Maths Tricks and Tips

Fibonacci Sequence

2014-09-27T22:57:00.004+05:30

The Fibonacci Series is 0,1,1,2,3,5,8,13,21,34,55,......

This Series is generated by adding two number before it :

for example :
5 is found by adding two numbers before it that is 2+3.

Similarly 8 is found by adding two numbers before it that is 5+3.

and so on..

The Fibonacci Sequence can be written as a rule such as
X_n=X_n-1 + X_n-2
For example : X_n-1=21 and X_n-2=13 Therefore X_n becomes 34.

When we make a square as shown below we get a Spiral shape :

Golden ratio : If we take two successive number Fibonacci number we get a known as Golden Ration which has the approximate value as 1.61803399....
Bigger the pair of Fibonacci Numbers, the closer the approximation.

The symbol for Golden ration is φ (phi).

Fibonacci in Nature:

The Fibonacci appears in the smallest, to the largest objects in nature. It is a way for information to flow in a very efficient manner.
some of the images below shows the Fibonacci sequence in Nature

Fibonacci sequence in Cancer Cell Division

Fibonacci sequence in Galaxy

(a+b)^2=a^2+2ab+b^2 But why?

2013-04-23T18:09:00.001+05:30

So Lets prove it :
Consider a line

Take any arbitrary point

So now we can say that length of line is a+b
Now we have to square a+b
that will be

lets add all the squares and rectangles

This will be a²

Now it is a²+b²

Here there are two rectangle in the above diagram

As we know that Area of rectangle is Length X Width

So it will be a x b

Now lets add every square and rectangle

a² + b² + a x b + a x bi.e., a² + b² + 2ab

Thus

(a+b)²=a² + 2ab+b²

Trick To Convert Fahrenheit and Celsius

2012-09-30T11:21:00.004+05:30

Formula :-

To Convert Celsius to Fahrenheit

°C x 9/5 + 32 = °F OR °F = (1.8 x °C) + 32

To Convert Fahrenheit to Celsius

(°F - 32) x 5/9 = °C OR °C = (°F - 32) / 1.8

Trick To Convert Celsius to Fahrenheit :

You will not get the accurate value of Fahrenheit using this trick.

Step 1 : Double the celsius

Step 2 : Add 30

For example

lets take 15 °C

Step 1 : Double the celsius 15 x 2 = 30

Step 2 : Add 30 so 30 + 30 = 60°F

So it is quite near to the actual answer 59°F

Now to convert Fahrenheit to Celsius

Step 1 : Subtract 30

Step 2 : Divide by 2

Program To Convert Celsius to Fahrenheit and Fahrenheit to Celsius

Fahrenheit to Celsius :

Degrees and Radians Calculator

2012-09-02T23:20:00.000+05:30

Converting from Degrees to Radians

Degrees can be converted to Radians with the help of this formula :

Radians = (Degrees x π) / 180

Converting from Radians to Degrees

Radians can be converted to Degrees with the help of this formula :

Degrees = (Radians x 180) / π

Symbol : 1 Degree = 1°
1 Radian = 1^c

1 Radian ≈ 57.2958°

1 Degree ≈ 0.017460317^c

Program to calculate Degrees to Radians and Radians to Degrees

Converting from Degrees To Radians

Converting from Radians to Degrees

Note : I have considered the value of π as 22/7 in the above program..

Factorial of a Number

2012-08-31T23:06:00.000+05:30

Definition : It just means to multiply a series of descending natural numbers. (symbol :- ! )

For Example : 1) 3! = 3 x 2 x 1 = 6

2) 5! = 5 x 4 x 3 x 2 x 1 = 120

Note : 1! is 1 and 0! is also 1

Factorials are use in many fields of Mathematics but particularly in Permutation and Combination.

Note : We can have factorial for negative numbers as well as decimal numbers but we need to get into a subject called the "Gamma Function"

Program to Calculate Factorial of a Number

Note : The above program may show result for decimal number which is incorrect

Prime Numbers

2012-08-31T00:02:00.001+05:30

Definition :- A Natural number greater than 1 which is divisible by one and itself is known as Prime Number.

