Definition:Real Number
Contents
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.
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
- James R. Munkres: Topology (2nd ed., 2000)... (previous)... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers
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
- James M. Hyslop: Infinite Series (1942)... (previous)... (next): $\S 2$: Functions
- Murray R. Spiegel: Theory and Problems of Complex Variables (1964)... (previous)... (next): $1$: Complex Numbers: The Real Number System: $4$
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$: Example $2$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 1$
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967)... (previous)... (next): Introduction: Special Symbols
- Ian D. Macdonald: The Theory of Groups (1968)... (previous)... (next): Appendix: Elementary set and number theory
- 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)... (previous)... (next): $\S 1.2$: Some examples of rings: Ring Example $4$
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28$
- Robert H. Kasriel: Undergraduate Topology (1971)... (previous)... (next): $\S 1.8$: Collections of Sets: Definition $8.4$
- T.S. Blyth: Set Theory and Abstract Algebra (1975)... (previous)... (next): $\S 1$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975)... (previous)... (next): 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): $\S 2 \ \text{(b)}$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996)... (previous)... (next): Appendix $\text{A}.1$: Sets
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000)... (previous)... (next): $\S 1.2.5$: An aside: proof by contradiction