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The study of whole numbers is one of the oldest lines of inquiry in mathematics which is still thriving today. After all, it is possible in pure mathematics for the relevance of work done thousands of years ago to be undiminished today. I want to talk today about what is possibly the prototypical example of such a piece of mathematics. I am referring to Euclid’s proof of the infinitude of prime numbers.
A divisor of a number n is another, smaller number, for which there is no remainder when n is divided by it. For example, three is a divisor of six, seven is a divisor of seven and thirty-nine is a divisor of one hundred and seventeen. A prime number is a number with exactly two divisors. Let’s think about what it means to have exactly two divisors. Firstly, we observe that every whole number has at least one divisor, namely the number itself. Secondly, every whole number except for ‘1’ has at least two divisors: The number itself and the number ‘1’. So, with the exception of the number ‘1’, being prime expresses the quality of having the fewest possible divisors. Why must there must be infinitely many such numbers?
What exactly are numbers? How can one or should one think about numbers or picture them? How many of them are there?
This is the first in a series of posts on numbers.
I’d like to avoid becoming bogged down in philosophy, but in my recent posts about the number line, I failed to actually explain what I meant by ‘numbers’. If you have read the number line posts this fact may have passed you by completely, but, in order not to get sidetracked, an assumption was made about the readers’ notion of number, namely that it was the same as my own. This could have led to some confusion, though of course I hope that it did not. Even if you have not read the posts, the idea of explaining what numbers are may still seem silly and not worth worrying about. By the end of this series of posts I hope to have convinced you otherwise. In this first post we won’t really be doing any maths, we’ll just be discussing the concept of number. In the next post we’ll start to talk about numbers on a more mathematical level.