Example:- 5 can only be divided evenly by 1 or 5, so it is a prime number.

But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number.

Fact: - The Number 2 is the only even number that is prime.

Here is The Simple Program To Calculate Whether A Given Number is Prime or Not

Real Numbers

2012-08-10T00:17:00.000+05:30

INTRODUCTION TO Real Numbers:-

The type of numbers, we normally use, such as

1, 15.82,-0.01, 314, etc.

Positive or negative, large or small, whole numbers or decimal numbers

are all Real numbers.

They are called "Real numbers”. because they are not

Imaginary numbers.

THE ALGEBRIC AND ORDER PROPERTIES OF |R:-

INTRODUCTION:-

We will give a short list of basic properties of addition and

Multiplication from which all other algebraic properties

which we can be derived as theorem.

We will use the conventional notation of(a+b)and

(a.b) when discussing the properties of addition and

Multiplication.

ALGEBRAIC PROPERTIES OF |R:-

On the set of R the real number there are two binary

Operatiions,denoted

by(+)and(.) called addition and multiplication, respectively.

These operations satisfy the following properties.

AXIOMS ABOUT ADDITION OF REAL NUMBERS:-

Ø a+b=b+a for all a,b in |R.(commutative properties of addition).

Ø 2- (a+b)+c = a+(b+c) for all a,b,c in |R (associative property of addition).

Ø 3- there exist an element 0 in |R s.t 0+a=a and a+0= a for all a in |R.(existence a zero element)

Ø 4-for each a belongs |R there exist an element -a in |R

s.t.a+(-a)=0 and (-a)+a=0 (existence of negative element).

MULTIPLICATION:-

Ø a.b = b.a for all a,b belongs |R (commutative property of multiplication).

Ø 2- (a.b).c = a.(b.c) for all a,b,c in |R(associative property in multiplication).

Ø 3- there exists an element 1in |R distinct from 0 s.t. 1.a = a and a.1 = a for all belongs to |R.(existence of a unit element).

Ø 4- for each a is not = in |R there exists an element 1/a in |R s.t.

a.(1/a)=1 and (1/a).a =1.(existence of reciprocals)

Ø 5- a.(b+c)=(a.b)+(a.c) and (b+c).a=(b.a)+(c.a) for all a,b,c belongs to |R

(Distributive property of multiplication over addition).

ROTATONAL AND IRRATIONAL NUMBERS:-

We regard the set |N of natural numbers as a subset of |R, by

Identifying the natural number n€|N with the n-fold sum of the unit

Element 1€|R. Similarly, we identifying “0” belongs |Z, “0” element of

“0”belongs to |R, and we identifying the n-fold of “-1” with the integer –n.

Thus, we consider |N and |Z to be subset of |R.

Element of |R that can be written in the form of b/a, where a,b belongs t o|Z and a is not equal to”0” are called rational numbers.

Notation:- Q

One consequence is that element of |R that are not in Q became known as irrational numbers, its mean that they are not rational integers.

THE ORDER PROPERTIES OF |R:-

There is nonempty subset real positive of |R called the set of positive real numbers, that satisfies the following properties.

Ø If a,b belongs to real positive numbers ,then a=b belongs to real positive numbers.

Ø If a,b belongs to real positive numbers, then a.b belongs to real positive numbers.

Ø If a belongs to real number, then exactly on of the following holds.

· a belongs to real positive numbers

· a is equal to 0

· -a is belongs to real positive numbers.

and is usually called the Trichotomy properties.

Ø If a belongs to real positive numbers, a>0 & say that a is positive (or strictly positive) real number.

Ø If a€|R+{0},a≥0 & say that a is nonnegative real number.

Ø If -a€|R+U{0}, a≤0 & say that a is no positive real number.

Note:-

The notation of inequality between two real numbers will how, be defined in term of then set |R+ of positive elements.

