Irrational numbers in reality I have a square tile which measures 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another will be $\sqrt 2$ metres. However  $\sqrt 2$ is an irrational number, could someone explain how it is possible for a non-terminating and non repeating number to represent a finite length in reality?
It's not the number $\sqrt{2}$ that's non-terminating; it's the decimal expansion of the number that's non-terminating. If you try to write down the entire decimal expansion of the number, you'll be writing forever, but the number itself is just a small number between $1.4$ and $1.5$.
A rational number can be defined as a certain type of function from integers to integers se follows. Take for example, $\frac{8}{14}$. I define it to be the function that assigns to each integer, the integer gotten by multiplying it by 8 then taking the quotient on division by 14. The following is the graph of this function which we call $\frac{8}{14}$



Now take the following two segments of a function which if you repeat them you get a rational number.

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Now you can get another two segments from these ones as follows. The first one is 1 copy of the first image followed by 2 copies of the second and the second one is one copy of the first image followed by 3 copies of the second to get

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Now again we can take two more images, one that can be gotten by taking one copy of the first followed by 2 copies of the second and 1 copy of the first followed by 3 copies of the second to get

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It's easy to see that this approaches a function that doesn't repeat. We've invented a new function from integers to integers. Now we can define the real numbers to be all functions of this sort and call the nonrepeating ones irrational numbers. It can in fact be shown that this particular nonrepeating function is $\sqrt3$.