Definition:Real Number/Digit Sequence

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This article is complete as far as it goes, but it could do with expansion.
In particular: 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 $k$
$\ds z = \sum_{j \mathop = 0}^n a_j b^j + \sum_{k \mathop = 1}^\infty d_k b^{-k}$

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


This article is complete as far as it goes, but it could do with expansion.
In particular: Add a link to a page justifying this -- based on Basis Representation Theorem and the similar result for the fractional part
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


  • Results about real numbers can be found here.


Sources

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