Why the Square Root of 2 is Irrational

Okay, this is a bit of a sidetrack to the overall goal of this blog, but it was a suggestion of something for me to do.  Plus, it’s quite a cool little thing to talk about.  In the previous entry about different ‘types’ of numbers, I said how \sqrt2 was irrational, which means we cannot write it like a fraction \frac{a}{b}, where a and b are whole numbers.  But how do we know this?  Can we be sure that this is the case?  Well yes, and I’m going to explain how.

 

In maths, you always like to be sure about things.  You can’t guess and say ‘Well this is probably true’; you need to be 100% certain.  There’s different methods of checking statements in maths, and I’m going to talk about one of these now, which we can use to show \sqrt2 is irrational.  What we do is take a statement which we think is false.  Then, we assume that it is true.  We then look at what this means, and hope that it contradicts something we already know to be true.  Unsurprisingly, we call this a ‘contradiction method’.  It seems a bit weird, but in the world of maths, it’s actually quite a useful trick.

 

Let’s look at a non-mathsy examples.  Someone, let’s call him Elmer, claims that ‘All rabbits have three eyes’.  Everyone else thinks that this is false, but Elmer is having none of this, so we need to show he’s wrong.  So, let’s assume that Elmer is correct, and all rabbits do have three eyes.  But what does this mean in the big picture?  Well, if we find a random rabbit, anywhere in the world, it will have three eyes.  So someone finds a random pet rabbit called Bugs, which only has two eyes.  This contradicts us assuming ‘all rabbits have three eyes’, which means it cannot possibly be true.  So Elmer is wrong.  He went a bit loony over it and hunted rabbits for the rest of his days.

 

So, let’s go to the big problem – showing \sqrt2 is irrational.  Just as a bit of a warning, there’ll be a bit of potentially scary maths here!  Since we think this is true, we take the opposite of this and assume that that is true.  So, we will assume that ‘\sqrt2 is rational’.  This means that \sqrt2 = \frac{a}{b}, with a and b being whole numbers..  We now square both sides, giving us:  \frac{a^2}{b^2} = 2.  Next, we multiply both sides by b^2, so we get the equation 2b^2 = a^2.  We haven’t done anything particularly ground breaking here, just a bit of simple rearranging.

 

Now we use a little subtle point.  When we write a fraction, we like to write it in ‘lowest terms’.  This means a and b can be divided by the same number, we should do this, since it gives us the same number anyway.  So, for example, we don’t write \frac{a}{b} = \frac{2}{4}, we write \frac{a}{b} = \frac{1}{2}.  Instead of \frac{a}{b} = \frac{4}{6}, we write \frac{a}{b} = \frac{2}{3}.  And so on.  One thing this means is that a and b both can’t be even (since if they were, they’d both be divisible by 2).  Bare this in mind.

 

We know that a^2 is even, since it is a number multiplied by 2.  This means that a is also even (if you don’t believe me on that one, check for yourself!).  Since a is even, this means we can write it as 2 multiplied by a different number, say a=2c.  Now we stick this into 2b^2 = a^2, getting 2b^2 = (2c)^2  So, 2b^2 = 2c \times 2c = 4c^2.  Divide by 2 gives us b^2 = 2c^2.  This means that b^2 is even and so is b.

 

This means that a and b are both even, which we know cannot be the case!  So we have contradicted the statement that ‘\sqrt2 is rational’, which mean that ‘\sqrt2 is irrational’.  This is what we wanted; yay!

 

If you didn’t follow all of that, don’t worry too much about it.  The rest of this blog won’t really focus on this, nor will it contain as many equations.  Also, it’s difficult to follow things like this if you don’t write it down yourself.  But if you did follow it, give yourself a pat on the back.

 

 

 

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