Is'nt it odd to become a square?

A)PYTHOGORAS:

He lived approximately around 500 B.C,almost around 200 to 250 years before Euclid,the next most famous mathematician(chronologically) of the western world.(The trouble given by them to us while studying had been in just the reverse order-there was only one pythogoras theorem,whereas Euclid was not satisfied until he compiled a complete docket of theorems and put them in the ascending order)

It is genuininely believed that Pythogoras went on a long trip to Asia Minor which included Persia and India and returned a learned mathematician as well as a religious freak.( a formidable combination even now; you can imagine what it would have been 2500 years back)

Nevertheless, he collected a group of followers and they were fascinated with the counting numbers and the various interesting relationships between them.One of the many such intersting observations made by them was the relationship between odd numbers and square numbers.(which was quite odd indeed!)

B)ODD TO BECOME A SQUARE:

All of us know what are odd numbers-1,3,5,7,9,11etc[Actually the only thing odd about them is that they are not divisible by 2,or so I thought, till I came across this following curiosity which makes them really odd indeed!]

We also know what are square numbers-1,4,9,16,25,36etc.Pythogoreans found that square numbers are nothing but the sum of that many number of odd numbers starting with 1.

The method they used was a system of placing dots to denote the integers progressively in a square form and counting how many additional dots are required to get the next square.

The same is illustrated in the following diagram and explanatory points given below

• The topmost table shows a progression of placement of dots to make squares. It can be seen that we are able to form from a 1 x1 square , a 2 x 2 square by adding 3 dots. From 2 x 2 square to 3 x 3 square by adding 5 dots and so on.

• Thus starting with 1, it can be seen that by adding successive odd numbers, we get progressively successive square numbers!{The sum of the first 2 odd numbers is 2 ² and the sum of the first 3 odd numbers is 3² and so on.}

• Algebraically also, it can be seen that it has to be so. Supposing we take a " a x a" square, and add dots as given in the above arrangement, we will need to add first ‘a’ dots along the x –axis and another ‘a’ dots along the y-axis. Since this will not close and complete the next square, we need to add ‘1’ more dot in the extreme right top corner to complete the next square. So what we have done is we have added:- "a² + 2a + 1" and obviously end up getting (a + 1)².

C)CONCLUSION:

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