Lets Talk Maths

Leap Year

2012-08-07T22:56:00.000-07:00

A leap year consists of 366 days. During Leap Year, we add a leap day, an extra day on February 29. Leap years are important because we need to keep our calendar in alignment with the Earth's revolutions around the sun. It takes the earth approximately 365.242199 days or 365 days, 5 hours, 48 minutes and 46 seconds to circle once around the sun. However the Gregorian calendar has only 365 days in a year, so if we didn't add a day on February 29 nearly every 4 years, we would lose almost six hours off our calendar every year. After 100 years, our calendar would be off by approximately 24 days!

How to find out that the given year is a leap year?

Any given year is a leap year if it is divisible by 4 but not by 100. If a year is divisible by 4 and by 100, it is not a leap year unless it is also divisible by 400.
For example, years such as 1992, 1996, 1988 and so on are leap years. They are divisible by 4 but not by 100. In case of century years, the 400 rule applies. Thus the century year 1900, though divisible by 4 is not a leap year. It is divisible by 100 but not divisible by 400. The century year 2000 is divisible by 4, is divisible by 100 but also divisible by 400. Thus it is a leap year.

Magic Square

2012-03-04T07:48:00.009-08:00

The ancient Chinese Magic Square was followed by the Lo Shu Square, which dates back to 2200 B.C. It symbolizes the natural order of the Universe, promoting logic, strategy and open-mindedness.

The Legend of Lo Shu
During ancient times in China there was a huge flood on the Lo River. The people tried to calm the River god’s anger by offering sacrifices, but each time they prepared an offering a turtle came up from the river and walked around the sacrifice, and the River god wouldn't accept the sacrifice. This happened several times, until one time, a child noticed curious markings forming a pattern on the turtle’s shell. After studying these markings the people realized the correct amount of sacrifices to make, that is 15. Then the river god was placated. The numbers in every row, up and down, across, or diagonally, add up to 15, which happens to be the number of days it takes for the new moon to become a full moon.
The odd and even numbers alternate in the periphery of the Lo Shu pattern; the 4 even numbers are at the four corners, and the 5 odd numbers (outnumbering the even numbers by one) form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (the number of days in each of the 24 cycles of the Chinese solar year). Since 5 is in the center cell, the sum of any two other cells that are directly through the 5 from each other is 10 (e.g., opposite corners add up to 10.
This pattern, in a certain way, was used by the people in controlling the river. The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.
The Square of Lo Shu is also referred to as the Magic Square of Saturn or Chronos.

Cultural significance of the magic square in India
Magic squares have fascinated humanity throughout the ages, and have been around for over 4,120 years. They are found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity and prevention of diseases.
The Kubera-Kolam is a floor painting used in India which is in the form of a magic square of order three. It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 72.

23 28 21

22 24 26

27 20 25

The 3x3 magic square was used as part of rituals in India from vedic times, and continues to be used to this day. The Ganesh yantra is a 3x3 magic square. A well known early 4x4 magic square in India can be seen in Khajuraho in the Parshvanath Jain temple. It dates from the 10th century.

7 12 1 14

2 13 8 11
16 3 10 5

9 6 15 4
This is referred to as the Chautisa Yantra, since each row, column, diagonal, 2x2 sub-square, the corners of each 3x3 and 4x4 square, the two sets of four symmetrical numbers (1+11+16+6 and 2+12+15+5), and the sum of the middle two entries of the two outer columns and rows (12+1+6+15 and 2+16+11+5), sums to 34. And Chautisa in Hindi means 34.

Tables made easy

2012-03-04T07:37:00.002-08:00

You can multiply most of the problems in your head when you know your multiplication table up to 5 times table. If you know your 2, 3, 4 and 5 times tables up to 10 times each number, then you also know part of your 6, 7, 8, and 9 times table. As you learn your higher tables, mental calculations become much easier. It is easy to learn your 11 times table and the twelve times table is not difficult. Six times 12 is six times 10 plus six times two. It is easy to add answers for 6 times 12 to get 72.
6 X 10 = 60; 6 X 2 = 12; 60 + 12 = 72

We see, how easy it is to learn your tables and how it can be done much more quickly than in olden days.
It is usual to learn tables up to the 12 times table. It is not hard to learn the 13 times table if you know the 12 times table. If you know that 12 times 3 is 36, just add another 3 to get 39. Twelve threes plus one more three makes 13 threes. If you know 12 fours are 48. Just add another four to get 13 times 4 equals 52.

