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I have a square stone slab 1 metre by metre, by the Pythagorean identity the diagonal from one corner to another is given as $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone explain how it is possible for a non-terminating (and non repeating) number to be represented as a fixed length in reality?

edit: I'm sorry if the tags are wrong

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What makes you think irrational numbers can't manifest themselves in reality? –  Daniel Rust Aug 12 at 13:05
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@DanielRust I have problem seeing an irrational number as a fixed length. How can it be represented by a finite length if the number doesn't end? –  seeker Aug 12 at 13:06
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Then you also have trouble seeing a non-terminating rational number like $1/3=0.333\ldots$ as a fixed length? Just because you have trouble 'seeing it' doesn't mean it can't occur. –  Daniel Rust Aug 12 at 13:07
    
@danielrust, I didn't say they don't occur. And yes I have trouble visualising 0.3333.. too –  seeker Aug 12 at 13:12
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Side note. Here is a number that is "non-terminating": 0.999999... Do you also have difficulties visualising it? It turns out to be another way to write 1. –  Jubobs Aug 12 at 19:10

7 Answers 7

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 $1.5$.

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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.

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Uncertainties everywhere. –  metacompactness Aug 12 at 19:29
    
quantum world... –  N0ir Aug 12 at 22:05
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$1\;meter=\frac{1}{\sqrt2}\;kellicap$. Suddenly it's the diagonal that's a single unit, and the sides have irrational length. But nothing has changed about the stone slab! Units are arbitrary. –  Brian S Aug 12 at 23:09

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).

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could someone explain how it is possible for a non-terminating (and non repeating) number to be represented as a fixed length in reality?

I think you got this wrong: the number $\sqrt{2}$ isn't represented by some length. Euclidean space (or physical reality) was there first. We use numbers to represent things in euclidean space. If a number system we choose can't represent some of those lengths, why would that change the length of the triangle side?

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Euclidean space doesn't have physical reality either. –  Buddha Aug 13 at 5:11
    
@Buddha: I never said is was. –  nikie Aug 13 at 6:09

The diagonal has irrational length. The length is fixed but its representation in a a base 10 system has an infinite number of digits. What does that mean? Well, if you used a scale that could measure lengths up to 3 decimal places, you'd find that the diagonal is a little longer than 1.414 but shorter than 1.415. If you used another scale that could measure up to 4 decimal places, the diagonal's length would fall between 1.4142 and 1.4143. You could keep using newer scales with finer precision but the length would never coincide with a mark on the scale. Because your scale is divided and repeatedly subdivided using decimal system you'd never find an exact match.

As another example consider the decimal number 0.3 that has only one digit. The same number when represented in base 2, i.e. as binary has an infinite number of bits (binary digits 0 and 1); 0.0100(1100...)

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Rational numbers are a mathematical concept. In a physical world there cannot be such thing as rational vs irrational lengths for two reasons.

The first is the question of units of measurement, but the OP is seemingly aware of it.

The second reason is that an exact value of a physical quantity doesn’t make sense without specifying a measurement procedure. Does a physical measurement procedure exist, for the length, that can discreet rationals from irrationals?

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The diagonal is long by 1 m + 4 dm + 1 cm + 4 mm + 213 µm + 562 nm + 373 pm + 95 fm...

You indeed accumulate "an infinity" of lengths, but the sum converges very quickly and is finite (not counting the fact that you quickly get to the size of the quarks).

The "non-repeatingness" is justified by the fact that any repeating number can be expressed as a fraction.

For instance, 1.4142141421414214142141421... is 141420/99999. (Take that number, multiply it by 100000 and subtract the original; all decimals cancel out.)

And $\sqrt2$ cannot be a fraction. If it were, let $\sqrt2=\frac pq$, with $p$ or $q$ odd (if both are even, you can simplify). Then $p^2=2q^2$, so that $p$ is even (the square of an odd number cannot be even). $p$ being even, $p^2$ is a multiple of $4$, so that $q^2$ is even, and $q$ is even !

As $\sqrt2$ cannot be a fraction, it cannot be a periodic number.

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I'm not sure what anything past the word 'quarks' has to do with answering the OP's question. –  Daniel Rust Aug 12 at 13:41
    
@Daniel Rust: feel free to downvote if that's your feeling. –  Yves Daoust Aug 12 at 13:49
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I think the first two paragraphs are useful, the impromptu proof that $\sqrt{2}$ is irrational just seems a bit misplaced - the OP already knows it is irrational. –  Daniel Rust Aug 12 at 13:59

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