Definition:Real Number

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Informal Definition

Any number on the number line is referred to as a real number.

This includes more numbers than the set of rational numbers as $\sqrt{2}$ for example is not rational.

The set of real numbers is denoted $\R$.


Formal Definition

Consider the set of rational numbers, $\Q$.

For any two Cauchy sequences of rational numbers $X = \left \langle {x_n} \right \rangle, Y = \left \langle {y_n} \right \rangle$, define an equivalence relation between the two as:

$X \equiv Y \iff \forall \epsilon > 0: \exists n \in \N: \forall i, j > n: \left|{x_i - y_j}\right| < \epsilon$


The real numbers are the set of all equivalence classes $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.


The set of real numbers is denoted $\R$.


Operations on Real Numbers

We interpret the following symbols:

Negative: $\forall a \in \R: \exists ! \left({-a}\right) \in \R: a + \left({-a}\right) = 0$
Minus: $\forall a, b \in \R: a - b = a + \left({-b}\right)$
Reciprocal: $\forall a \in \R \setminus \left\{{0}\right\}: \exists ! a^{-1} \in \R: a \times \left({a^{-1}})\right) = 1 = \left({a^{-1}}\right) \times a$ (it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$)
Divided by: $\forall a, b \in \R \setminus \left\{{0}\right\}: a \div b = \dfrac a b = a / b = a \times \left({b^{-1}}\right)$


The validity of all these operations is justified by Real Numbers form Field.


Real Number Line

From Set of Real Numbers is Equivalent to Infinite Straight Line, the set of real numbers is isomorphic to any infinite straight line.

The real number line is an arbitrary infinite straight line each of whose points is identified with a real number such that the distance between any two real number is consistent with the length of the line between those two points.

RealNumberLine.png

Thus we can identify any (either physically drawn or imagined) line with the set of real numbers and thereby illustrate truths about the real numbers by means of diagrams.


Axiomatic Definition

Definition

Let $\left({R, +, \cdot, \le}\right)$ be a Dedekind complete totally ordered field.

Then $R$ is called the (field of) real numbers.


Real Number Axioms


Also defined as

Some sources additionaly specify that $\left({R, \le}\right)$ be densely ordered.

This condition, while conceptually important, is superfluous, by Totally Ordered Field is Densely Ordered.


Sources

Also denoted as

Variants on $\R$ are often seen, for example $\mathbf R$ and $\mathcal R$, or even just $R$.


Also see

  • Results about real numbers can be found here.


Sources