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 $1.5$.

In reality, an exact side length of one meter does not exist, either. Nor does an exact square shape. Also note that the digit sequences as such are irrelevant as they depend on the units involved - with a suitable unit, the diagonal is maybe one kellicap long and the side length is irrational.

I will give you the same answer I gave to a friend some years ago (I don't know if it's right... How can we know? Is this question about mathematics?):

Irrational numbers are the result of calculations, not of measurements with rulers. These calculations are based on axioms that were extrapolated from experience and influenced by human intuition.

We can use Euclid's geometry in the real world very well up to a point, but it was discovered that Euclidean geometry is not always the simplest to be used in the real world (and if one insists in using it, many physical theories become much more complicated). Geometry started as somewhat of a physical theory, since its axioms are based on experience and on human intuition of how things "should be" in the real world. The reason why so many doubted the parallel postulate is because it involved extending a line segment "indefinitely", and this is not something we can test empirically, even for a single case (and if I remember rightly, the ancient Greeks believed the "Universe" was finite). (See Non-Euclidean geometry). Even the notion of perimeter of a "physical object" doesn't make sense in "reality" (see Coastline paradox , and if one also think about the discovery of atoms and about many other new theories and discoveries that may appear in the future, things start to become really complicated if you want mathematics to be in accord with "reality"...). What is "reality"? We try to model "reality", but how can you be sure that your model is in accord with "reality"? I think this is impossible, but at least sometimes we can find useful approximations (Euclidean geometry and Newtonian mechanics are nowadays considered to be just approximations). One of the beautiful things about mathematics is that many times mathematicians don't care much about "reality", and their ideas find applications in physics anyway. Is there any situation in physics where we need a better approximation than 1.4142135623730950488016887242 for $\sqrt{2}$? And the fact that some things don't make sense to a human doesn't mean they can't be true in nature, because "nature has no obligation to make sense to you" (this was the favorite answer of an anonymous guy on the Internet when people complained about quantum mechanics and general relativity don't making sense).

A question related to yours was asked today: Calculus in a discrete universe

Sorry for my English (it's not my native language).

We would like to conjecture that two important mechanisms are involved with a mathematical description of the material world:

• Abstraction, giving rise to Science
• Idealization, giving rise to Mathematics

Schematically:

Abstraction

Etymology. Perfect passive participle of abstraho ("draw away from"). Certain properties of the whole thing are preserved in the process of Abstraction:

• abstract
• abstractus, Latin

We shall argue that Abstraction is not a mathematical but rather a physical activity. It's already done by our senses. Our eyes can see the light, as it is casted back from a piece of paper. The same piece of paper can be felt by our fingertips. And when it is crumpled up, the sound of it will be heard by our ears. But eyes cannot hear sound, fingertips cannot see light. All these single perceptions of our senses have to work together. And even if we are not handicapped, the end-result is still an abstraction of reality as a whole, a part of it. None of our senses is capable, for example, to see ultraviolet colors, as some insects probably can.
But why should attention be restricted to the creations of Nature? Why not take a look at our own creations: human made Technology? Some cameras are capable to "see" in the infrared domain. Our radio telescopes are even capable to "see" the radio frequencies of far away galaxies. Far more common and well-known everyday abstractions of reality are performed, however, with measuring devices like rods for the abstraction of lengths, clocks for the abstraction of time intervals. But these measuring devices have become more and more self supporting these days. When coupled with digital computers, human interaction is hardly needed anymore. All such apparatus make an abstraction of reality, which is thus a physical and not a mental process.

Idealization

This raises an obvious question: where does"real" mathematics start then? Answer: with the next step: Idealization. Idealization could be characterized as the true mathematical activity. Idealization is where imagination and phantasy come in. And it turns out that infinity is often a keyword accompanying this process.
Many challenging idealizations are found in theories of Physics. In "The Theory of Heat Radiation" by Max Planck, Wien's Displacement Law (chapter III) can only be derived under the following conditions: if the black radiation contained in a perfectly evacuated cavity with perfectly reflecting walls is compressed or expanded adiabatically and infinitely slowly. Idealized Carnot engines are used in Thermodynamics for defining that stunning but indispensable quantity, called Entropy. And the list goes on and on. How about ideal, frictionless movement in mechanics? How about ideal pendulums, which can only exist through a sine with (almost) zero amplitude. As soon as physicists have devised their mathematical model, then it can be said that idealization has been accomplished a great deal. One should become alerted as soon as the following phrases are being uttered: "perfect", "ideal", "zero", but especially "infinitely", like in "infinitely slow" or "infinitely thin". It can safely be concluded that Infinities are invariably associated with Idealizations.
Concerning mathematics, among the most classical examples of idealization, without doubt, is good old Euclidean Geometry - where we should start to consider geometry in its original setting: classical Greek philosophy. Remember utterings like: a point has no size, a line is infinitely thin, parallel lines intersect at infinity. The concept of an irrational number wouldn't have emerged if Euclidean geometry hadn't been there in the first place.

So what is $\sqrt{2}$ ? It's an idealization. It's an idealization of numerous abstractions, abstractions of numbers like $1.414213562373$ or $1.14$ or $99/70$ , as measured for example with a rod when trying to determine the length of the hypotenuse of a right triangle with legs of length $1$ meters.
$\sqrt{2}$ doesn't exist in the real world. But neither does an ideal triangle. All you can have in reality is "wooden triangles with legs not exactly $1$ meters and a main angle not exactly right".
A mathematics with such triangles would be extremely clumsy, so we are happy that idealized triangles can be imagined. It may be concluded that your "fixed length" is an idealization, an illusion as well. This resolves the "paradox" that a "fixed length" could not be represented by the infinitely many decimals of an irrational number. Both the length and the number are not real.