## Posts tagged ‘Descartes’

### Math Clocks

Jiminy. The folks at Clock Zone make a math class wall clock — and I would like to be the first to publicly chastise buy.com, amazon, and anyone else who is selling it. It contains at least two mathematical errors:

SPOILER ALERT: In my rant below, I identify the errors in the clock. If you’d like to identify them for yourself, don’t read any further.

I say “at least” two errors because there may be more. The obvious errors are for 9 (the expression assumes that the exact value of π is equal to the common approximation 3.14) and for 7 (because the equation is quadratic, *x* = 7 is only one of the answers; the other possible answer is *x* = ‑6).

More generally, I have an issue with any of the algebraic equations that are meant to represent integers. For instance, the equation 50/2 = 100/*x* has solution *x* = 4, but I believe that it is incorrect to say that the equation itself is equal to 4. So perhaps the clock has four errors, if you consider the algebraic equations for 4 and 10 to be erroneous, as I do.

This clock is meant to be a math joke. Edward de Bono in *The Mechanism of the Mind* (1969) suggested that when a familiar connection (such as seeing the numerals 1‑12 on a clock) is disrupted, laughter occurs as a new connection (seeing mathematical expressions instead of numerals) is made. Sadly, math jokes are supposed to make you laugh… yet this clock makes me want to cry.

To ease the pain, I did a little research and uncovered several clocks of famous mathematicians. I present them here for your enjoyment.

Leonardo da Pisa:

Kenneth Appel:

Rene Descartes:

Karl Friedrich Gauss:

### The Girl With the Dragon Tattoo

*The Girl Who Played with Fire* is the second volume in the late Stieg Larsson‘s *The Millenium Trilogy*. (Of course, you probably already knew that, since virtually everyone in North America has read this book. I mean, *someone* had to buy those 20 million copies, right?)

In this book the heroine, Lisbeth Salander, gets absorbed in recreational mathematics. She stumbles across a theorem about perfect numbers that, surprisingly, was proved by Euclid. (This is surprising because Euclid did most of his work in geometry, and a proof of his theorem about perfect numbers would rely on algebra and number theory.) The theorem appeared as Proposition IX.36 of Euclid’s *Elements.*

Stieg Larsson writes:

…a perfect number is always a multiple of two numbers, in which one number is a power of 2, and the second consists of the difference between the next power of 2 and 1. This was a refinement of Pythagoras’ equation, and [Lisbeth] could see the endless combinations:

6 = 2

^{1}(2^{2}– 1)28 = 2

^{2}(2^{3}– 1)496 = 2

^{4}(2^{5}– 1)8128 = 2

^{6}(2^{7}– 1)She could go on indefinitely without finding any numbers that would break the rule.

What Lisbeth does not state, but what is required for Euclid’s theorem to hold, is that 2^{k}(2^{k – 1} – 1) is a perfect number if and only if 2^{k – 1} is prime. She doesn’t state this — but her list of “endless combinations” only includes examples for which this is the case.

I don’t begrudge Larsson for this omission. After all, how can you be mad at the first author to sell more than one-million e-books on Amazon, especially when his most popular works were published posthumously? Besides, adding too much math to a popular fiction novel might make it a little less popular. I’m just happy that so many readers will be exposed to a little of the mathematical beauty that makes me love numbers.

Here’s a perfect quote from Descartes:

Perfect numbers, like perfect individuals, are very rare.

And a perfect joke:

Teacher: What is 14 + 14?

Student: 28.

Teacher: That’s good!

Student: Good? It’sperfect!