Now that you know your 13 times table, learn the 14 times table. Factor 14 to 7 times 2. Then you multiply the number by 7 and double the answer.
7 X 1 = 7; double = 14; 14 X 1 = 14
7 X 2 = 14; double = 28; 14 X 2 = 28
7 X 3 = 21; double = 42; 14 X 3 = 42
7 X 4 = 28; double = 56; 14 X 4 = 56
7 X 5 = 35; double = 70; 14 X 5 = 70
7 X 6 = 42; double = 84; 14 X 6 = 84
7 X 7 = 49; double = 98; 14 X 7 = 98
7 X 8 = 56; double = 112; 14 X 8 = 112
7 X 9 = 63; double = 126; 14 X 9 = 126
7 X 10 = 70; double = 140; 14 X 10 = 140

Practice these easy steps and you will learn your tables without even trying hard. You will find that you can multiply and divide directly by 13, 14 and 15. This will give you an advantage in your Math Class.

Fibonacci sequence in sunflower

2012-03-01T21:08:00.007-08:00

The arrangement of seeds on the head of a sunflower is a very good example of the Fibonacci sequence. By closely observing the seed configuration we notice two series of curves, one winding in one direction and one in another. The numbers of spirals are not the same in each direction. If we count the number of spirals given in the figure, we notice the number is 21 and the number of spirals in the opposite direction is 34.

Coincidentally, this number is within the Fibonacci sequence. The numbers of spirals are not the same in each direction. In general they are either, 21 and 34, or 34 and 55, or 55 and 89 or 89 and 144. In principle all the sunflowers show a number of spirals that are within the Fibonacci sequence.
For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is because of the efficiency during the growth process of plants.
In many cases, the head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill eventually all the space. Each new seed appears at a certain angle in relation to the preceding one. In order to optimize the filling, it is necessary to choose the most irrational number there is. This number is exactly the golden mean. The corresponding angle, the golden angle is 137.5 degrees. This angle has to be chosen very precisely. Even the variation of 1/10 of a degree destroys the optimization completely. When the angle is exactly the golden mean, two families of spirals (one in each direction) are then visible. Moreover, generally the petals of flower are formed at the extremity of one of the families of spiral. This explains why the number of petals corresponds on average to a Fibonacci number.
We learn that Nature is fortunately a much better Mathematician than most of us.

Multiplication by 11

2012-02-16T23:48:00.001-08:00

We all know that it is easy to multiply a single digit number by 11. If you want to multiply 4 by 11, you simply repeat the digit for the answer, 44. If you multiply 6 by 11, you get 66.
Did you know it is easy to multiply two digit number by 11?
Multiplying a two-digit number by 11.
To multiply 16 by 11, add both the digits of the given number, 1+6 = 7. Now place this answer , 7, in between the two digits, in this case giving you 176.
16 X 11 = 176
These calculations are easy to do in your head. If somebody asks you to multiply 52 by 11, you could immediately say, "five hundred and seventy two".
That was easy! Now, what happens if the digits add to ten or more? It is not difficult - you simply carry the one (tens digit) to the first digit. For example, if you were multiplying 78 by 11, you would add 7+8=15.
Put the 5 between the digits and add the 1 to the 7 to get 858.
Let's try another example. To multiply 95 by 11.
9+5 = 14
Now carry 1 over 9.
1 plus 9 gives you 10. We work with 10 the same way as we would with the single digit number. Ten is the first part of our answer, 4 is the middle digit and 5 remains the final digit.
So, 95 X 11 = 1045
This approach also applies to numbers such as 22 and 33. Twenty-two is 2 times 11.
To multiply 17 by 22, you would multiply 17 by 2 to get 34.
Now 3+4 = 7.
Using the shortcut that we have learned, place 7 between the 3 and 4.
We get, 374
We see, 17X22 = 374

Practice some problems for yourself and you will be surprised how fast you can call out the answer.

Finding age trick

2012-02-14T00:37:00.000-08:00

Math tricks are a lot of fun. Students love them in the classroom. We are more open to learning when we are having fun in the process. Math teachers often use number tricks in the classroom to encourage enthusiasm among students. You can use some of these tricks with your friends as well.

A simple trick to find someone's age.
Step 1) Ask your friend to multiply the first number of his age by 5.
Step 2) Now ask him to add 3 to the result.
Step 3) Tell him to double this figure.
Step 4) Finally, ask him to add the second number of his age to the resulting figure and have him tell you the answer.
Step 5) Now you deduct 6 from the answer and you will have his age.

I am sure your friends will be impressed.

Number Trick

2012-02-13T00:23:00.001-08:00

An interesting trick of number 421. What is so special about the number 421?
Number 421 is the smallest prime number formed by the powers of two in logical order from right to left.

Follow the given steps and you will notice an interesting loop of 4...2...1...
1) Select any whole number.
2) If it is an even number,divide by 2; If it is an odd number, multiply by 3 and add 1 to it.
3) Keep on repeating the process mentioned in step 2. We'll see that after a while the value 4,2,1 repeats itself.

Example: Lets pick a number, say, 23.

23 is an odd number; so multiply by 3 and add 1 to it;
(23 × 3) + 1 = 70
70 is an even number; so divide by 2;
70 / 2 = 35
35 is an odd number; so multiply by 3 and add 1 to it;
(35 × 3) + 1 = 106
106 is an even number; so divide by 2;
106 / 2 = 53
53 is an odd number; so multiply by 3 and add 1 to it;
(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
. . . . . . . . . . . . . . . . .
We see that the loop 4..2..1 goes on and on.

Funny Calculations

2012-02-12T07:16:00.000-08:00

You can play this funny Math trick with your friends. Ask them to do the following calculations and you can predict their date of birth.
1) Add 18 to your Birth month.
2) Multiply by 25.
3) Subtract 333.
4) Multiply by 8.
5) Subtract 554.
6) Divide by 2.
7) Add your Birth date.
8) Multiply by 5.
9) Add 692.
10) Multiply by 20.
11) Add only the last two digits of your Birth year.
12) Subtract 32940 to get your complete date of birth.

Suppose the answer is 102382. This implies that your friend's date of birth is October 23, 1982 (MM/DD/YY).
(Note: If the answer is not right, your friend did not follow the directions correctly or lied about his Birthday.)

Fibonacci Numbers

2012-02-07T19:41:00.000-08:00

The numbers in the following integer sequence are called the Fibonacci numbers.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.
1 = 1+0
2 = 1+1
3 = 2+1
5 = 3+2
8 = 5+3
13= 8+5
21= 13+8
34= 21+13
55= 34+21

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. His book, ‘Liber Abaci’ in 1202, introduced this sequence to Western European Mathematics. In one of the chapters of his book, Fibonacci posed a well known problem of rabbit population.
Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

To solve this problem we will consider the following steps:
- At the beginning of the experiment, there is one pair of rabbits
- At the end of the first month, the two rabbits have mated but have not given birth,therefore there is still only 1 pair.
- At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
- At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
- At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
- At the end of fifth month, every pair that was alive two months ago gives birth. This makes three new pairs for a total of eight.
Continuing in this manner, to forecast the number of pairs of rabbits many months from now simply add together the number of pairs that were alive in the preceding two months.
Now, do you know of any series of numbers which begins with one and one and continues by adding the preceding two numbers to get the next? Of course you do. It's the Fibonacci series.

Golden Spiral in Nature

2012-02-07T00:59:00.000-08:00

"Nature hides her secrets because of her essential loftiness, but not by means of ruse".
~Albert Einstein

Maths in Da Vinci's Art

2012-02-05T07:49:00.004-08:00

Leonardo Da Vinci’s most famous illustration is the Vitruvian Man. The figure illustrates the Golden proportion (1:1.618), also known as the ‘Divine proportion’. The Vitruvian Man's drawing is based on the correlations of ideal human proportions. It is accompanied by notes based on the work of the famous first century Roman architect, Vitruvius. He was also the author of ‘De Architectura Libra X’; Vitruvius described the human figure, which was divinely created, as being the principal source of proportion. Leonardo’s drawing is traditionally named in honor of the architect. It is a pen ink drawing of a male figure whose outstretched limbs touch the circumference of the circle and the edges of a square. His naval falls in the exact center of the circle. The figure is surrounded by hand written text. Leonardo’s presentation looks dynamic. The male figure adopts two different positions within the same image. He is within the circle and the square, simultaneously. The figure’s active arms and legs suggest movement and liveliness. If the distance from the top of the head to the naval is 1, then the naval distance from the naval to the bottom of the foot is 1.618. Also if the distance from the right shoulder to the tip of left finger is1, then the total height of the human body (head to toe) is 1.618.
Modern scientists believe that divinely proportioned individuals, both males and females, appear not only attractive but tend to be strong and physiologically healthy.

The beautiful rectangle and the spiral

2012-02-04T06:43:00.000-08:00

The Golden Spiral was first discovered by Pythagoras. The spiral is derived from the golden rectangle. When you connect a curve through the corners of these concentric rectangles, the golden spiral is formed.
This shape is found in a lot of things in nature: the Nautilus shell, the face of a sunflower, the curve of an elephant tusk, the horns of wild sheep, the spiral in a pineapple, curl in our eyelashes, our finger prints, our DNA, the shape of the Galaxy and much more. Man attempted to imitate this beauty of nature. We can see that the Sphinx and Pyramids of Giza all lie on a Golden Spiral.

