Definition:Real Number/Number Line Definition
Definition
A real number is defined as a number which is identified with a point on the real number line.
Real Number Line
From the Cantor-Dedekind Hypothesis, 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 numbers is consistent with the length of the line between those two points.
Notation
While the symbol $\R$ is the current standard symbol used to denote the set of real numbers, variants are commonly seen.
For example: $\mathbf R$, $\RR$ and $\mathfrak R$, or even just $R$.
Equality of Real Numbers
Two real numbers are defined as being equal if and only if they correspond to the same point on the real number line.
Also known as
When the term number is used in general discourse, it is often tacitly understood as meaning real number.
They are sometimes referred to in the pedagogical context as ordinary numbers, so as to distinguish them from complex numbers
However, depending on the context, the word number may also be taken to mean integer or natural number.
Hence it is wise to be specific.
Also see
- Results about real numbers can be found here.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.1$. Sets: Example $2$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers