Episode 10: Numbers, infinite and music

This week the topic was types of numbers and infinity. We interviewed Dorothy Ker, who’s a musician and composer. We talked about the way Dorothy uses maths to inspire her creativity, as well as the types of maths that composers and musicians use.

Interesting links:

• "A gentle infinity" - One of Dorothy's compositions
• Amelia and the Mapmaker, the project on the Poincaré conjecture
• Jorge Luis Borges, on Wikipedia
• Marcus Du Sautoy on Borges for BBC Radio 4's Great Lives
• Recounting the rational numbers, at The Math Less Travelled

Puzzle: Which are there more of: whole numbers, or square numbers? (If you think the answer is obvious, try counting them).

Solution:

It may seem obvious to say that there are more whole numbers than square numbers - if you start counting, by the time you reach 20 you’ve counted 20 whole numbers but only 4 square numbers, and the square numbers only get further apart as you go up the number line. The set of all square numbers is contained in the set of all whole numbers, and it’s definitely smaller in some sense, as not all whole numbers are square.

However, since there are infinitely many square numbers, it’s possible to count them in the same way you count the whole numbers. Each square is paired up with its own square root - 1 with 1, 4 with 2, 9 with 3 and so on - so there are countably infinitely many square numbers, and since for any whole number I can find a corresponding square number by simply squaring it, these sets are considered to be the same size.