Definition:Real Number/Digit Sequence
Make a point to note what happens when the number ends in an infinite string of $9$s
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Definition
Let $b \in \N_{>1}$ be a given natural number which is greater than $1$.
The set of real numbers can be expressed as the set of all sequences of digits:
- $z = \sqbrk {a_n a_{n - 1} \dotsm a_2 a_1 a_0 \cdotp d_1 d_2 \dotsm d_{m - 1} d_m d_{m + 1} \dotsm}$
such that:
- $0 \le a_j < b$ and $0 \le d_k < b$ for all $j$ and $k$
- $\displaystyle z = \sum_{0 \mathop \le j \le n} a_j b^j + \sum_{k \mathop \ge 1} d_k b^{-k}$
It is usual for $b$ to be $10$.
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 $R$.
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
Add a link to a page justifying this -- based on Basis Representation Theorem and the similar result for the fractional part
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- Results about real numbers can be found here.
Sources
- 1935: E.T. Copson: An Introduction to the Theory of Functions of a Complex Variable ... (next): Chapter $\text {I}$. Complex Numbers: $1.1$ The introduction of complex numbers into algebra
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.38$. Decimals
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 1$. Introduction
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction