Wednesday, 31 August 2011

e, Bah Gum

e: The Story of a Number, Eli Maor

The test of lucid writing isn’t that it is easy to understand but that it is as easy to understand as it can be. The writing in this book is not always easy to understand, but it’s still some of the most lucid I’ve ever come across. Less laudably, it was strangely repetitive too. This appears on page 124:
This makes the spiral a close relative of the circle, for which the angle of intersection is 90°. Indeed, the circle is a logarithmic spiral whose rate of growth is 0…
And this on page 134:
This property [of intersecting any straight line through the pole at the same angle] endows the [logarithmic] spiral with perfect symmetry of the circle indeed the circle is a logarithmic spiral for which the angle of intersection is 90° and the rate of growth is 0.
That aside, I can recommend this book highly as a history and survey of the most overlooked of the three great mathematical constants. The most recently recognized too, but then there’s an obvious reason for all that. π and φ have simple definitions: the ratio of a circle’s circumference to its diameter and the ratio x/y such that (x+y)/x = x/y. e, the base of natural logarithms, doesn’t have such a simple definition: it’s the limit of the equation (1+1/n)n as n = , and begins 2·7182182... That misleading double “182” is an artefact of its representation in base 10: e is not only irrational, like φ, which means its digits never begin repeating, but transcendental too, like π. But if e became familiar to mathematicians thousands of years later than π, it got a symbol of its own at nearly the same time. As David Blatner describes in The Joy of π, the symbol π was popularized, but not invented, by the great Swiss mathematician Leonhard Euler (pronounced “Oiler”), but Euler seems to have both invented and popularized e. Maor lays to rest an old story:

Why did he choose the letter e? There is no general consensus. According to one view, Euler chose it because it is the first letter of the word exponential. More likely, the choice came to him naturally as the first “unused” letter of the alphabet, since the letters a, b, c, and d frequently appear elsewhere in mathematics. It seems unlikely that Euler chose the letter because it was the initial of his own name, as has occasionally been suggested: he was an extremely modest man and often delayed publication of his own work so that a colleague or student of his could get the credit. (ch. 13, ‘eix: “The Most Famous of All Formulas”’, pg. 156)
But Euler certainly deserved to have a mathematical constant named in his honor, if for no other reason and there are certainly lots of other reasons than his discovery of the relationship explored in this chapter: eix = -1, which “appeals equally to the mystic, the scientist, the philosopher, the mathematician”. Rather like this book as a whole, and though some of it was well beyond me, it’s a model of pop math, from the mathematically rigorous its examination of the catenary, or the shape made by a hanging chain, for example to the culturally quirky. I’ve often read before that Jakob Bernoulli, one of a Swiss family that was the mathematical equivalent of the Bachs, asked for a logarithmic spiral to be carved on his tombstone with the words Eadem mutata resurgo: “Even though changed I rise again”. But I read for the first time here that the engraver got it wrong out of ignorance or laziness and used an Archimedean spiral instead. Not only that, I got to see the tombstone itself. That’s dedicated research, and though dedicated research doesn’t guarantee a good book, it’s part of what makes this book so good.

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