of 8:

(−1) × (−8) = 8
(−2) × (−4) = 8)

So ALL the factors of 8 are:

1, 2, 3, 4, and 8 AND −1, −2, −3, −4, and −8

Example:-Let us find the factors of 12:-

3 and 4 are factors of 12 since 12 = 3 x 4
Similarly 2 and 6 are also factors of 12 since 12 = 2 x 6,
And 12 = 1 x 12, Therefore 1 and 12 are factors of 12 also.
So factors of 12 are 1, 2, 4, 6, and 12.

INCOMPLETE ANSWER Why ???

because multiplying negatives make a positive, −1, −2, −3, −4, −6 and −12 are also factors of 12:

(−1) × (−12) = 12
(−2) × (−6) = 12
(−3) × (−4) = 12

So ALL the factors of 12 are:

Similarly, we can find factors of any numbers in mathematics using the above methods or Prime Factorisation Methods.

______________________________________________________

Multiples in Mathematics

2020-05-01T15:34:00.000+05:30

In mathematics, Multiples are what we get after multiplying the number by an integer like 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...(not a fraction).
The product of any quantity and an integer are known as Factors. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n like 0, 1, 2, 3, 4, ..., which is called the multiplier If a is not zero, this is equivalent to saying that b/a is an integer since the division of zero is not possible.
The above definition can be also written in many ways like:-
' On Multiplying numbers the obtained product is called a multiple of that numbers such as 2 x 3 = 6 so 6 is multiple of 2 as well as 3 also'
The Multiples of any number are always greater than or equals to given Numbers and it is different from Factors. Multiples also not to be confused with multiplication generally.

Anyone can write multiples easily with the help of Their tables in such a way ( Here i am writing few examples of multiples like 1, 2, 3, 5, 6, 10, 100, ... ) :-

Multiples of 1

1 × 0 = 0, so 0 is a multiple of 1
1 × 1 = 1, so 1 is a multiple of 1
1 × 2 = 2, so 2 is a multiple of 1
1 × 3 = 3, so 3 is a multiple of 1
1 × 4 = 4, so 4 is a multiple of 1
1 × 5 = 5, so 5 is a multiple of 1
1 × 6 = 6, so 6 is a multiple of 1
1 × 7 = 7, so 7 is a multiple of 1
1 × 8 = 8, so 8 is a multiple of 1
1 × 9 = 9, so 9 is a multiple of 1
1 × 10 = 10, so 10 is a multiple of 1
and so on

Multiples of 2

2 × 0 = 0, so 0 is a multiple of 2
2 × 1 = 2, so 2 is a multiple of 2
2 × 2 = 4, so 4 is a multiple of 2
2 × 3 = 6, so 6 is a multiple of 2
2 × 4 = 8, so 8 is a multiple of 2
2 × 5 = 10, so 10 is a multiple of 2
2 × 6 = 12, so 12 is a multiple of 2
2 × 7 = 14, so 14 is a multiple of 2
2 × 8 = 16, so 16 is a multiple of 2
2 × 9 = 18, so 18 is a multiple of 2
2 × 10 = 20, so 20 is a multiple of 2
and so on

Multiples of 3

3 × 0 = 0, so 0 is a multiple of 3
3 × 1 = 3, so 3 is a multiple of 3
3 × 2 = 6, so 6 is a multiple of 3
3 × 3 = 9, so 9 is a multiple of 3
and so on

Multiples of 5

5 × 0 = 0, so 0 is a multiple of 5
5 × 1 = 5, so 5 is a multiple of 5
5 × 2 = 10, so 10 is a multiple of 5
5 × 3 = 15, so 15 is a multiple of 5
5 × 4 = 20, so 20 is a multiple of 5
5 × 20 = 100, so 100 is a multiple of 5
5 × 30 = 150, so 150 is a multiple of 5
5 × 400 = 2000, so 2000 is a multiple of 5
and so on

Multiples of 6

it is not necessary to start from 0 or 1 or 2 or 3 etc , we can write according to our choice.