Definition:-

Ø Let a,b be element of |R

If a-b €|R+, then we write a<b or b<a.

If a-b€|R+ U{0},then we write a≥b or b≤a.

ABSOLUTE VALUE AND THE REAL LINE:-

From the trichotomy property, if a€|R, then exactly one of the following holds.

a€|P,a=0,-a=|P.

Assured that if a€|R and a ≠0, then exactly one of the number a&-a is positive.

The absolute value of a≠o is defined to be the positive one of these two numbers. The absolute value of 0 is defined to be 0.

DEFINITION:-

The absolute value of a real number a, denoted by |a| is defined by

|a|= {a, if a>0,

{0, if a=0,

{-a,if a<0.

For example,|5|=5 and |-8|=8.

THE REAL LINE:-

DEFINITION:-

Let a€|R and e>0.Then the e- neighborhood of a is the set Ve(a):={x€|R |x-a|<e}.

THE PROPERTY OF SUPREMUM:

DEFINITION:-

S be a non empty subset of r,m€|R is called a least upper bound of s if

Ø M is an upper bound of s, i.e. x≤ m¥ x€ s.

Ø Any upper bound, k€ s is > or = m ,i.e.m≤k

Ø Example:-s={1,2,3}

Therefore 3 is supremum.

GREATEST LOWER BOND (glb) (INFIMUM):-

“S” be a nonempty subset of |R, k€|R is called infimum or glb of “s” if.

Ø “k” lower bound of “s” i.e. k≤x¥x€s.

Ø If “l” Is lower bound of “s” then l≤k.

For example.

S={1,2,3}

Therefore 1 is infimum of greatest lower bound.

COMPLETENESS AXIOM:

If A is nonempty has infimum then it has a uniq infimum.

ARCHEMEDIAN’S PRINCIPLE:

For any the real number” x”, there exist a natural number “n”

Such that x<n.

There is infinity many irrationals between two distinct real numbers.

DENSENESS PROPERTY:-

Between two distinct real number there exist many irrationals.

- Submitted by Aman Singh

Circumference of a Circle

2012-07-12T20:06:00.000+05:30

Here is the formula to calculate circumference of a circle

2 $π r OR$ $π d$

$Where r is the Radius of a Circle and d is the Diameter of a Circle...$

$For example : Let the Radius of a Circle be 5cm$

$So, Circumference of a circle = 2 x 3.141592 x 5$

$i.e., 31.41592 cm$

$Here is a Program to Calculate Circumference of a Circle...$

Enter the Radius of a Circle

Circumference of a Circle is

Note: Here π = 3.141592

Perimeter of a Triangle

2012-07-11T20:27:00.002+05:30

Here is the formula to find the Perimeter of a Triangle....

Perimeter of a Triangle = a + b + c

where a, b and c are the side lengths of the triangle.

For example : Suppose the length of the sides are 5cm, 6cm and 10cm

Perimeter of the Triangle = 5 + 6 + 10

i.e., 21 cm

Here is the Program to Calculate the Perimeter of a Triangle..

Perimeter of a Rectangle

2012-07-11T19:21:00.001+05:30

Here is the formula to find the Perimeter of a Rectangle...

Perimeter of a Rectangle = 2a + 2b

where a and b are the breadth and height respectively...

For example : Suppose the breadth of a rectangle is 5 cm.

and height of a rectangle is 6cm.

so the Perimeter of a Rectangle = 2x5 + 2x6

i.e., 22cm.

Here is the program to calculate the Perimeter of a Rectangle...

Perimeter of a square

2012-07-11T18:33:00.001+05:30

Here is the formula to find the Perimeter of a Square...

Perimeter of a Square = 4 x (Length of a Side)

For example : Suppose the length of a side is 5 cm.

so the perimeter of a square = 4 x 5.

i.e., 20 cm.

Here is the program to calculate the Perimeter of a Square..

