# Definition:Real Number/Digit Sequence

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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 $k$
- $\ds z = \sum_{j \mathop = 0}^n a_j b^j + \sum_{k \mathop = 1}^\infty d_k b^{-k}$

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

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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