3 × 6 = 18, so 18 is a multiple of 6
4 × 6 = 24, so 24 is a multiple of 6
5 × 6 = 30, so 30 is a multiple of 6
6 × 6 = 36, so 36 is a multiple of 6
and so on

Multiples of 10

10 × 0 = 0, so 0 is a multiple of 10
10 × 1 = 10, so 10 is a multiple of 10
10 × 2 = 20, so 20 is a multiple of 10
10 × 3 = 30, so 30 is a multiple of 10
10 × 4 = 40, so 40 is a multiple of 10
10 × 5 = 50, so 50 is a multiple of 10
10 × 6 = 60, so 60 is a multiple of 10
( intentionally I left 70 , which is also multiple)
10 × 8 = 80, so 80 is a multiple of 10
10 × 9 = 90, so 90 is a multiple of 10
10 × 10 = 100, so 100 is a multiple of 10
and so on

We can write multiples of any number in mathematics up to any time according to our requirements.

Multiples of 100

100 × 0 = 0, so 0 is a multiple of 100
100 × 1 = 100, so 100 is a multiple of 100
100 × 2 = 200, so 200 is a multiple of 100
100 × 3 = 300, so 300 is a multiple of 100
100 × 4 = 400, so 400 is a multiple of 100
100 × 5 = 500, so 500 is a multiple of 100
100 × 6 = 600, so 600 is a multiple of 100
100 × 7= 700, so 700 is a multiple of 100
100 × 8 = 800, so 800 is a multiple of 100
100 × 9 = 900, so 900 is a multiple of 100
100 × 10 = 1000, so 1000 is a multiple of 100
100 × 18 = 1800, so 1800 is a multiple of 100
100 × 90 = 9000, so 9000 is a multiple of 100
100 × 100 = 10000, so 10000 is a multiple of 100
and so on we can write so many times...

Basic Properties of Multiples

(1) 0 ( Zero) is a multiple of every number.

Such as:-

1 × 0 = 0, so 0 is a multiple of 1
2 × 0 = 0, so 0 is a multiple of 2
3 × 0 = 0, so 0 is a multiple of 3
10 × 0 = 0, so 0 is a multiple of 10

(2) Every number is a multiple of 1 ( One).

Such as:-

1 × 0 = 0, so 0 is a multiple of 1
1 × 1 = 1, so 1 is a multiple of 1
1 × 2 = 2, so 2 is a multiple of 1
1 × 3 = 3, so 3 is a multiple of 1 etc

(3) Multiples of a number are always infinite (we can write to them easily).

Such as:-

100 × 0 = 0, so 0 is a multiple of 100
100 × 10 = 1000, so 1000 is a multiple of 100
100 × 20 = 2000, so 2000 is a multiple of 100
100 × 18 =1800, so 1800 is a multiple of 100
100 × 19 = 1900, so 1900 is a multiple of 100
100 × 10 = 1000, so 1000 is a multiple of 100
100 × 18 = 1800, so 1800 is a multiple of 100
100 × 90 = 9000, so 9000 is a multiple of 100
100 × 1000 = 100000, so 100000 is a multiple of 100

and so on we can write so many times...
(4) The product of any two or more factors is the multiples of each factor.
Such as:-

2 x 5 x 1 = 10,

Therefore, 10 is the multiple of both 1, 2, and 5.

3 x 1 = 3,

Therefore, 3 is the multiple of both 3 and 1.

30 = 2 x 3 x 5 x 1,

Therefore, 30 is the multiple of 1, 2, 3 and 5.

36 = 2 x 2 x 3 x 3 x 1 ,

Therefore, 36 is the multiple of 1, 2, 3, 4, 6, 9, 12, 18 and 36.

(5) Every number is a multiple of Itself.

Such as:-

1 × 1 = 1, so 1 is a multiple of 1
3 × 1 = 3, so 3 is a multiple of 3
2 × 1 = 2, so 2 is a multiple of 2
5 × 1 = 5, so 5 is a multiple of 5
10 × 1 = 10, so 10 is a multiple of 10
100 × 1 = 100, so 100 is a multiple of 100
2012 × 1 = 2012, so 2012 is a multiple of 2012
6666 × 1 = 6666, so 6666 is a multiple of 6666

Note :- In general, when a and b are both integers, and b is a multiple of a, then a is called a DIVISOR of b.We can also says that a divides b.

______________________________________________________

What is Mathematisation & Why ?

2020-05-01T11:59:00.008+05:30

What is mathematisation???????

If we consider a very uncommon example may help us work out some sort of answer to this type of question. It is an example of counting objects /articles /things. We may count our belongings the number of students, the number of fingers in our hand, and the number of anything we want to think about more precisely, more clearly and of course with counting numbers we get to quantify things, do arithmetic. In common the word 'mathematisation' to refer to the mental processes which produce mathematics from ancient times.