Calculate Area of Square, Area of Parallelogram, Area of Trapezoid, Area of Circle and Area of Ellipse

2012-07-08T17:14:00.005+05:30

Here are some formulas to calculate Area of Square, Area of Parallelogram, Area of Trapezoid, Area of Circle and Area of Ellipse...

Note :- a = Side of a square.
             b = Breadth.
             h = Height.
             r = Radius

Formulas

Square = a²

Parallelogram = b*h

Trapezoid = 1/2* (b₁ + b₂)*h

Circle = π r²

Ellipse = π r₁ r₂

Click on the Buttons to find Area of Square, Area of Parallelogram, Area of Trapezoid, Area of Circle and Area of Ellipse.

1) Click on the button to find the

2) Click on the button to find the

3) Click on the button to find the

4) Click on the button to find the

5) Click on the button to find the

Note : I have considered the value of π upto 3.14159265 in the above programs.

Kaprekar Number

2012-07-04T12:07:00.000+05:30

D.R.Kaprekar

Kaprekar Number 6174

Rules :

Select any four digit number, but do not select the numbers like 1111, 2222, 3333,...

Arrange the digits in decreasing order.

Now arrange the digits in ascending order.

Subtract the smaller number from the larger number.

Repeat the steps until you get 6174 in repetition.

For Example :- Lets take a number 1234

Step 1:- Arranging the digits in descending order we get 4321.

Step 2:- Arranging the digits in ascending order we get 1234.

Step 3:- Subtract 1234 from 4321 = 4321 - 1234 = 3087.

Step 4:- Arranging the digits in descending order we get 8730.

Step 5:- Arranging the digits in ascending order we get 0378.

Step 6:- Subtract 0378 from 8730 = 8730 - 0378 = 8352.

Step 7:- Arranging the digits in descending order we get 8532.

Step 8:- Arranging the digits in ascending order we get 2358.

Step 9:- Subtract 2358 from 8532 = 8532 - 2358 =6174

Step 10:- Arranging the digits in descending order we get 7641.

Step 11:- Arranging the digits in ascending order we get 1467.

Step 12:- Subtract 1467 from 7641 = 7641 - 1467 = 6174.

You will find that all four digit number using kaprekar rule will reach the number 6174

Birthday Date Calculation

2012-07-03T22:59:00.000+05:30

Birthday Date Calculation Trick

Step 1: Add 18 to your birth month.

Step 2: Multiply by 25.

Step 3: Subtract 333.

Step 4: Multiply by 8.

Step 5: Subtract 554.

Step 6: Divide by 2.

Step 7: Add your birth date.

Step 8: Multiply by 5.

Step 9: Add 692.

Step 10: Multiply by 20.

Step 11: Add only the last two digits of your birth year.

Step 12: Subtract 32940 to get your Birthday.

Angel Number

2012-07-02T20:09:00.015+05:30

Angel Number 421

Step 1 :- Select any whole number

Step 2 :- Suppose a whole number is even then divide it by 2.
               If it is odd then multiply by 3 and add 1.

Step 3 :- Repeat the step 2 till you get the loop value 4,2,1 in repetition.

For Example :- Let us consider a whole number 15.

As 15 is an odd number so (15 × 3) + 1 = 46.

Now 46 is an even number so (46 / 2) = 23.

Again 23 is an odd number so (23 × 3) + 1 = 70.

70 is an even number so (70 / 2) = 35.

35 is an odd number so (35 × 3) + 1 = 106.

106 is an even number so (106 / 2) = 53.

53 is an odd number so (53 × 3) + 1 = 160.

160 is an even number so (160 / 2) = 80.

80 is an even number so ( 80 / 2 ) =40.

40 is an even number so ( 40 / 2 ) = 20.

20 is an even number so ( 20 / 2 ) = 10.

10 is an even number so ( 10 / 2 ) = 5.

5 is an odd number so ( 5 × 3 ) + 1 = 16.

16 is an even number so (16 / 2) = 8.

8 is an even number so ( 8 / 2) = 4.

4 is an even number so ( 4 / 2 ) = 2.

2 is an even number so ( 2 / 2 ) = 1.

1 is an odd number so (1 × 3 ) + 1 = 4.

4 is an even number so ( 4 / 2 ) = 2.

2 is an even number so ( 2 / 2 ) = 1.

                     .
                       .
                     .
                       .
                     .
                       .
                     .
                       .

so the number 4 , 2 , 1 goes on............

Note :- The angel number 421 is the smallest prime formed by the two powers in logical order from right to left.

Trigonometric Formula

2012-06-30T21:10:00.009+05:30

Some useful Trignomtric Formulas :

Sin A =

opposite / hypotenuse

Cos A =

Adjacent / hypotenuse

Tan A =

opposite / Adjacent

Law of Sines : Sin A/a = Sin B/b = Sin B/c

Law of Cosines :

C²=a² + b² - 2ab cos C

Speciality of 0...

2012-06-21T19:25:00.003+05:30

Speciality of '0'

0 is a very unique number and it's speciality can started in following points :

1) '0' is a whole number as well as Integer.

2) '0' can never possess + or - sign because it is been invented for our convenience.

for e.g., :- 0 - 7 = -7, 0 + 7 = 7, -7 + 7 = 0

0 x 7 = 0 , 7 x 0 = 0, -7 - 0 = -7

7 - 0 = 7, 7 + 0 = 7, 7 - 7 = 0

0/7 = 0 but 7/0 ≠ 0 because

it will give answer as '∞' i.e., infinity and which is not an appropriate answer. Hence it is clear from the above

examples that '0' has been invented just for our convenience while solving mathematical operation and nothing

else.

3) Every number loses it's value or it becomes zero whenever it is multiplied with '0' or it is dividing

zero..

4) It is the only number which is the largest number from negative infinity and the smallest from the

positive infinity. It always remains in the center of real number line....

How to find whether a large number is divisible by 13 ?

2012-06-16T21:08:00.003+05:30

How to find whether a large number is divisible by 13 ?

Lets take an example : Is 377 divisible by 13 ?

Here is trick

37 - 9(7) = 37 - 63 = -26

As -26 is divisible by 13 we can say that 377 is also divisible by 13...

Lets take another example : Is 2054 divisible by 13 ?

Therefore

205 - 9(4) = 205 - 36 = 169

Again

16 - 9(9) = 16 - 81 = -65

As -65 is divisible by 13 ... The number 2059 is divisible by 13...

Note : Multiply the unit's place with 9 as in both the above examples....

How to find whether a number is divisible by 11 ?

2012-06-15T21:51:00.001+05:30

How to find whether a number is divisible by 11 ?

Lets take a simple example.. Does 11 divides 154 ?

Here is a trick...

Arrange 154 in the following manner

(1 + 4) - 5 = 5 - 5 = 0 Since 11 Divides 0 the number 154 is divisible by 11...

Lets take second example.. Does 11 divides 2607 ?

Arrange the number in the following manner...

(2 + 0) - (6 + 7) = 2 - 13 = -11

Since 11 divides -11 the number 2607 is divisible by 11...

Now lets take a large number 3469917

Again arrange the number

( 3 + 6 + 9 + 7 ) - ( 4 + 9 + 1 ) = ( 25 ) - ( 14 ) = 11

Since 11 divides 11 the number 3469917 is divisible by 11....

Note : If 11 does not divides the result(11 in the above example) then your number is not divisible by 11..

How to find whether a number is divisible by 9 ?

2012-06-14T11:10:00.002+05:30

How to find whether a number is divisible by 9 ?

Lets take a small example ... Is 243 divisible by 9 ?

Here is a trick 2 + 4 + 3 = 9 since 9 is divisible by 9 the number 243 is divisible by 9..

Lets take another example 281475..

So adding the numbers we get

2 + 8 + 1 + 4 + 7 + 5 = 27 as 27 is divisible by 9 the number 281475 is also divisible by 9...

Note : You can also add the result further... for e.g., the above result is 27 so 2 + 7 = 9 which is divisible by 9

Fun with Calculator.......

