Irrational numbers are number with infinite and non-repeat decimal representation. Probably the most common example of an irrational number is *π*, the *Ludolphian number*. We can estimate the value of *π* to *3.1415* but is has much more digits, infinite amount of non-repeating digits. We know at least 12 trillion digits of *π* by now. It’s probably not very useful to know all these digits but you now, everyone has its own hobby.

## How do we Know Irational Numbers Even Exist?

Maybe it’s some non-sense crazy mathematicians invented to confuse ordinary people. Well, irational numbers really exist in real world and we can simply prove it. Let’s suppose isosceles triangle

with given lengths: *|AB| = 1* and *|BC| = 1*. What’s the length of the side *AC*? We can compute it using the Pythagorean theorem:

^{2}+ 1

^{2}} = √{1 + 1} = √2

So the length of the hypotenuse *AC* is equal to *√2*. What number is *√2*? Is it an integer? Clearly not. Is it a rational number? Well, maybe. If it is a rational number, it can be written in the form of fraction, i. e. there must be two integers *a, b* for which the equation

holds. Now, each integer is either *odd* or *even*. That’s simple.

- We can prove that square of even integer is again an even integer, i. e. if
*n*is an even integer then*n*is again even. E. g.^{2}*4*. Both^{2}= 16*4*and*16*are even numbers. And the converse holds too: if*n*is even, then^{2}*n*is even.- How we can prove it? Every even integer
*n*can be written in the form*n = 2k*for some integer*k*. E. g. the number 32 can be written as*32 = 2 · 16*. Thus*k = 16*. Then the square of even number*n*is equal to^{2}*(2k)*. We can simplify it to^{2}*4k*and then we can adjust it to^{2}*2 · 2 · k*. We can see by now that the expression^{2}*2 · 2 · k*is an even number.^{2}

- How we can prove it? Every even integer
- Then, if
*n*is an odd integer, then*n*is odd too. E. g.^{2}*7*. Both^{2}= 49*7*and*49*and odd integers. The converse holds too, if*n*is odd, then^{2}*n*is odd.- The prove is similar. Each odd integer
*n*can be written as*n = 2k + 1*. The square of*n*is then equal to*n*. We can simplify i to^{2}= (2k + 1)^{2}*(2k + 1)*and them modify it to the final form:^{2}= 4k^{2}+ 4k + 1*2(2k*.We can see that this is an odd number.^{2}+ 2k) + 1

- The prove is similar. Each odd integer

We can use this knowledge to prove that *√2* is an irrational number. We’re trying to find integers *a, b* such that

We can safely assume that the numbers *a* and *b* have no common divisor. Meaning that the fraction cannot be simplified. E. g. the fraction *12/8* can be simplified to *3/2* and it’s still the same number. The numbers *12* and *8* had the common divisor *4*. This innocent precondition will be crucial.

We can compute squares of both sides of equation:

We can simplify it little bit:

And adjust it:

^{2}= 2b

^{2}

Now:

*2b*is clearly an even number since it’s in the form^{2}*2k*where*k = b*.^{2}- Since the right side of the equation is an even number, the left side must be also even. I. e.
*a*is an even number and moreover the number^{2}*a*itself is an even number. Thus there must be integer*k*such that*a = 2k*and from this we have

^{2}= 4k

^{2}.

We know that *a ^{2} = 2b^{2}* thus we can write

^{2}= 4k

^{2}= 2b

^{2}

Now we have the equation *4k ^{2} = 2b^{2}* which we can simplify to the final form

^{2}= b

^{2}.

From the last equation we can derive that the number *b ^{2}* is even and thus the number

*b*itself is even. As a corollary we now have both numbers

*a*and

*b*even.

But… but wait. We assumed that the number *a* and *b* have no common divisor. Our first assumption was the fraction cannot be simplified because the number *a* and *b* have no common divisor. And if both numbers *a* and *b* are even then it means they *have* a common divisor, the number *2*. This violates our precondition and thus there are no integers *a* and *b* such that

which proves that the *√2* cannot be a rational number since it cannot be expressed as a fraction. So there are numbers in this world that cannot be expressed as a fraction and you find them anywhere around you. Just keep looking :-).