Mathematisation is a word that strikes a chord among most of the students, especially those who have been tensed by the void traditional or mechanical manner in which mathematics has been taught from ancient times and has been used by most of us to capture what they would consider as the heart of the mathematical enterprise – the thinking and the reasoning process of learner.

Therefore in the precious mathematization of a problem or area of study consists of applying mathematical concepts/innovative ideas to that problem or field so as to think more precisely or clearly about things.

Firstly, French-born economist and mathematician cum Nobel Laurate Gerard Debreu introduces an account of events that relates to the origins of interdisciplinary in economics. In the analysis of Market trend, Economics or commercial activity somehow allocate numbers to products/object/things for purchase/sale or in other words for transactions. The rules of the business may not always be clear but numbers ( mathematical symbols ) are somehow assigned to costs and revenues to optimize their gain or loss.

Not only in recent times but from decades spread of mathematised economic (economical) theory was helped even by its esoteric character in business and management. Since its messages cannot be deciphered by modern welfare economists like Nobel laurates Amartya Sen who do not have the proper key, their evaluation is entrusted to those who have access to the code. But acceptance of their technical expertise also implies acceptance of their values of welfare and development economics in the era of mathematisation.

The beauty of mathematics is that it is clear thinking with a logical approach. A person is tempted to add, however, that mathematics is clear thinking plus the mental tools used to clarify our thinking. So algorithms, mental tools or structures are created like number systems, Decimal system which makes sorting out things and ideas about things much more precise.

In recent times a variety of scientific fields have seen increasing mathematisation throughout the last four decades. Driven by the construction of large-scale data processing facilities in the All Engineerings and Branches of Natural sciences (including Medical science also) and the introduction of digital tools and models in the human and social sciences, contemporary research practices are increasingly dependent on mathematics.

The fields such as Artificial Intelligence in computer science, cognitive neuroscience, linguistics, Aerospace Engineering, Climate science & Weather forecast, Economics, and network science, research is being mediated through the development, integration, and application of mathematical models on the basis of mathematisation. Although, this dependency on applied mathematics (Like Statistics & Operation Research, etc) and more broadly on computation and digitalisation has so far received little attention by scientists as well as philosophers of science along with mathematisation.

In nutshell, we may hope that a serious, ingenious, study of mathematisation will help all of us in mainly two respects: it will help to connect the inner mathematical experience with its outer objective form and, in the classroom, it will help us to handle the individuality and spontaneity of students who are coming to terms with the most impersonal and rigorous subject we ask them to learn.

Mathematisation is a word that strikes a chord among most of the students, especially those who have been tensed by the void traditional or mechanical manner in which mathematics has been taught from ancient times and has been used by most of us to capture what they would consider as the heart of the mathematical enterprise – the logical thinking and the reasoning along with positive attitude, or More precisely, mathematisation is the act of putting a structure onto a structure.

____________________________________________________

Order of Operations ( BODMAS OR PEDMAS )

2020-05-01T01:09:00.001+05:30

We have a popular Phrase about Mathematics is that "Mathematics is the mother of civilization and culture. " Therefore the Order of Operations ( BODMAS OR PEDMAS ) is a well-defined collection of rules / Instructions based on Observations about which standard operation/procedure to perform first in order to evaluate given mathematical expressions in mathematics so the solution is unique ( Not Different).

"Operations" mean things like addition, subtraction, multiply, divide, squaring, Parentheses, etc. BODMAS is in mathematics an operation/procedure to solve problems. For orders of operation following rule is commonly followed.

'BODMAS' is an acronym used in mathematics and in it 'B' stands for Bracket, 'O' stands for Of, 'D' stands for Division, 'M' represents Multiplication, 'A' Represents Addition, and
'S' represents Subtractions. The 'BODMAS' is used to explain or solve the order of operation of a mathematical expression.

In some area (regions) of the world, the 'BODMAS' is also known as 'PEDMAS' which 'P' stands for stands for Parentheses, 'E' stands for Exponents, 'D' stands for Division, 'M' represents Multiplication, 'A' Represents Addition, and

'S' represents Subtractions.

Illustrations for Order of Operations ( BODMAS OR PEDMAS )

For solving any problems, Do things in Brackets First, if more than one bracket is present than the following order may be followed, -, ( ), { }, [ ].