“He that holds fast the golden mean,
And lives contently between
The little and the great,
Feels not the wants that pinch the poor,
Nor plagues that haunt the rich man's door.”~William Cowper

The Beauty Of A Rectangle

2012-02-03T02:36:00.000-08:00

Look around, you see rectangular objects everywhere, for instance, computer screen, laptops, Desks, Books, Newspaper, cell phones, greeting cards, gift wrappers, paintings, calendars, doors, windows, closets, calculators, towels, bed sheets, photographs, color boxes, note books, dining table, mirrors, credit cards and the list goes on and on. Have you ever wondered why so many people have been drawing and building things that are rectangular shaped?

The ancient Greeks discovered that there is a specifically shaped rectangle that is most pleasing to the eye. It is not too thick, not too thin but just right. The rectangle possessing this characteristic is called the “Golden Rectangle”. If you simply draw what you believe to be the most beautiful rectangle, then measure the lengths of each side and finally divide the longest length by the shortest. You’ll probably find the ratio is somewhere around 1.6, which is called the golden ratio, rounded to the nearest tenth.
A golden rectangle is one whose side lengths are in the golden ratio, or approximately 1:1.618.
In the world of mathematics, the numeric value 1.618 is called "phi", named for the Greek sculptor Phidias. He sculpted many things including the bands of sculpture that run above the columns of the Parthenon. The exterior dimensions of the Parthenon in Athens, built in about 440BC, form a perfect golden rectangle.

Besides being “beautiful”, the golden rectangle has an intriguing characteristic. If you draw a golden rectangle and draw a line inside it to divide that rectangle into a square and another smaller rectangle, that small rectangle will amazingly be another golden rectangle. You can do this again with this new golden rectangle and you’ll once again get a square and yet another golden rectangle.

Leonardo Da Vinci called the golden ratio the “divine proportion”. His Mona Lisa has many Golden rectangles throughout the painting. By drawing a rectangle around her face, we can see that it is golden. If we divide that rectangle with a line drawn across her eyes, we get another golden rectangle, meaning that the proportion of her head length to her eyes is golden. There are other golden rectangles that can be drawn on the rest of her body like from her neck to the top of her hands.
Da Vinci created other pieces that were also drawn according to the golden ratio such as The Last Supper, Old Man and the Vitruvian Man.

I conclude with Plato’s quote “The good, of course, is beautiful, and the beautiful never lacks proportion”.

Cool Multiplication

2012-02-01T05:46:00.000-08:00

Here's an easy way to find out square of a two digit number.
I) Let's square a two digit number that ends in 5. (without using your calculator :))
For example, we would like to square the number 65.
Step 1: Multiply the first digit of a number with the next number that comes after it. In this case 7 comes after 6.
6x7=42
Step 2: Now place 25 after it.
4225

Well, 65X65 = 4225
By the way, 5 is my favourite number. What's yours?

II) Let's square any two digit number.
For example 27.
Step 1: Look for the nearest ten boundary, which is 30 (27+3=30).
Step 2: Now go down 3 numbers from 27.
=24
Step 3: Multiply 24X30 = 720 (answer 1)
Step 4: 27 is 3 from nearest ten boundary 30
Now square of 3 = 9 (answer 2)
Step 5: Add (answer1) and (answer2)
720+9 = 729

27X27 = 729

Find out on which day you were born?

2012-01-27T05:33:00.000-08:00

Have you ever wondered which day of week you were born? I guess yes! So lets find out using simple math to find the day of the week on which you were born.
Finding the month number!
We know that January has 31 days. A week has 7days so every date in February will be 3 days later than the same day in January. (28 is 4 weeks exactly. 28+3=31). For instance, if January 1st of any given year falls on Saturday, February 1st of that year will be 3 days later. That is on Tuesday. Following the same pattern for all the months, the following table is calculated.
Jan: 0, Feb: 3, Mar: 3, Apr: 6, May: 1, June: 4, July: 6, Aug: 2, Sept: 5, Oct: 0, Nov: 3, Dec: 5
The Process
Step 1- Write down your date of birth. Let us say 12th March 1983.
Step2- Number associated with the month from the above table is 3 .
Step 3- Write down the date , 12.
Step 4- Write down last two digits of the year, 83.
Step 5- Let us find out the number of leap years. Divide the last two digits of the year by four. 83/4=20 remainder 3.
Step 6- Now add all the four numbers from Step 2 to Step 5. 3+12+83+20=118.
Step 7- Divide 118 by 7 = 16 Remainder 6. Remainder tells you the day.
For finding out the day, refer to the following table:
Sunday: 0
Monday: 1
Tuesday: 2
Wednesday: 3
Thursday: 4
Friday: 5
Saturday: 6
From above, 6 is associated with Saturday so the answer is Saturday.
If you know any other method to find the day of a given date, you are most welcome to share with us.
~By Sangeeta N