Here’s Mathematician KP Hart’s Math Question and Answer for Friday, December 18th!
Counting, Part 3
For Counting Part 1, click here. Part 2, click here.
On the 29th of November 1873 Georg Cantor wrote a letter to Richard Dedekind with the question whether real numbers could be counted. Counted in the sense that we used in the previous two posts: can we couple the natural numbers and the real numbers in such a way that every natural number is teamed up with exactly one real number and, conversely, every real number is attached to exactly one natural number The previous sentence paraphrases Cantor’s wording of the question; nowadays we would ask whether there is a bijective map between both sets, but that term did not exist in Cantor’s days.
A little over a week later, on the 7th of December 1873, Cantor had solved the problem: the answer was “no”, there is no such coupling, there is no bijective map between the set, N, of the natural numbers and R, the set of natural numbers.
In the meantime, Cantor and Dedekind had proved that you can count the so-called algebraic numbers. A number is algebraic if it is a solution of an equation of the form anxn+…+a1x+a0=0, where the ai are whole numbers. Among these are the square roots of natural numbers, as solutions of x2-k=0, higher-order roots, and many more.
The proof is pretty and constructive: every equation has a height (a term coined by Cantor), defined as n+|an|+&hellip+|a1|+|a0|; so the degree plus the sum of the absolute values of the ai. The equation x=0 has height 2 and x2-2=0 has height 2+1+2=5. If you try to determine all equations of height 2, 3, 4, 5 and 6; then you will realize (and be able to prove) that there are only finitely many equations of a certain height.
By extension we can assign a height to algebraic numbers too: the height of α is the smallest possible height of an equation that has α as one of its solutions. The number 0 has height 2 and both √2 and -√2 have height 5. Because there are only finitely many equations of any given height and because an equation of degree n has at most n solutions there are only finitely many algebraic numbers of a given height. This will let us count the algebraic numbers: first sort them by height and for each height the numbers are already ordered by their position on the real line and we read them left to right.
Next time we will see how Cantor showed that the set of real numbers is uncountable.
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Read all of KP Harts math questions here!
About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.
I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.
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