Here’s Mathematician KP Hart’s Math Question and Answer for Friday, December 4th!
Counting, Part 2
For Counting Part 1, click here.
Today we will count the rational numbers. The rational numbers, those are the numbers that can be written as a fraction of whole numbers. That can be done in many ways, for example the fractions ½, 2/4, 3/6, 4/8, … all represent the same number. For every rational number we choose the fraction with the smallest possible numerator and denominator (and whole numbers get denominator 1). So, from the above sequence of fractions only ½ remains. Furthermore, in case the rational number is negative, we put the minus sign in front of the fraction. Thus, we write -½ and not 1/(-2).
Now we perform a nice trick: we read the minus sign and the slash as digits. How? By working in the duo-decimal system. We replace the -, when present, by A and the / by B. In the duo-decimal system the A is the digit for `ten’ and the B represents `eleven’. For example, ½ becomes 1B212(the 12 indicates that we work duo-decimally). In the decimal system we get 1×144+11×12+2×1=27810, that natural number corresponds to ½ (the 10 indicates decimal representation). The natural that goes with -½ is A1B212 and that becomes 10×1728+1×144+11×12+2×1, or 1755810. Calculate for yourself which natural numbers go with 0/1, 1/1, and -1/1.
What we did is couple every rational number with a natural number and in such a way that different rational numbers are coupled with different natural numbers. Of course not every natural number is coupled with a rational number: the slash and the denominator make it so that all natural numbers that we get are at bigger than 12310 (and 0/1 goes with 13310). But that is no big problem: we can count the set of numbers that we get because they are all natural numbers. Call that set Q, the we can couple every natural number with a number in Q: 0 with the minimum of Q, 1 with the next number in Q after that, 2 with the next one after that, and so on. In this way every natural number is coupled with one rational number and conversely every rational number with one natural number.
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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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