Interesting numbers

I’ve been revisiting my copy of the Penguin Book of Curious and Interesting Numbers (last published 1997, but still available from lots of online sources). There are 210 pages, but it’s not until page 81 that the author even gets beyond number 10: lower numbers tend to have many more interesting properties than higher ones, and not just mathematically. For example, the Pythagoreans associated 6 with marriage and health because it is the product of their first even and odd numbers, which they thought of as female and male. And why is 40 interesting? Because it is a biblical expression for a long period of time: 40 days in the wilderness, 40 years of wandering in the desert. And also, following on from my last blog on the Imperial System, because there are 40 rods, perches or poles in a furlong of 220 yards.

Three pages alone are devoted to zero. In ancient civilisations, zero as a number didn’t exist: after all, numerals represent numbers of things, so if you have nothing it didn’t make sense to give that a number. Our number system, with 0 as an extra numeral, originated in India, and now of course the world would be a very different place without it! “Was zero a number? Was it a digit? If it stands for nothing, then surely it is nothing? Yet, if you add a harmless zero to the end of a number, you multiply it by 10!”

I like the entry for 39, which has no special properties at all. “This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting. It is therefore also the first number to be both simultaneously interesting and uninteresting.”

There are large entries for the fundamental numbers 0, 1, \pi, eie will not be familiar to Studies students but, at its simplest, the function y = ex is the only one where the gradient of the curve is always the same as the y value, and hence is at the heart of the mathematics of exponential growth – the rate of change being related to the “population” at all times. And i will be familiar to HL students as \sqrt{-1}. These fundamental numbers are related in Euler’s identity (“the most beautiful equation“) e^{i\pi}+1=0 (explanation below).

Some of the “interesting” facts about numbers are very obscure. For example, 26 is listed for two reasons: (a) it is the smallest non-palindromic number (a palindrome reads backwards the same as forwards) whose square, 676, is palindromic. And (b), 26 is equal to the sum of the digits of its cube. 263 = 17576, and  1 + 7 + 5 + 7 + 6 = 26.

\begin{array}{l} {e^{i\pi }} = \cos \pi  + i\sin \pi  =  - 1 + i \times 0 =  - 1\\ \therefore {e^{i\pi }} + 1 = 0 \end{array}


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