# What is a Real Number?

I believe I'm over thinking it, but I want to be 100% sure.

A Real Number is any number, correct? Whether it be an integer or something else.

It's the set $\mathbb R$ from $(-\infty, +\infty)$ correct?

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Well, not really. i is a number, but isn't a real number. Also, it's vague what you are saying. Reals numbers are defined as having the obvious properties of rational numbers, but with one added axiom called completeness. – simplicity Nov 2 '11 at 23:57
@simplicity : But $i$ is not in the interval $(-\infty,+\infty)$. – Michael Hardy Nov 3 '11 at 0:51
where does i lie? – MathMathCookie Nov 3 '11 at 1:58

A common way to think about the real numbers as the set of every point on the number line between $-\infty$ and $+\infty$, including irrational numbers (like $\sqrt{2}$) and transcendental numbers (like $\pi$ and $e$).

For most people this is totally sufficient, though this doesn't really come close to telling you what they are, and it's circular anyway (how do you define the number line..?)

Perhaps a more illuminating way to think about the real numbers is as the rational numbers (i.e. the numbers of the form $p/q$ for $p,q$ integers and $q\neq 0$) with all of the 'gaps' filled in. In fact, that's the layman's description of one of the formal ways of specifying the real numbers -- as the completion of the rationals.

When we say that the rationals aren't complete, we mean that there are sequences of rational numbers that converge to (get closer and closer to) a number that isn't a rational. The sequence

$$\frac{1}{1}, \frac{14}{10}, \frac{141}{100}, \frac{1414}{1000}, \dots$$

is an example of such a sequence -- every number in the sequence is rational, but the limit is the irrational number $\sqrt{2}$.

This can't happen with the real numbers, because 'all the gaps are filled in'. Every sequence of real numbers that converges is guaranteed to converge to a real number.

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+1, for an interesting explanation indeed! – NoChance Nov 3 '11 at 0:04
You might want to add that the bulk of the real numbers are numbers that you can't actually define in anyway. Indescribable numbers. – simplicity Nov 3 '11 at 0:15
Thanks, a very detailed and informative explanation! – switz Nov 3 '11 at 0:23
"Every sequence of real numbers that converges is guaranteed to converge to a real number" - maybe a better phrasing is every cauchy sequence is convergent, you can't say "Every sequence of real numbers that converges" since if $\sqrt{2}$ is 'not there' the sequence you havw in your answer (is cauchy) not convergent – Belgi Aug 9 '12 at 6:45

It seems most people are complicating the answer you are looking for.

Yes, you are correct, it is the set of all numbers on the number line, i.e any number in (-∞, +∞).

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That's not true. If you don't assume completeness you will get every number in the number line. But, with bits missing out like irrational numbers. – simplicity Nov 3 '11 at 0:46
If you ask yourself what is the number line, you'll see this is self-referential. – Adam Zalcman Nov 3 '11 at 0:51
@AdamZalcman: It doesn't matter. The OP is not asking for a definition, only clarification, so it is permitted to be self-referential. (Just like dictionary entries often are circular if you follow them deep enough, but still help in our understanding of a term.) – ShreevatsaR Aug 9 '12 at 6:57

Yes, the integers and fractions are real numbers, as are things that cannot be expressed as fractions, such as $\sqrt 2$ and $\pi$. That is, the real numbers means the entire number line. On the other hand, $\infty$ and $-\infty$ are not real numbers.

The reason they are called "real numbers" as opposed to simply "numbers" is that higher mathematics works with various other kinds of things we call "numbers" too -- but those things do not correspond to points on the number line. You can safely ignore this until you need to learn about complex numbers (or transfinite ordinal numbers, or cardinal numbers, or p-adic numbers or whatever).

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You can think of the set of real numbers as the set of all rational numbers closed under taking the limits of Cauchy sequences, i.e. all rational numbers plus all limits of Cauchy sequences you can build in the set.

However, not all numbers belong to the set of real numbers, see http://en.wikipedia.org/wiki/Complex_number.

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You can argue that complex numbers aren't numbers. I personally see complex numbers as just matrices. Which, in my view aren't numbers. – simplicity Nov 3 '11 at 0:06
That's semantics, not mathematics. – Chris Taylor Nov 3 '11 at 0:12
@simplicity: You might consider a new moniker. (Yes, I am aware of matrix representations of complex numbers and quaternions) – The Chaz 2.0 Nov 3 '11 at 0:12
@simplicity: We already know which algebraic structure we're in, so there is no possibility of confusion with other notions of number and your point about semantics does not obtain. Moreover, as Henning states in chat, you yourself called $i$ a "number" in your first comment to the question. Ultimately, mathematics at this sort of level all boils down to study of various algebraic structures, wherein the ones most closely connected with the original notions of numbers and the trending generalizations thereof receive the label of number. – anon Nov 3 '11 at 0:31
Some might argue that real numbers aren't numbers. They might see real numbers as just series. Which, in their view wouldn't be numbers. – Adam Zalcman Nov 3 '11 at 0:32

Your definition is a bit loose...See this link for a more accurate definition and some examples: