gently musing about marginalia and related issues

Whenever Wimbledon comes along, I am pleasantly reminded of a question I was asked at school.

The question was simple. If you have 128 people playing in a knockout tournament, how many matches will it take to complete the tournament? Assume no draws or replays.

When we were asked the question, everyone knew the traditional way to get the answer. 128 people. 64 pairs. 64 matches in the first round. Then 32. Then 16. Then 8. Then 4, then 2, then 1. Giving us 64+32+16+8+4+2+1 or 127.

Later on that day, a few of us got into conversation with the teacher, and an alternative route was broached. 128 people. How many winners? One. So how many losers? 127. So how many matches will that take? 1 loser generated per match. 127 matches.

How I loved the simplicity of that approach. Just work out the number of losers and you will have the number of matches. I could have danced all night.

I felt the same way when I first learnt about decimals. Why some decimals “terminate”. Why others “circulate”. How beautifully they do this.

The way I was taught it was something like this. Think of a fraction. Convert it into a decimal. Think of another fraction. Convert it into a decimal. Do this a dozen times with different fractions, and observe the results.

It soon became clear that most fractions didn’t terminate cleanly, they “recurred”. Fractions like 1/3, 2/7, 3/11. Some fractions, on the other hand, ended cleanly, fractions like 1/2 and 2/5 and 3/10. It didn’t take too long to see that the only fractions that terminated were those where the denominator contained factors of 2 and/or 5.

2 and 5. In a base 10 world. Oh yeah. That bored me. So what did I find interesting about decimals? Again, it took a question.

And the question went something like this. Other than those with 2 or 5 as denominator, all fractions that have a prime number as denominator recur when expressed as decimals. The length of the recurring number or numbers is called the period of recurrence. So for example 1/3, or 0.333333…. has a period of 1, 1/11 or .090909…. has a period of 2. What is the maximum period of recurrence of any fraction where the denominator is a prime other than 2 or 5?

Which led to a lovely meandering journey. To convert a fraction into a decimal you have to divide the numerator by the denominator. You keep carrying the residue over until it is zero (in which case the fraction terminates). Or you keep carrying over and over and over.

When would a fraction circulate? When it hits the same residue again and starts the same sequence of residues as a result.

So how long before the same residue must be encountered? That depends on how many different residues can be had before that. What is the maximum number of different residues? One less than the denominator prime.

Discovering that 1/97 indeed has a period of recurrence of 96 filled me with glee. 96 different residues before the same residue is encountered. How beautiful.

There are so many others, little stories and techniques and tricks and tips that have helped me keep a passionate amateur interest in mathematics, particularly in the theory of primes.

The trouble with being an amateur like me is that you start getting idealistic about the subject. And you believe in things like elegance and simplicity. Which means that I’m one of those guys who still believes that there must be an elegant solution to Fermat’s Last Theorem. A solution that could not be jotted down in a book about Diophantine equations, but a short and elegant solution nevertheless.

My love for mathematics exists because others put their love of maths into me first. They spent time with me and explained things to me and taught me and filled me with wonder and amazement.

It’s the same thing with so much in life. Whatever you love, whoever you love. Love has to be shared in order to grow.

So when I think about copyright and patent and stuff like that, when I think about music and art and stuff like that, I think about love. The love that a creative artist puts into his or her creation. And how love has to be shared in order to grow.

One thought on “gently musing about marginalia and related issues”

  1. How beautifully one can summarise part of the essence of ones blog-writings in a last paragraph :-)

    Thanks for the tip on Bricklin’s book yesterday. It added to listening to him on a recent TWiT MBW podcast and made me realise I indeed want to read it.

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