		2 × (4 + 5)	=	2 × 9	=		18
		2 × (4 + 5)	=	8 + 5	=	13	(wrong) WHY ????

Since 2 is not multiplied by both terms in brackets so the second term is 5 written instead of 10 so the answer is wrong.

Exponents(Powers, Roots) before Multiply, Divide, Add or Subtract

		5 × 2²	=	5 × 4	=	20
		5 × 2²	=	10²	=	100	(wrong)

WHY ????

Multiply or Divide before you Add or Subtract on solving

		12 + 5 × 3	=	12 + 15	=	27
		12 + 5 × 3	=	17 × 3	=	51	(wrong)

WHY ????

Otherwise, just go left to right

		45 ÷ 5 × 3	=	9 × 3	=		27
		45 ÷ 5 × 3	=	45 ÷ 15	=	3	(wrong)

WHY ????

For example, Thus, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20.

if more than one bracket is present than the following order may be followed,-, ( ), { }, [ ].

The technique for learning or Remember is such as, How Do I Remember It All ... ? BODMAS ! It's so simple......

B	Brackets first
O	Orders (i.e. Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

How Do I Remember It All ... ? PEDMAS ! It's so simple.....

P	Parenthesis first
E	Exponents(i.e. Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

-----------------------------------------------------------------------------------------------------------

Division in Mathematics

2020-04-29T19:59:00.002+05:30

The beauty of Mathematics depends on basic four arithmetic Operations along with some functional operation:-

Addition ( + )
Subtraction ( - )
Multiplication ( x )
Division (÷) So, we shall discuss the fourth operation Division.

DIVISION

Generally, Division symbol (÷)having a horizontal line a dot below and another at above , OR " / " ( Read as a fraction Bar or Solidus) is often used to indicate mathematical division from ancient times. The result is expressed with equals " = " symbols.

it is a basic arithmetic operation that represents the operation of repeated subtraction of objects from a given collection. The result of the division is also called a Repeated Subtraction.

The division is considered as Reciprocal of multiplication and can be learned as follows:-

4  ÷ 1 = 4 means how many times one can be subtracted from 4 to get Remainder 0.
4
-1 ( Once subtracted )
------------
3
-1   ( Twice subtracted )
-------------
2
-1   ( Thrice subtracted )
------------
1
-1    ( Fourth times subtracted )
------------
0 ( Remainder)
-------------------
  Since 1 is repeatedly subtracted FOUR times so answer of 4  ÷ 1 is 4.

The simplest way of viewing division is in terms of QUOTIENT: from the quotient perspective,

15 / 5

means the number of 5s that must be added to get 15. In terms of partition,

15 / 5

means the size of each of 5 parts into which a set of size 15 is divided. For example, 15 apples divide into five groups of THREE apples, meaning that fifteen divided by five is equal to Three. This is denoted as

15/ 5 = 3

, or

15 / 5 = 3

.
The number / Polynomial which divided is known as Dividend and the obtained result is called Quotient, The number from divided is called as Divisor. In the example, 15 is the dividend, 5 is the divisor, and 3 is the quotient, and Remainder is zero.

-5 ( Once subtracted )

------------

-5 ( Twice subtracted )

-------------

-5 ( Thrice subtracted )

------------

0 ( Remainder) Therefore 15 /5 = 3.

BY DIVIDING NUMBERS ZERO ( 0)

1 ÷ 0 = ???? means how many times ZERO can be subtracted from 4 to get Remainder 0.

-0 ( Once subtracted )

------------

1 ( Remainder)

-0 ( Twice subtracted )

-------------

1 ( Remainder)

-0 ( Thrice subtracted )

------------

1 ( Remainder)

-0 ( Fourth times subtracted )

------------

1 ( Remainder)

-0

---------

1 ( Remainder)

IN ALL CASES REMAINDER IS SAME TO THE DIVIDEND 1 SO IF WE USE THIS PROCESS UNCOUNTED NUMBER OF TIMES THEN Remainder IS ALWAYS 1 ( DIVIDEND ), SO DIVISION OF ZERO IS NOT POSSIBLE OR RESTRICTED.

1 ÷ 0 = ???? ANSWER - NOT POSSIBLE.

By Division of zero to any positive numbers, is not possible.

