# 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?

• What makes you think irrational numbers can't manifest themselves in reality? – Dan Rust Aug 12 '14 at 13:05
• @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 '14 at 13:06
• 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. – Dan Rust Aug 12 '14 at 13:07
• 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 '14 at 19:10
• What more details are you looking for? The purpose of the bounty is unclear, perhaps because the question is not quite clear itself. Nothing in reality is accurate, there is nothing of length "exactly" $1$ meter (even the official rods by which "meter" is defined shrink and expand when time and space conditions change); let alone a stone slab which is exactly a perfect square whose sides are exactly $1$ meter. What other answers do you look for? – Asaf Karagila Jan 24 '15 at 14:53

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.

• Uncertainties everywhere. – Paracosmiste Aug 12 '14 at 19:29
• quantum world... – OutFall Aug 12 '14 at 22:05
• $1\;meter=\frac{1}{\sqrt2}\;kellicap$. 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 '14 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).

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:

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.

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?

• Euclidean space doesn't have physical reality either. – Buddha Aug 13 '14 at 5:11
• @Buddha: I never said is was. – nikie Aug 13 '14 at 6:09

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?

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...)

Unless I'm mistaken, I think the issue you are addressing doesn't pertain so much to the problem of representation accuracy of the physical world as to the apparently paradoxical nature of having an infinitely-long representation of a finite amount, even when that amount is itself an idealization as finely grained as we want it to be.

In the case of a square and its diagonals, it's not true that the diagonal length is irrational. It is however true, if the square's side length is represented by a rational number, that the diagonal length is represented by an irrational number. This shows that irrationality is not a characteristic of the measured object, but of the measurement (the number). It is an issue deriving from the representation system we are using. Other representation systems do not have that particular issue, for instance, descriptive geometry.

Something similar can be said about the infinite decimal expansion on the representation of the quantity we usually refer to as $\sqrt{2}$. It just so happens that a particular representation of a number is based on the idea of an infinite number of integer places, followed by a decimal point and then an infinite number of decimal places. In this sense, all numbers expressed with this representation have an infinite decimal expansion, like ${15}/{2} = 7.500(...)$. For practical reasons, as it doesn't impact the representation's accuracy, we omit the leftmost zeros at the integer part and the rightmost zeros at the decimal part, but the infinite places are always there. This infinity is a characteristic of the representation, not of the represented number, so there is no contradiction in having an infinitely-long representation of something finite.

Note that your $1$ meter measurement itself is not perfectly accurate. Even if we reach the atomic level, we can never make a 2 series of atoms measure the exact same length.

1. We will need highly advanced tools to do so.
2. We will have to use the exact measurement as the internationally approved metal rod for 1 metre.
3. We will need to use the same element, and keep the same number of atoms, amount of internal chemical force, magnitude of gravity at place of measuring, etc.
4. Atoms are continuously vibrating, so we can only take a mean length.
5. Einstein's physics says that we can not do two things at the same point on time, or have the exact same placement of a finite space at two different points in time and regions of space.

This leads us to conclude that neither $1$ metre, nor $\sqrt2$ metre, is exactly in the physical world as in the euclidean one.

The important fact here is that you can't have EXACLTY a $1$ meter slab because there is (even if very small) an "uncertainty" with the length of the slab. In fact we could measure the length of the slab infinitely many times but it won't never be $1$ exactly (for more details see this numberphile video). So if we can't have a length $1$ we can't have $\sqrt2$ and mathematics is saved.

