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 $k$
$\displaystyle z = \sum_{0 \mathop \le j \le n} a_j b^j + \sum_{k \mathop \ge 1} d_k b^{-k}$

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: $\mathbf R$ and $\RR$, 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

  • Results about real numbers can be found here.


Sources