BY DIVISION OF ONE (1)

1 ÷ 1 = 1

2÷1 = 2 - (1 + 1 ) = 2 ( Twice subtracted)

3 ÷1 = 3 - (1 + 1 + 1 ) = 3 ( Thrice subtracted)

4 ÷ 1= 4 - (1 + 1 + 1 + 1 ) = 4

5 ÷1= 5 - (1 + 1 + 1 + 1 + 1 ) = 5

6 ÷ 1= 6 - ( 1 + 1 + 1+ 1 + 1 + 1 ) = 6

7 ÷1= 7 - (1 + 1 + 1+ 1 + 1 + 1 + 1 )= 7

8 ÷1 = 8 - (1 + 1 + 1+ 1 + 1 + 1 +1 + 1 ) = 8
9 ÷ 1 = 9 - (1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1) = 9
10 ÷ 1= 10 - (1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1 +1 ) = 10 ( Ten times subtracted)

Any number (x ) ÷ 1 = any number ( x)

...

By multiplication of ONE to any numbers, results are always the same number so ONE ( 1 ) Iis known as an identity for the division.

BY MULTIPLYING DIFFERENT NUMBERS

8 ÷ 2 = 4

12 ÷ 2 = 6

18 ÷ 3 = 6

100 ÷ 10 = 10

21 ÷ 7 = 3

similarly, more results can be verified.

--------------------

Multiplication in Mathematics

2020-04-29T17:20:00.005+05:30

The beauty of Mathematics depends on basic four arithmetic Operations along with some functional operation:-

Addition ( + )
Subtraction ( - )
Multiplication ( x )
Division (÷) so we shall discuss the third operation multiplication.

MULTIPLICATION

Generally, "X " ( Popularly Read as a cross) OR 'Dot ' . ' symbol is used from ancient times. The subtraction is written using anyone from X OR. OR * between the terms; that is, in symbolically. The result is expressed with equals " = " symbols.

it is a basic arithmetic operation that represents the operation of repeated addition of objects from a given collection. The result of the multiplication is also called a Repeated addition.

multiplication can be learned as follows:-

The multiplication of numbers is another way of REPEATED ADDITION; that is, the multiplication of two numbers is equivalent to adding as many times one of them, the multiplicand, as the value of the other one, is known as the multiplier.
The multiplier can be written first traditionally and multiplicand second These are called factors.

For example, 2 multiplied by 3 (often written as 3 X 2 OR 2 X 3 and spoken as "3 times 2" OR

" 2 times 3") can be calculated by adding 3 copies of 2 together:

3 + 3 = Think about result ( 2 x 3 = 6 )

OR

adding 2 copies of 3 together:

2 X 3 = 2 + 2 + 2 = 6

For example, 4 multiplied by 3 (often written as

3\times 4

and spoken as "3 times 4") can be calculated by adding 3 copies of 4 together:

3\times 4=4+4+4=12

Therefore 3 and 4 are known as factors of 12.

If in ANOTHER way that we have four fruits and we have got four more then Total fruits are such as

4 x 2 = 4 + 4 = 8
OR
2 X 4 = 2 + 2 + 2 + 2 = 8

BOTH results are the same it is an important property of multiplication.

BY MULTIPLYING SAME NUMBERS ZERO ( 0)

Zero MULTIPLIED by ZERO always equals to ZERO
0 X 0 = 0
0 X 1 = 0

0 X 2= 0 + 0 = 0

0 X 4 = .... Think and Write as addition as four zeroes together

0 X 5= 0 + 0 + 0 + 0 + 0 = 0

0 X 6 = .... Think and Write as addition as six zeroes together

...

If we multiply zero to any number result is always Zero. ( Important Fact of Mathematics)

BY MULTIPLYING SAME NUMBERS ONE (1)

1 X 1 = 1

1X 2= 1 + 1 = 2

1 X 3 = 1 + 1 + 1 = 3

1X 4 = 1 + 1 + 1 + 1 = 4

1 X 5= 1 + 1 + 1 + 1 + 1 = 5

1 X 6 = 1 + 1 + 1+ 1 + 1 + 1 = 6

1 X 7 = 1 + 1 + 1+ 1 + 1 + 1 + 1 = 7

1 X 8 = 1 + 1 + 1+ 1 + 1 + 1 +1 + 1 = 8
1 X 9 = 1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1 = 9
1 X 10= 1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1 +1 = 10

...