You can prove that $$\sqrt2$$ is irrational using statements only about real numbers once you define what a real number is and those operations on it, without even resorting to the Pythagorean theorem from which you can deduce that the diagonal of a square of unit length has length $$\sqrt2$$ in Euclidean geometry. I don't like a proof using the Pythagorean theorem because people treat it as an undefined concept defined over $$\mathbb{R}^2$$ and just assume that the distance from the origin to any point which I will now call the distance of that point satisfies certain properties:

• The distance of every point in $$\mathbb{R}^2$$ is a nonnegative real number
• For every nonnegative real number $$r$$, $$d(r, 0) = r$$
• For every point $$(x, y)$$, $$d(x, -y) = d(x, y)$$
• For any points $$(x, y)$$ and $$(z, w)$$, $$d(xz - yw, xw + yz) = d(x, y)d(z, w)$$

from which they deduce that $$\forall x \in \mathbb{R}\forall y \in \mathbb{R} d(x, y) = \sqrt{x^2 + y^2}$$. It turns out that once you construct the real numbers properly which requires dropping the assumption that all real numbers are rational and show that it's a complete ordered field as I will do later, you actually can define $$d(x, y)$$ as $$\sqrt{x^2 + y^2}$$ and prove that it in fact does satisfy those properties. Although we can't make accurate measurements, the Euclidean metric still exists so $$d(1, 1)$$ still exists and equals $$\sqrt2$$ which is irrational.

Dropping the topic of distance in $$\mathbb{R}^2$$ entirely, we can construct the rational numbers directly from the integers so you might think the rational numbers are all the numbers there are. So far, that's no problem. People also think $$(\mathbb{R}, 1, 0, +, \times, \leq)$$ is a complete ordered field which is still no problem. For myself though, I actually feel the need to explicitly construct them and show that it's a complete ordered field. The problem only happens when you combine those claims and define a rational number as a real number that can be expressed as $$p\div q$$ for some integer $$p$$ and nonzero integer $$q$$ then add the assumption that all of them can be expressed that way. The statement that there is no rational solution to $$x^2 = 2$$ can be reduced to a statement of pure number theory and be proven in a system of pure number theory and there seems to be no problem with the statement about number theory it's reduced to.

We can also construct the dyadic rationals, the numbers that are terminating in binary then construct the real numbers from the Dedekind cuts of them and show that it's a complete ordered field which is unique up to isomorphism. That shows that if you want to define the real numbers with those operations to be any complete ordered field, it will be isomorphic to that one. Defining real numbers by Cauchy sequences seems to be a very hard and confusing topic because I would be like, how should I do it. Once they've already been constructed in the easier way, I don't find it nearly as hard to then prove that all Cauchy sequences approach a real number. From the property of being a complete ordered field, you can also show that $$\sqrt2$$ exists in that field, and there is no reason to assume that for all numbers in that field, there exists an integer $$p$$ and a nonzero integer $$q$$ such that $$p\div q$$ is that number. In fact, by Cantor's diagonal argument, there are irrational numbers. Since it can also be proven that there is no rational solution to $$x^2 = 2$$, that means $$\sqrt2$$ is irrational.

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.

• I'm not sure what anything past the word 'quarks' has to do with answering the OP's question. – Dan Rust Aug 12 '14 at 13:41
• @Daniel Rust: feel free to downvote if that's your feeling. – Yves Daoust Aug 12 '14 at 13:49
• 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. – Dan Rust Aug 12 '14 at 13:59

All the problem is that we are living in a finite universe and we have to deal with Einstein's relativity. As mentioned by others mathematics is concerned with idealization. With perfect precision, if it would exist, we could draw a triangle with $\sqrt{2}$ diagonal. In many physical problems, requiring infinite precision or realization of an ideal phenomenon requires infinities to be involved in reality. Since this is not possible, at least in this universe, realization of an ideal $\sqrt{2}$ is a simple contradiction.

Another nice example is the impulse response of a linear time invariant system from signal processing. If the system is continuous time, then to characterize the system response to all sort of inputs, we need to see its response to a dirac delta function. Why? because dirac delta is flat over the whole spectrum. Such a function is even ill defined mathematically, if we don't use a nice abstraction for example measure theory. The question is then how to realize dirac delta function in reality. Now again we need something infinite, which we unfortunately don't have.

