While it’s one of the most well-known and well-trodden proofs among proofs, the irrationality of $\sqrt{2}$ shouldn’t be lacking on a blog about mathematics and physics. So, here it goes.
What is irrationality?
For those who aren’t too familiar with mathematical jargon, let’s first discuss what it means to be irrational. Obviously, we’re not talking about the psychological attribute but the mathematical one.
You might remember primary school when you had to learn about fractions such as
\begin{equation} 1 = \frac{4}{12} + \frac{2}{3}. \end{equation}
A practical application of a fraction is when you were reading a recipe for a delicious dish with a certain ratio of water and rice, which, even if you might not be aware of it all the time, can be written as a fraction, representing the ratio between water and rice. In fact, a fraction is a ratio.
For instance, in order to cook the perfect, fluffy rice without the need to pour off excess water when the rice is cooked, the ratio is that for 1 cup of rice, you add 1.5 cups of water1 So, the fraction is $\frac{1}{1.5}$.
Of course, it’s conventional to write a fraction using whole numbers (integers) only, so, $\frac{1}{1.5} = \frac{2}{3}$. Just multiply the numerator and the denominator by two. In other words, for 2 cups of rice, add 3 cups of water.
If we use our calculator, we get $\frac{2}{3} = 0.666\dots$ There is no end to this number, but the number can be perfectly written down as a ratio: $\frac{2}{3}$.
Of course, $\frac{2}{3}$ is the same as $\frac{4}{6}$, or $\frac{10}{15}$, or $\frac{200}{300}$, since, and this is crucial, all the other fractions (ratios) are simply multiples of our original fraction: they can all be simplified to their ‘simplest’ form, $\frac{2}{3}$. A slightly more technical way of saying this is that the fraction $\frac{2}{3}$ is the form in the lowest terms of the fraction $\frac{200}{300}$. It’s very important to remember this.
Now we’ve arrived at what irrational numbers are.
Premise 1. A number is irrational when it cannot be written as a ratio in lowest terms.
Two of the more well-known examples of irrational numbers are $\pi$ and $\sqrt{2}$. If we use our calculator, we can see how there seems to be no numerical repetition in them. This is a quality that irrational numbers possess.
Proof by contradiction
So, how do we prove that $\sqrt{2}$ is irrational, i.e. it cannot be written as a ratio? We do this by contradiction: if the opposite of a statement is demonstrably false (and there are really only two options), then the statement itself must be true. In a previous article, we used the same strategy to prove that ‘minus minus is plus’.
We will do that here, too.
Even and odd
Premise 2
We will also use the fact that some number multiplied by 2 equals an even number. Take any number, odd or even, multiply that by 2, and you will get an even number. This isn’t rocket science, really, as a characteristic of an even number is that it’s divisible by 2. In our proof, we will use the symbol $k$ for ‘some number, any number, odd or even’ being multiplied by 2.
Premise 3
If you take the square of an odd number, the result is always odd. If you take the square of an even number, the result is always even. Conversely, if you take the root of an odd number, the result is always odd. The same idea goes for even numbers. Check in your head to see if that’s true (it is). We will provide a proof for that in another post.
Okay, ready? Let’s go.
Proof that the square root of 2 is irrational
Anti-Premise 1. Suppose, by contradiction, that $\sqrt{2}$ can be written as some ratio in lowest terms: some (whole) number $a$ divided by some other (whole) number $b$ in lowest terms.
In other words, suppose
\begin{equation} \sqrt{2} = \frac{a}{b}. \end{equation}
To make life a little bit easier, we get rid of the square root by squaring both sides of the equation:
\begin{equation} 2 = \frac{a^2}{b^2}. \end{equation}
If we rearrange this, we get
\begin{equation} a^2 = 2b^2. \end{equation}
Now, we see that $a^2$ is an even number as $b^2$ – whatever that number is – is multiplied by 2. It also means that $a$ cannot be an odd number – it’s even. Remember premise 3?
Conclusion 1: $a$ cannot be odd – it’s even.
We can then also state that $a = 2k$, where $k$ is some number, any number, odd or even. If we substitute that into equation (4), we get
\begin{equation} (2k)^2 = 2b^2. \end{equation}
If we rearrange that, we get
\begin{equation} b^2 = \frac{(2k)^2}{2}. \end{equation}
If we simplify this in one extra step, we get
\begin{equation} b^2 = \frac{4k^2}{2} = 2k^2. \end{equation}
This means that irrespective of what number $k^2$ is, because it’s multiplied by 2, the result is an even number. In other words, $b^2$ is an even number, which also means that $b$ is an even number.
Conclusion 2. $b$ cannot be an odd number – it’s also an even number.
Looking at conclusions 1 and 2, we arrive at
Conclusion 3: $\frac{a}{b}$ is not a ratio in the lowest whole numbers as $a$ and $b$ can still be divided by 2.
This is a contradiction. The fraction $\frac{a}{b}$ cannot both be the lowest fraction and not be the lowest fraction. Conclusion 3 contradicts Anti-Premise 1. Therefore, there is no fraction $\frac{a}{b}$ in lowest terms that exists that can be equal to $\sqrt{2}$. Hence, the original statement Premise 1 is true.
Credentials of the Babylonian tablet clay tablet showing the root of 2: Photograph by Bill Casselman under CC BY-SA 3.0, and the Yale Babylonian Collection as the original holder of the tablet. A black and white rendition of Casselman’s own photograph of the Yale Babylonian Collection‘s Tablet YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras’ Theorem for an isosceles triangle. The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888…(30 x square root of 2).
- Rinse the rice thoroughly to remove the starch and dust for a nice fluffy texture. Add water by 1.5 times the used volume of rice. Add salt if you must. Bring the water to the boil as quickly as possible. Bring down the heat but keep the water bubbling softly. Give it one good stir. Put the lid on and don’t remove it for eight minutes. Don’t look inside; the water needs to stay in the pan. After eight minutes, shut off the heat and let it rest for another eight minutes. Still, don’t look. Keep the lid on the whole time. That’s it.[↩]