By multiplication of ONE to any numbers, results are always the same number so ONE ( 1 ) Iis known as multiplicative identity.

BY MULTIPLYING DIFFERENT NUMBERS

7 X 3 = 7 + 7 + 7 = 21 (" SEVEN multiplied by THREE equals TWENTY ONE")

THIS CAN BE ALSO WRITTEN AS

7 X 3 ___
21__

--------------------

S.N.	Symbol	Name of Symbol	Definition	Illustration
1	P(A)	Probability Function	Probability of any event A	P(A) = 0.2 or any number between 0 to 1 0 ≤ P (A ≤ 1
2	P(A) = 0	Complete Uncertainty	The event cannot happen	On tossing a coin we shall get prime number.
3	P(A) = 1	Complete certainty	The event will happen definitely	On tossing a die we shall get a number less than 7.
4	P (A ∩ B)	Probability of intersection of events	Probability of Event A and B	P (A ∩ B) = 0.5
5	P (A ⋃ B)	Probability of union of events	Probability of Event A or B	P (A⋃B) = 0.5
6	P (A / B)	Conditional Probability Function	Probability of Event A when B already occurred.	P (A / B) = 0.2 or any number between 0 to 1
7	P (B / A)	Conditional Probability Function	Probability of Event B when A already occurred.	P (B / A) = 0.6 or any number between 0 to 1
8	μ	Mean or population mean	Mean of population values	Mue = 10
9	f(t) or f(x)	pdf	Probability Density function	P (a ≤ t ≤ b) = ∫ f (t) dt, where t is any variable can be replaced by x or y etc.
10	F (t)	cdf	Cumulative distribution function	F(t) = P (T≤ t)
11	E (X)	Expectation Value	Expected value of any random variable x	E (X) = 10
12	E (X / Y)	Conditional Expectation Value	Conditional Expected value of any random variable X given Y	E (X / Y) = 3
13	E (Y/ X)	Conditional Expectation Value	Conditional Expected value of any random variable Y given X	E (Y / X) = 2
14	Sigma 2 (σ 2)	Var(X)	Variance of random variable x	Var(x) = 16
15	∑	Sigma (Capital)	Summation	Sum of all values in the range.
16	∑∑	Sigma (Capital) Sigma (Capital)	Double Summation	Sum of all values in the range.
17	Range	Range	Difference between maximum and minimum value	For series 2, 4, 6, 10, 10, 10, 20,40 Range= 40-2 = 38
18	MR	Mid-range	½(Maximum + Minimum )	For series 2, 4, 6, 10, 10, 10, 20,40 MR= ½(2 + 40) =21
19	Mo	Mode	Value that frequently occurs in data / population / series. Or Maximum number of repetitions	For series 2, 4, 6, 10, 10, 10, 20,40 Mode = 10
20	X bar	Sample Mean / average / Arithmetic Mean	Sum of all observations / No. of observations	For series 2, 4, 6, 10, 10, 10, 20,40 Mean = 1/8 (2+4+6+10+10 +10+20+40)
21	Std(X) or σ(x)	Standard deviation	Standard deviation of random variable x = square root of variance	For any observation, if Var(x) = 16 then Std(X) = 4
22	Median	Median	Mid value of x	For series 2, 4, 6, 10, 10, 10, 20,40 Median = 10
23	cov (x, y)	Covariance	Covariance of any random variable x and y	cov(x, y) = 4
24	corr (x, y)	Correlation	Correlation of any random variable x and y	Corr (x, y) = 4.5
25	ρ(x, y)	Correlation	Correlation of any random variable x and y	ρ (x, y) = 4.5
26	Q	Quartile	% population are below this average on the basis of Q1, Q2, or Q3.
27	Q1	First Quartile / lower	25 % population are below this average or first quartile
28	Q2	Second Quartile / Median	50 % population are below this average
29	Q3	Third Quartile / Upper	25 % population are above this average or 75 % population are below this average