I think Cantor is perfectly the right person to resolve this matter but we don't have him either, due to same reasons, no infinite time life span.

If you're trying to find mathematical sense in the physical world, then let me ask you this:

How do you measure length?

Do you count the number of molecules? atoms? quarks?

Tell me what you basic unit is, and I will ask you to represent $\frac12$ with it.

By the way, these numbers are called irrational because they don't make (rational) sense.

In fact, in order to generate an irrational number, you need to apply an infinite number of arithmetic operations on one or more rational numbers.

For example, $\sqrt{2}=\sum\limits_{n=0}^{\infty}\frac{(-1)^n(2n)!}{(4^n)(n!)^2(1-2n)}$... That doesn't make much sense either now, does it?

• The Latin word irrational comes from is ratio, which just mean the same as in English. An irrational number is one which is not the ratio of two integers, not one that doesn't make sense. – egreg Feb 20 '15 at 17:21

Measurement can never be accurate how advance the technologies may be.. In this case, Let's assume that you are measuring the diagonal value by a scale. The diagonal is of $\sqrt2$. That means value lies in between 1 and 2. If you have a more advanced measuring device you can measure up to 1.4 and 1.5. The actual values lies between these two. A more advance measuring device will result in between 1.40 and 1.42..

To answer your question, length of the slab is finite. But there is always next decimal value up to which you can measure.

Conclusion: Irrational numbers are the subset of Real numbers always.

For mathematical sense in the mathematical world, perhaps any non-terminating decimal, rational or irrational could be viewed as a limit that converges but never becomes a fixed value in the same way the limit of 1/x as x approaches positive infinity. Here is a case of a finite limit at infinity, but remember this limit, zero, is never realized on the graph of the function. There is no coordinate (+infinity, zero) that belongs to the function for zero is not a number. So, the mathematical "heretics" who reject the idea of infinity outright, possibly have to reject non-terminating decimals for the same reason they reject the idea of hyper-reals. Instead of having an endless string of numbers to the left of the decimal point, you have an endless string of numbers to the right of the decimal point. Infinite series and sequences proves convergence of certain infinite sums, but there is arguably no way to tell if this limit is actually being approached infinitesimally like the limit stated earlier or if the limit is realized. If you reject infinities in some sense at least you may have to reject the irrationals and non-terminating rationals as rigorously real numbers. The good news? The number line will still look continuous. A detectable discontinuity would require removing any two points and all those in between, indeed infinitely many. Since there are no consecutive numbers even among strictly terminating decimals, you'll never see what you're missing! ...and maybe because it wasn't there to begin with. Slapping an equals sign between an infinite sequence that converges and its limit may actually overlook an indeterminate aspect of the nature of anything we call a number that isn't strictly finite, so anything that doesn't end in a final string of zeroes. -heretic

If I was going to get upset about the 'reality' of any real numbers that are irrational, the square root of $$2$$ would not be on the top of my list.

Put on your "I am Pythagoras with ruler and compass in hand' hat. Using mind experiments you construct all the positive rational numbers on a 'ruled line'. Working in the plane, you construct the length $$\sqrt 2$$ and mentally place it as an offset from $$0$$ on the line. You know it is something new, but you can get arbitrarily close approximations with rational numbers.

You don't know about decimal representations and when you are told how they just 'go on and on' you yawn - you've already come to terms with the 'reality' of the situation.

As you look into the nature of real numbers you might learn that there is something called a computable number; the $$\sqrt 2$$ is a computable number.

Here is an irrational number that would be on my list of 'hard to fathom' irrational numbers.

Keep flipping a coin with outcomes $$H = 1$$ and $$T=0$$,

$$\quad (a_1, a_2, a_3, \dots)$$

$$\tag 1 a = \sum_{n=1}^\infty \frac{a_n}{2^n}$$

Now that is not easy to imagine, but since we don't want 'gaps' in $$\mathbb R$$ it is included as an idealization.