YOUR MONDAY MATH with Mathematician KP Hart: COUNTING, Part 1

Mathematicians and other children enjoy, occasionally, to brag. Often the object is to trump the other by mentioning a large number: “I have more cookies/marbles/songs on my MP3-player than you.’ “Oh yeah? How many?” And then the large numbers appear: ten, hundred, thousand, tenthousandhundredmillion, …, until someone shouts `uncountable’. How much that is never decided but it usually ends the debate – except when there is a mathematician involved. She will ask what`uncountable’ means. A possible outcome of the ensuing discussion might be that `uncountable’ should mean `infinitely many’, more in particular: `as many as there are natural numbers’. Whereupon the mathematician grins and says “Oh, but that is still countable; as many as there are real numbers, now that is uncountable.” And instead of going “right, whatever”, let’s have that mathematician explain what she means by that.

Imagine you have two small boxes of chocolate sprinkles (a popular Dutch breakfast item, used on sandwiches mostly) and you need to know which holds more sprinkles. You don’t know if all the sprinkles have the same weight, so weighing will not be conclusive. You could count the sprinkles in both boxes but that requires a lot of attention as the numbers could be quite big. As the question only asks about more or fewer you could also do the following: each time pick one sprinkle from each box (and eat them). Continue until one of the boxes is empty: the one that is empty first had fewer sprinkles. If they become empty at the same time then they contained the same number of sprinkles.

This is the way you can investigate mathematically which of two sets has more elements. In this post we already saw a few examples of this: there are as many natural numbers as there are even natural numbers. Number the sprinkles in a (large) box of milk chocolate ones using the natural numbers and use the even natural numbers to number a box of pure chocolate sprinkles. Now follow this strategy: at step n you pick sprinkles number n and 2n from the milk and pure boxes respectively. This will take an infinite amount of time but we can predict that `in the end’ both boxes will be empty at the same time: every milk-sprinkle corresponds to exactly one pure sprinkle and conversely every pure sprinkle has exactly one milk counterpart. In the words of this post: we have made a bijection between the boxes of sprinkles.

In a similar way you can show that there are as many natural numbers as there are whole numbers. The whole numbers consist of the natural numbers together with the negative numbers -1, -2, -3, … Here is a way to couple the elements of both sets:

(0,0), (1,-1), (2,1), (3,-2), (4,2), …, (2n,n), (2n+1,-(n+1)), …Next post, we look at the rational numbers. Meanwhile: can you make a formula for the correspondence between the natural numbers and the whole numbers? That is, a formula f(n) so that you can write it as (0,f(0)), (1,f(1)), … (n,f(n)), ….