2012-06-13T00:54:00.000+05:30

Take a calculator

Then type the number 12345679

Note : Do not type 8 in the above number..

Multiply it with the table of 9..

For e.g., 12345679 X 9

               12345679 X 18

           12345679 X 27

     12345679 X 36

and so on.....

See the result.......

How to find whether a number is divisible by 6 ?

2012-06-11T20:41:00.000+05:30

To check whether the number is divisible by 6 requires two step

Step 1 : To check whether the number is divisible by 2.

Step 2 : Sum up the digit and check whether it is divisible by 3.

For example : 138

Step 1 : 138 is divisible by 2 as 8 is divisible by 2....

Step 2 : 1 + 3 + 8 = 12 which is divisible by 3....

As the number satisfy both the steps the number is divisible by 6

Note : To know more about numbers divisible by 3 click here...

Lets take a bigger number 27402

Step 1 : 27402 is divisible by 2 as 2 is divisible by 2 in the unit's place...

Step 2 : 2 + 7 + 4 + 0 + 2 = 15 as 15 is divisible by 3...

As the number satisfy both the steps the number 27402 is divisible by 6...

Note : Even if one of the above step is missing then the number is not divisible by 6...

How to find whether a large number is divisible by 3 ?

2012-06-09T21:50:00.001+05:30

Lets consider a small number 234 is divisible by 3.

So here is a trick 2 + 3 + 4 = 9

Since 9 is divisible by 3

Therefore 234 is divisible by 3....

Now consider a large number 27934576592477693217

So

2 + 7 + 9 + 3 + 4 + 5 + 7 + 6 + 5 + 9 + 2 + 4 + 7 + 7 + 6 + 9 + 3 + 2 + 1 + 7 = 105

You can further add 1 + 0 + 5 = 6

As 6 is divisible by 3

Therefore 27934576592477693217 is divisible by 3...

Multiply Using Line Method

2012-06-07T23:32:00.010+05:30

Multiply two numbers using line :

Lets take an example 45 X 23 :

Step1 : Draw 4 lines shown in the Diagram.

Step 2 : Now again draw 5 lines as shown in the Diagram..

Step 3 : Now draw 2 lines which intersects both 4 and 5 lines...

Step 4 : Follow the Diagram....

Step 5 : Make two partition As shown below ..

Step 6 : Now count the red dots on left side, Middle (top + bottom) and right side...

Step 7 : Now add all the numbers in the following manner

     8        ......{Left side}
   2   2         .......{Top + Bottom}
1   5    .......{Right Side}
                                    _______________
                                       1 0    3    5     ........{Answer}

So our answer is 1035......

Square of Number in 100's

2012-06-07T19:23:00.001+05:30

Choose a number greater than 100.

For e.g.,

Lets start with 103

so

Step 1: Square the last two digits 03 X 03 = 09

Step 2: Add the last two number : 103 + 03 = 106

Step 3: Place the number in this format Step2 result Step1 result =10609 is our answer

Take Second example 112

so

Step 1: Square the last two digits 12 X 12 = 144

             Note : Only consider the last two digit of Step 1
                        and then the 100 ^th place will be the carry.

Step 2: Add the last two number 112 + 12 = 124 + 1(carry) =125

Step 3: Place the number in the format Step 2 result (125) Step 1 result (44) =12544 is our answer....

Note: This may needs lots of practice to do further calculations.

Enter Length of First side
Enter Length of Second side
Enter Length of Third side
Perimeter of the Triangle is

Enter the Radius of a Circle
Circumference of a Circle is

Enter the Breadth of a Rectangle
Enter the Height of a Rectangle
Perimeter of a Rectangle is

Square = a²
Parallelogram = b*h
Trapezoid = 1/2* (b₁ + b₂)*h
Circle = π r²
Ellipse = π r₁ r₂

Fahrenheit:
Celsius :

Degrees :
Radians :

Radians :
Degrees :

Enter Number
Result