# Why must we distinguish between rational and irrational numbers?

The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians make a distinction between these two types of numbers? Why are integers special anyway, other than being historically significant?

Is there any property that sets rational or irrational numbers apart, other than the way they are written in our number system?

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Why do we make distinctions between $7$ and $12$ in the first place, except for historical significance? – Hagen von Eitzen Dec 9 '12 at 20:11

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You have a point. In United States law, it was eventually confirmed that "separate but equal" was not really equal at all. – Will Jagy Dec 9 '12 at 22:00
Did you ever wonder how to charcterise the real numbers among the complex numbers (without using real/imaginary parts, complex conjugation and such, which notions need real numbers to be defined in the first place)? – Marc van Leeuwen Dec 9 '12 at 23:03
"Rational numbers can be written as the ratio of two integers, and irrational numbers can't." I'm pretty sure that's not quite how it's defined; you're assuming all numbers are real. – Joe Z. Dec 10 '12 at 23:51

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Well, if you don't want the Pythagoreans to drown you, it is important not to make this distinction... – Asaf Karagila Dec 12 '12 at 20:26

## 7 Answers

Here's one example of where the difference between rational numbers and irrational numbers matters. Consider a circle of circumference $1$ (in any units you choose), and suppose we have an ant (of infinitesimal size, of course) on the circle that moves forward by $f$ instantaneously once per second. Then the ant will return to its starting point if and only if $f$ is a rational number.

Maybe that was a little contrived. How about this instead? Consider an infinite square lattice with a chosen point $O$. Choose another point $P$ and draw the line segment $O P$. Pick an angle $\theta$ and draw a line $L$ starting from $O$ so that the angle between $L$ and $O P$ is $\theta$. Then, the line $L$ passes through a lattice point other than $O$ if and only if $\tan \theta$ is rational.

In general the difference between rational and irrational becomes most apparent when you have some kind of periodicity in space or time, as in the examples above.

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If "moves forward by $f$" means "travels distance $f$ along the circle", then I think your claim is false since the circumference of the circle is irrational. In this case, my feeling is $f$ need be a rational multiple of $\pi$. – Austin Mohr Dec 10 '12 at 2:23

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@AustinMohr: My first reaction was the same as yours, but no, the answer specifies "a circle of circumference 1", so the circumference is not irrational. – ruakh Dec 10 '12 at 4:29

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@ruakh Thanks for the clarification. I guess it's time I learned to read... – Austin Mohr Dec 10 '12 at 4:44

One thing is in how you construct them. Starting from the natural numbers (and $0$) you construct the integers by saying that $\mathbb{Z}$ is the smallest set that contains the naturals and is a group under addition. Similarly, the rationals $\mathbb{Q}$ is the smallest set containing $\mathbb{Z}$ that forms a group under multiplication (when $0$ is taken out). The reals $\mathbb{R}$ can then be constructed by defining it to be the smallest set containing $\mathbb{Q}$ in which every bounded set has a least upper bound.

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Thanks. A follow-up: How are the natural numbers constructed, or do we only start from them arbitrarily? – mage Dec 9 '12 at 20:19

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Start with nothing (a.k.a. $0$), and keep adding one... – Zhen Lin Dec 9 '12 at 20:19

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@ZhenLin: That will eventually construct each individual natural number (if you live long enough), but not the set of natural numbers. That is why one needs an axiom of infinity. – Marc van Leeuwen Dec 9 '12 at 22:56

Another example where they differ. Take any polynomial $a_nx^n + a_{n-1}x^{n-1} + \ldots a_1x + a_0$ with integer coefficients. You can easily find all rational roots of that polynomial: Any rational root $\frac{p}{q}$ (with $p$, $q$ relativly prime integers) must satisfy: $p$ divides $a_0$ and $q$ divides $a_n$. So there are finitely many possibilities which you can check by hand.

It's not easy in general to find irrational roots.

