Not going too far – just far enough
October 17, 2014
“Just how far would you like to go?” the enigmatic Frank asks in the sleeve notes to the Dylan album John Wesley Harding. “Not too far,” comes the reply. “Just far enough so we’s can say that we’ve been there.”
Reading maths at the higher levels is akin to poetry – dense, abstract, often impenetrable, albeit with its own rewards. But there are plenty of books out there that offer a digestible taste of the good stuff without talking down to the reader. The Dover series of maths and science books are a case in point, offering insights into a complex and sometimes daunting world but written in an engaging way – not taking you too far, but just far enough.
One of my favourites is Excursions in Number Theory, by Ogilvy and Anderson (ISBN 0486257789), not least because number theory itself is often conceptually simple to grasp – prime numbers, modulo arthimetic, series, perfect numbers … each can readily be explained and each provides a lifetime’s worth of study.
Take the primes, the building blocks of the number system. They hide within them one of the great unsolved conundrums of mathematics, the Riemann Hypothesis, and are also fundemental to many powerful and beautiful results.
But the chapter I enjoy the most doesn’t concern numbers, but people. It is the chapter on calculating prodigies, those gifted individuals who can manage prodigious calculations in their head. The US-born Zerah Colburn, for example, is recorded as giving the square root of 106,929 before the number could be fully written down (the answer, by the way, is 327). Zacharias Dase took less than a minute to mentally multiply 79,532,853 by 93,758,479 (you can work that one out for yourselves).
But my heart goes out to one William Shanks, a grafter rather than a prodigy, who devoted over 20 years of his life to calculating by hand the value of pi to 707 decimal palces. In the late 19th century, Shanks didn’t have the benefit of a hand calculator, and even a summary of his methods covered 87 pages. In 1949, long after Shanks’s death, one of the early electronic computers spat out the first 2,000 digits of pi in only three days – and when the result was studied, it was discovered Shanks had made an error at around the 500th decimal place. Twenty years of calculations wasted! It was, you may feel, a Shanks-less task.