Definition:Real Number

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 $\R$.

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

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: $\forall a \in \R \setminus \left\{{0}\right\}: \exists ! a^{-1} \in \R: a \times \left({a^{-1}})\right) = 1 = \left({a^{-1}}\right) \times a$ (we often write $1/a$ or $\displaystyle \frac 1 a$ for $a^{-1}$)
• Divided by: $\displaystyle \forall a, b \in \R \setminus \left\{{0}\right\}: a \div b = \frac a b = a / b = a \times \left({b^{-1}}\right)$

The validity of all these operations is guaranteed by the fact that the real numbers form a field‎.

Real Number Line

• It can be shown (and intuitively understood) that the set of real numbers is isomorphic to any infinite straight line in space.

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.

• It can be shown that the real number line is a metric space.

Hence the real number line is also a topological space.

• It can be shown that the real number line is a vector space.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$: Example $2$
• Seth Warner: Modern Algebra (1965): $\S 1$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.1$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $4$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(b)}$