S.N.	Symbol	Name of Symbol	Definition	Illustration
1	{ }	Set	A well-defined collection of objects / elements are known as a set.	A = {1,17,19,24}, B = {3, 7,19,24}, OR C= Set of word MATHEMATICS = {M, A, T, H, E, I, C, S}
2	A = B	equality	both sets have the same members (orders may be different)	A = {1,2,3,4} and B = {1,2,3,4}
3	A ∪ B	union	Those objects which belong to set A or set B.	If A = {1,17,19,24}, and B = {3, 7,19,24}, then A ∪ B = {1, 3, 7, 17, 19, 24}
4	A ∩ B	intersection	Those objects that belong to set A and set B both.	If A = {1,17,19,24}, and B = {3, 7,19,24}, then A ∩ B = {19,24}
5	A ⊆ B	subset	A is a subset of B or set A is included in set B.	{1,2,3,4} ⊆ {1,2,3,4}
6	P ⊂ Q	proper subset or a strict subset	“P is a subset of Q, but Pis not equal to Q”.	{1,2, 4} ⊂ {1,2,3,4}
7	P ⊄ Q	not a subset	"The first (P) set is not a subset of another or second (Q ) set "	{1,2, 4} ⊄ {1,22, 44}
8	P ⊇ Q	superset	P is a superset of Q or set P includes set Q	{1,14,28} ⊇ {1,14}
9	P ⊅ Q	not a superset	set P is not a superset of set Q	{19,14,208} ⊅ {19,23}
10		power set	All subsets of any set like A
11	P (A)	power set	All subsets of any set like A
12		complement	All the objects that do not belong to given set A
13		complement	All the objects that do not belong to given set A
14	A – B	Set A-minus Set B Or relative complement	Objects that belong to set A and but not to set B	If A = {1,2,3,4} and B = {1, 2, 5, 7} then A-B = {3, 4}
15	A / B	Set A minus Set B or relative complement	Objects that belong to set A and but not to set B	If A = {1,2,3,4} and B = {1, 2, 5, 7} then A-B = {3, 4}
16	B – A	Set B minus Set A Or relative complement	Objects that belong to set B and but not to set A	If A = {1,2,3,4} and B = {1, 2, 5, 7} then B- A = {5, 7}
17	B / A	Set B minus Set A or relative complement	Objects that belong to set B and but not to set A	If A = {1,2,3,4} and B = {1, 2, 5, 7} then B- A = {5, 7}
18	a ∈ A	belongs to OR elements of	Element is member of a given set	If A = {1,2,3,4} then 1 ∈ A, 2 ∈ A, 3 ∈ A, 4 ∈ A
19	a ∉ A	Not belongs to OR not element of	Element is NOT a member of given set	If A = {1,2,3,4} then 5 ∉ A
20	A ∆ B	Symmetric difference of sets	Set of all those elements that belong to A or B but not to their intersection. (A ∪ B) – (A ∩ B) OR (A - B) ∪ (B - A)	If A = {1,2,3,4} and B = {1, 2, 5, 7} then A ∆ B = {3,4,5,7}
21	A ⊖ B	Symmetric difference of sets	Set of all those elements that belong to A or B but not to their intersection. (A ∪ B) – (A ∩ B) OR (A - B) ∪ (B - A)	If A = {1,2,3,4} and B = {1, 2, 5, 7} then A ⊖ B = {3,4,5,7}
22	(x, y)	Ordered Pair of x and y	Set of two elements whose order is fixed.
23	A × B	Cartesian Product	it is a set of all possible ordered pairs from A and B
24	B × A	Cartesian Product	Set of all ordered pairs from B and A
25	n (A)	The cardinality of set A	The total number of elements of the set A	A = {1, 2,3,9,14}, n(A) =5
26	\|A\|	The cardinality of set A	The total number of elements of the set A	A = {1, 2,3,9,14}, \|A\|=5
27	(Phi)	Empty Set	The set having No elements.
28	(Zeta)	Universal Set	Superset of all sets

Roman Numerals	Value
I	1
V	5
X	10
L	50
C	100
D	500
M	1000

Mathematics

Mathematical Symbols - Calculus and Analysis

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols

These are very important Basic Tools frequently used by us not only in Calculus and Analysis but also in Physics, Chemistry, Economics as well as in other branches of applied mathematical sciences, humanities, and engineering.

Table of important Mathematical Symbol of Calculus and Analysis

Mathematical Symbols - Logical and General Reasoning

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematical symbols - Logical and Reasoning

These are very important Basic Tools frequently used by us not only in computer, Set theory, statistics and Probability but also in Operation Research, Management as well as in other branches of pure & applied mathematical sciences and engineering.

