Astounding Math Stories

Astounding Math Stories

I am in the process of selling my old collection of Astounding Science Fiction on Ebay. It occurred to me that math needs an Astounding Math magazine. It could contain short descriptions of some mathematical facts that are really weird that could awaken people’s interest in math. Well, geeky people anyway.

These astounding facts also illustrate some important ideas in math. For one thing, some of them are not astounding when you get the right representation or proof (for example e^{iπ} = –1). And some are simply frauds (infinite cardinals).

Each of the examples below will be fleshed out and given references. The first one, the Perrin function, has already been posted on the Astounding Math Stories website. Comments and suggestions are earnestly desired.

Perrin pseudoprimes

The Perrin function P(n) is a certain easily-defined function on the natural numbers with this property:

For all integers n e^{iπ} = –1

This is Astounding. Who would have thought that the numbers e, i and π would be related in this way? Well, actually, it is not hard to understand why it is true if you use Euler’s formula in the context of the Argand representation. And the fact that Euler’s formula works is not very difficult to understand. (Perhaps Euler’s formula is nevertheless Astounding. Feynman thought it was.) This is an excellent example of the ratchet effect: An amazing or incomprehensible statement about math suddenly becomes totally obvious and you can’t understand why you didn’t understand it before!

Infinite cardinals

There are “as many” integers as there are rational numbers. This is Astounding!

Really? In fact, this statement is a fraud. It depends on defining the cardinality of a set in terms of bijections (not a fraud) and then referring to the cardinality of the integers or rationals in terms of words like “many”.

What is happening is that, for finite sets, the cardinality function on sets Card(S) means the same as #(S). the number of elements of S. On infinite sets, the cardinality function does not have some of the familiar properties of the number of elements of a finite set; in particular, it can happen that S is a proper subset of T but Card(S) = Card(T). Unless you specifically state, “When I say S has as many elements as T, I mean Card(S) = Card(T)”, then you are deliberately using the word “many” with a nonstandard meaning that the listener may not know. This is like the politician who told his audience that his opponent was a “sexagenerian”. Playing on someone’s ignorance is FRAUD.

Composite numbers
It is possible to prove that an integer n is composite without knowing any nontrivial factors of n. This is Astounding! How could you show it is composite without finding a nontrivial factor? This is a consciousness-raising example that shows that just because you can’t think of how to do something doesn’t mean it is impossible.

To show that n is not a prime, all you have to do is find an integer a for which a^n – a is not divisible by n (Fermat’s Little Theorem). That is not even very hard to prove.

Humongous numbers

As far as we know, π(x) − li(x) changes sign for the first time at some humongous number. 1.397×10^316 is a LARGE number. Other even huger numbers are Moser’s number and Graham’s number. One could also refer to Ackerman’s function. All these are consciousness raising examples that show how big numbers can get.

Many years ago the Little Girl Next Door asked me what the next number after a trillion is. I said, a trillion and one. She was Disgusted.

Function Spaces

The Astounding thing about function spaces is that each point in a function space is a function. A whole function, like sin x or the Riemann Zeta function. Not its values, not its formula, but the whole thing. This story is going to be difficult to write, but if I can carry it off it may help the student along the way to understanding the concept of an encapsulated mathematical object.

Other topics:

The monster group

1^2+2^2+3^2+…+24^2 = 702 and the Leech Lattice

41 and 163

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2 thoughts on “Astounding Math Stories”

  1. I prefer the formulation

    e^{iπ} + 1 = 0

    because it links 5, rather than 4, fundamental numbers.

  2. Let the elliptic modular function, j have a q-series (Fourier series)
    with gthe kth term = a[k]q^k, then

    sum a[k]^2, k=1..24

    to see why the Monster encodes gthe universe.

    Recall the only non-trivial square pyramid has side 70.

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