A Simple Proof that the Square Root of Two is Irrational.

A Simple Proof that the Square Root of Two is Irrational.

This is one of my favourite proofs in mathematics, and a truly ancient one. The number \(\sqrt{2}\) comes from the triangle above and both have been studied for thousands of years, dating back to the Babylonians. The ancient Greeks were the first to show that this number can’t be expressed as a fraction, which is what irrational means. I’ll go into more detail later.

Being able to go so far back is what makes mathematics unique. The ideas and foundations of ancient science, medicine, cosmology and so on have been discarded by advances made in those fields. In mathematics however, results like this one are still studied and appreciated.

There are three parts to this post;

  • What are rational and irrational numbers?
  • What is a proof by contradiction?
  • The actual proof.

What are numbers?

The simplest answer is to just say one, two, three and so on. Mathematicians call these the natural numbers, given how intuitive they are. I have one nose, two eyes and ten toes. We can use them for counting and adding.

What if we get bored of plain old addition and want to experiment with the wacky world of subtraction? Well sometimes it works and sometimes it doesn’t. Five bananas minus two bananas is three bananas. But what is two bananas minus five bananas? How can you have minus three bananas? By being in debt of course! Minus three bananas means I owe somebody three bananas.

We now have a new set of numbers called the integers. They are zero, the natural numbers and their negative counterparts. We can do addition and subtraction to our hearts content. Multiplication also comes for free but as soon as we look at division we find problems. We don’t have enough numbers. What is one pizza divided between six people? A slice of pizza! Well let’s keep things mathematical and say a sixth of a pizza.

Numbers like one sixth are called fractions or ratios. They are written as one integer divided by another. The set of all such numbers is called the rational numbers.

We have already redefined the idea of a number twice and we are not yet finished. Each time we tried to do something with our numbers, we had to expand our concept of number. First it was subtraction, then it was division and soon taking square roots will give us new kinds of numbers. This is a very powerful and recurring theme in mathematics.

Rational and Irrational Numbers

Rational numbers have two simple properties.

  1. A rational number can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b\) is positive. There are in fact many ways to do this for any rational number. For example \(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=…\) Furthermore there is a unique fraction where the bottom half of the fraction is as small as possible. This is important later.
  2. A rational number can be written as a decimal expansion that eventually repeats permanently. For example \(\frac{1}{9}=0.09090909…\) where the \(09\) repeats for ever. The repeating part can be any number of digits, and the start of the number doesn’t need to repeat. For example \(\frac{1}{7}=0.142857…\) where six digits repeat or \(\frac{301}{300}=1.003333…\) where the initial 1.00 is not important.

Irrational numbers have neither of these properties. For example \(\sqrt{2}=1.4142135623…\) will continue forever, with no repeating pattern of digits.

Proof by contradiction

Proof by contradiction is a classical mathematical tool. Have you ever seen a science fiction movie with a parallel universe? Maybe fish fly, cats chase dogs and cows make coffee not milk. Well a proof by contradiction is the mathematical equivalent of exploring a parallel universe.

It starts with an assumption. We assume that our intended result is false. If our intended result is indeed false, we will have entered a parallel universe. We then look at the consequences of our assumption. This is like stepping on a butterfly in a time travel story. The hope is we will find a mathematical absurdity such as \(0=1\) or anything that is false. This shows us that our parallel reality is indeed not real.

Which means we have a mistake somewhere in our proof. Unless we slipped up during our ‘exploring’, the only possible mistake is the assumption. If our assumption is false, the result must be true.

Show time: The square root of two is irrational.

We start by assuming \(\sqrt{2}=\frac{p}{q}\) where the denominator is the smallest possible. If we square both sides we get \(2=\frac{p^2}{q^2}\). Multiplying by \(q^2\) gives us \(2q^2=p^2\). We define \(n\) to be this number, i.e. \(n=2q^2=p^2\).

We find our contradiction by looking at the last digits of \(p\) and \(q\). First consider the square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and so on. Notice that none of the first few squares end in 2, 3, 7 or 8. In fact no square number ends with these digits.

Last digit of \(p\) 1 2 3 4 5 6 7 8 9 0
Last digit of \(\large p^2\) 1 4 9 6 5 6 9 4 1 0

What about \(\large 2q^2\)?

Last digit of \(q\) 1 2 3 4 5 6 7 8 9 0
Last digit of \(\large q^2\) 1 4 9 6 5 6 9 4 1 0
Last digit of \(\large 2q^2\) 2 8 8 2 0 2 8 8 2 0

What is the last digit of \(n\)? The second table tells us it must be 0, 2 or 8. The first table tells us that of these three options only 0 is possible. This tells us that \(n\) is divisible by 10 and therefore by 5. If 5 divides \(p^2\) then 5 also divides \(p\). Similarly if 5 divides \(2q^2\) 5 also divides \(q\). This means the numerator and denominator of our fraction \(\sqrt{2}=\frac{p}{q}\) are both divisible by five.

This is our required contradiction. We chose p and q so that q was the smallest possible denominator. But we now learn that we can divide p and q by five to get another fraction with even smaller denominator. This gives us a denominator that is smaller than the smallest possible. This logical inconsistency shows that our starting assumption that \(\sqrt{2}\) is rational is a false assumption. Therefore \(\sqrt{2}\) is irrational and our proof is complete.

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