Table of important Mathematical Symbol of Logical and Reasoning

Mathematical Symbols - Roman Symbols (One to Ten Thousand)

Mathematical Roman Symbols are very important to learn the number system in mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Roman Symbols

Rules for writing Roman symbols

Table of important Mathematical Symbol of Roman Symbols

Table of important Mathematical Symbol - Roman Symbols Greater than 3000

⊽CCC

We can use the above rules to write some numerals/numbers in Roman symbols with the help of following illustrations and vice versa are such as:

Illustrations 1:- Write the Roman Symbols for the following numerals/numbers, with justifications?

Illustrations 2:- Write the Roman Symbols for the following numerals/numbers, with justifications?

Illustrations 3:- Write the Roman Symbols for the following numerals/numbers, with justifications?

Illustrations 4:- Write the Roman Symbols for the following numerals/numbers, with justifications?

Mathematical Symbols - Statistics and Probability

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Generally, Mathematical Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Statistics and Probability

Table of important Mathematical Symbol of Probability and Statistics

On having knowledge of the above Mathematical symbols of probability and statistics, we can easily enhance our subject ideas.

___________________________________________________________

Mathematical Symbols - Set Theory

Mathematical Symbols are very important to learn mathematics, on using symbols we can easily understand the concepts of topics.

Mathematical Symbols can be categorized on their importance in the following ways:

Mathematical symbols - Set Theory

Mathematics Formula Algebra - Exponents And Powers

Mathematics Formula Algebra

Exponents and Powers

Laws of Exponents and Powers

Laws of Exponents - 1

On Multiplying Power with the Same Base

Laws of exponent - 2On Dividing Power with the Same Base

Laws of Exponents - 3 On Multiplying Power with the Same Exponents

Laws of exponent - 4On Dividing power with the Same Exponents

Laws of exponent - 5On taking Numbers with Exponents ZERO

Laws of exponent - 6On taking Power of Power

Mathematics Symbols - Trigonometry

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Trigonometric

These are very important Basic Tools frequently used by us in Geometry as well as in other branches of mathematics, physics, sciences, and Engineering.

We know that Trigonometry is a branch of mathematics that deals with sides and angles of triangles.

Trigonometry is frequently used in Physics and other sciences so we can say without trigonometry imagination of physics and engineering sciences is not possible.

Note:- Degree, Gradians and Radians are units of measurements of angles in various systems.On having knowledge of the above Mathematics symbols - Trigonometric we can easily enhance our subject ideas also using standard relations and Trigonometric Identities.

Mathematics Formula Algebra - Identities

Mathematics Formula Algebra

Binomial Algebraic Identity

Trinomial(Polynomial) Algebraic Identity

Similarly, we can use the above-mentioned identities to factorize algebraic polynomials and evaluate values like examples 3 and 4.

From the above illustration 8, we can conclude that identity second is obtained by first with substitution of b by (-b). Similar results can be obtained for five and six, seven & eight.

Aims for mathematics education at the elementary stage & Secondary stage

At Elementary Stage

At Secondary Stage

The general aim of Mathematics Education

Mathematics symbols - Geometric

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Geometric

Mathematics symbols - Algebraic

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics symbols - Algebraic

Mathematics symbols - Basic

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Mathematics Symbols - Number

Mathematics Symbols are very important to learn mathematics, on using symbols we can easily understand.

Mathematics Symbols can be categorized on their importance in the following ways:

Laws of exponent - 2
On Dividing Power with the Same Base

Laws of Exponents - 3
On Multiplying Power with the Same Exponents

Laws of exponent - 4
On Dividing power with the Same Exponents

Laws of exponent - 5
On taking Numbers with Exponents ZERO

Laws of exponent - 6
On taking Power of Power

These are very important Basic Tools frequently used by us in Geometry as well as in other branches of mathematics, physics,
sciences, and Engineering.

Note:- Degree, Gradians and Radians are units of measurements of angles in various systems.
On having knowledge of the above Mathematics symbols - Trigonometric we can easily enhance our subject ideas also using standard relations and Trigonometric Identities.

(A ) BASIC TYPES OF NUMBERS

Natural Numbers (N)

Whole Numbers (W)

Integers Numbers (I or Z)

Rational Numbers (Q)

Irrational Numbers

Real Numbers (R)

Imaginary Numbers, Complex Number, etc

(B) OTHER STANDARD TYPES OF NUMBERS

Whole Numbers (W)

OR
Complete Numbers

Highest Common Factor (HCF)
or
Greatest Common Divisor ( GCD)