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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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18  
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

6 Answers 6

up vote 33 down vote accepted

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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There are also some interesting properties of polynomial rings that are similar to those of integers, specifically with respect to factors (useful for finding roots). I personally don't know much about it, but find it interesting and hope to learn more. –  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

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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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

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 non Riemann integrable function.

Moreover, the fact that the rationals are countable helps in proving the Lindelöf Covering theorem.

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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