One of my favorite XKCD comics is Purity: http://xkcd.com/435/
And number theory is the most pure of the math subjects.
As we learn numbers, we at first learn that they represent things we can count. But that breaks down once you encounter fractions or the number zero. So then it is things you can measure. But that breaks when you get to negative numbers. If you come up with an analogy that encompasses negative numbers – financial ledgers – debts and credits for example, then that breaks when you introduce imaginary numbers.
So what the heck are numbers anyway? Well in math a number is defined by its properties and what operations can be performed on it. That is getting pretty abstract.
I bought this book on Monday: The Peguin Dictionary of curious and interesting numbers. Since then I’ve been learning about all sorts of properties i didn’t know about.
Whole numbers, integers, rational number and irrational numbers you’ve probably heard of. Transcendental and imaginary as well unless you slept through math class. Odd, even, and prime are likely still familiar terms.
Here are some that might be new to you:
- perfect, tri-perfect and semi-perfect
- abundant, super abundant and deficient
You know of square and cubed numbers, but what of triangular, tetrahedral, hexagonal and so forth? And those just scratch the surface of the various sequences and series the most famous might be Fibonacci’s, but it is only one of many.
There there are the numbers that might be prime – Fermat numbers, Mersenne primes, etc.
The thing I like most about these is that knowing this serves almost no purpose whatsoever.