Definition:Real Number
1. Improve informal description. 2. Construction of reals between 0 and 1 as set of digit sequences not ending in zeros or not ending in nines. 3. Construction from Dedekind cuts. 4. Abstract definition as Dedekind completion of rationals. 5. Axiomatic definition.
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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: $\forall a \in \R \setminus \left\{{0}\right\}: \exists ! a^{-1} \in \R: a \times \left({a^{-1}})\right) = 1 = \left({a^{-1}}\right) \times a$ (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.
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
- 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
- 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): $\text{II}: \ 28$
- 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): $\S 2 \ \text{(b)}$
- 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