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– eacousineau Dec 10 '12 at 2:21

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Most important properties that polynomials (in one variable over field) share with integers are: division algorithm, being principal ideal domain, and being unique factorisation domain. You can learn a lot about it in e.g. Hungerford's Algebra. – rafaelm Dec 10 '12 at 2:37

From an algebraic perspective, if you believe in the "naturalness" of $\mathbb{R}$ or $\mathbb{C}$, then $\mathbb{Q}$ sits naturally inside them as the minimal subfield of characteristic zero. Topologically it is also worth noting that $\mathbb{Q}$ is dense in $\mathbb{R}$. These are just two properties that illustrate that $\mathbb{Q}$ is really a natural and interesting set.

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$\mathbb{Q}$ sits inside of every field of characteristic zero (isomorphically). That makes it very important from an algebraic perspective. – asmeurer Dec 10 '12 at 1:34

Why are integers special anyway, other than being historically significant?

From a technical perspective, it is good to know that we can perform exact computations on rationals. On irrational numbers, you have to approximate unless you restrict yourself to a suitable subset of irrational numbers, like e.g. an extension like $\mathbb Q[\sqrt2]$ or the algebraic numbers $\bar{\mathbb Q}$. But even when performing computations with such fields, many operations are internally formulated on rational numbers, which in turn are formulated on integers.

So in a sense, anything you can express with rational numbers is something which you can compute in a straight-forward way (although still using rationals based on arbitrary length integers, as opposed to floating point numbers) without loosing exactness. Anything involving reals has to fail there: you cannot even enter a single irrational real number unless you do so using a formula describing how to compute it from rational numbers.

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The Dirichlet function, which holds different values for rationals and irrationals is an example of a function defined for each real x, but continuous nowhere. This wouldn't have been true were the set of rationals not dense. This function is a straightforward example of a nonintegrable function.

Moreover, the fact that the rationals are countable helps in proving the Lindelof Covering theorem. http://en.wikipedia.org/wiki/Lindel%C3%B6f's_lemma

Also, the construction of the field of fractions of an integral domain is motivated from the construction of rationals from the integers.

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I think irrational numbers are for specifying an amount of something accurately(or far more specific if not accurate) up to infinitely small scale and can never be reached by getting the ratio of any 2 whole numbers(integers). Examples: amount of liquid, exact volume of a container, exact length of a rope around a drum, density of moisture in the air. (Though, when computing these quantity, we get rational numbers. But that can't be easily know.)

Rational numbers occur when counting some whole/countable objects like apple, orange, houses, trees, ballpens, etc, are involve, which getting the count of an object as a whole(ignoring their sizes) are more important than getting their volume, or the space they occupy nor their weight.

Example: "how many coins do you have right now?" ask for a rational number while "how much profit you earned in your bank last year" ask for a quantity that has a whole number count and a an amount less than a whole number to specify exactness.

Non-integer rational numbers still involves whole numbers. Example: I have apples double your number of apples, or your apples are just 50% of my apples. We know that 50% comes from 50/100 or 1/2. If we know these, 50% or 0.5 is obviously a rational number. They are accurate if expressed as fractions compared to floating point notation which involves rounding off when writing them to save space.

I think, rational is more on finite measurement while irrational is more on infinite measurement(infinitely small approximations). I'm just answering based on my opinion and have no complete info about these two set of numbers. But for informal explanation that is my answer. I hope it helps.

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This is not so true. Saying that there are exactly 4 of something is no less exact than saying that the ratio between the circumference and diameter is $\pi$. – mixedmath♦ Dec 10 '12 at 7:56
@mixedmath, have you received the email I sent to the six mods for whom I could find email addresses? If not, could you please send me an email from some account you use? Note, it is just, as requested, the original text and IP address for one of the cheaters in the "sticks and stones" idiocy. – Will Jagy Dec 10 '12 at 19:19
@Will: I have - but if in the future you need to contact me, you can send me email at mixedmath [at] gmail [dot] com. – mixedmath♦ Dec 10 '12 at 20:04
@mixedmath, good, sent anyway so my browser remembers your address. – Will Jagy Dec 10 '12 at 20:25
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