Tuesday, April 03, 2007

1, 2, 3... Infinity

I was just going through some old newspapers, in one of which there was an article about the fear children have for mathematics. In a survey, an overwhelming majority of the children indicated that mathematics is their most hated subject. The article then happily concludes this to be the reason why so few students get into the IITs and IIMs. Idiots!

Well firstly – the reason very few people get into the IITs is because there are very few seats! If the IITs had a billion seats, all of us would have been engineering graduates. Except my friend Anirban, who would have still studied Chemistry at JU!
Secondly, a love for mathematics does not make you good at it. It just helps you appreciate it better. Best example – ME! I loved all branches of Mathematics all through school and did spectacularly badly in most of them. It is a lot like my love for music. I love music but can’t sing to save my life!
Thirdly, mathematics does not require a higher level of intelligence than say, history. It is just that success in solving a calculus problem can be demonstrated more charismatically than writing an equally successful treatise on history. “Pythagoras, come to the blackboard and prove what you said about triangles.” This is so much easier to execute while Herodotus and Aristotle are slogging it out on tablets about their theories of history and philosophy.

I think if kids do math problems in a non-threatening environment with no apparent ‘stake’, then they don’t develop any fear for the subject, approach it with a bit of fun and end up learning quite a bit! Otherwise you end up like my aunt (pishi – father’s sister) who refuses to come anywhere near anything that has numbers written on it and blithely blames my father for not being able to teach her!

Ironically, the reason why I never developed a fear for numbers was a steady stream of games my father (the same guy as described above!) taught me. A little beyond me initially, I learnt them with a bit of effort but once I did, they were not only a source of entertainment but very effective to counter Math test fears!
The most common one is a factorization of car numbers. That is, finding out the numbers that can exactly divide the four digit car numbers. Most of them are easy enough and you can easily find their factors but once in a while you come across a 2021 and it takes a while to realize it is actually 43 x 47.
Our first car – a vintage model Ambassador Mark II – was number 1419. That is, 3 x 11 x 43. Still remember that!

An off-shoot of this was the methods to found out if a particular number can be divided by another. The easy ones are –
(1) If a number ends with an even number, then it can be divided by 2.
(2) If a number ends with 5, then it can be divided by 5.
(3) If the digits of a number add up to a number divisible by 3/9, then the number can be divided by 3/9.

Then there are a couple of slightly tricky ones.
How do you find out if a number is divisible by 11? You add up the digits in the odd-places and then add the digits in the even-places, if the difference between the two sums is 0 or 11, then the number is divisible by 11. That is, for 259246779 – you add up 2+9+4+7+9 (odd places) = 31. Then you add up 5+2+6+7 (even places) = 20. The difference is 11, so the number can be divided by 11.

How do you find out if a number is divisible by 7? There is a method to this as well but it is so complicated that it is probably easier to divide the damn number by 7. I remember the method vaguely – having read it in a Russian book (by Y. Perelman) - but would not hazard it without checking.
Oh-kay. I just found the method here. Read it and explain to me if you can!

A trick my apparently math-phobic wife taught me was how to remember 9-times table. I had haughtily told her that I knew the 89-times table as well but she pointed out the trick was intended for pre-schoolers and not people with 18 years of formal education.
The trick is – you hold up your ten fingers in front of you and start from the left.
For 9 x 1, you hide the first left finger (little finger on your left hand). You get 0 fingers to the left of it and 9 fingers to its right. So, 9 x 1 = 09. For 9 x 2, you hide your second left finger (ring finger of your left hand). You have 1 finger to the left of it and 8 fingers to its right. So, 9 x 2 = 18.
You carry on hiding the relevant finger and you have 0/9, 1/8, 2/7, 3/6… 8/1 and 9/0 fingers to the left and right of the hidden finger. And that is the 9 times table!

One of the first math games I learnt went something like this.
Take any three-digit number. Write the same number next to it, making it a six-digit number. For example, 372372.
Do you think 7 can exactly divide this number (that is, without leaving any remainders)? Unlikely, but you may feel that it is possible since I am playing this game with you! It is. In the example, it becomes 53196.
Do you think 11 can exactly divide this number you have got now? Feeling similar to above! Again, it is! It becomes 4836.
Now, what are the chances this number is divisible by 13? Even unlikelier, right? But as you would have guessed by now, it is divisible by 13 as well – and the final result is 372, the original result you started off with.
The explanation is quite simple, actually. Writing a three-digit number side-by-side produces the same result if you multiply it by 1001. That is, 372 x 1001 = 372372. And 1001 = 7 x 11 x 13. Which explains why the six-digit number can be exactly divided by 7, 11 and 13. We effectively multiplied a number by 7, 11 and 13 and divided it by the same.

That’s the beauty of mathematics. There are so many cool ‘parlour tricks’ to play with kids, its quite a wonder that so many of them don’t start liking it. Maybe they would, if they are taken through it in a non-CBSE kind of way.
But then, I am probably speaking a bit sooner than I should. For all you know, my son will be throwing up on his Geometry paper in a few years time!

I took the title from a lovely book on popular science by George Gamow, covering theory of numbers, structure of atoms, theory of relativity and other pretty deep scientific topics explained really well.
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