Positional Numeral Systems

Definition: Positional Numeral System

A positional numeral system or place value system consists of a natural number $b>1$ and a set of $b-1$ distinct symbols $D=\{d_0,d_1,\dots ,d_{b-1}\}$.

We call $b$ the base or radix and we call $d_0,d_1,\dots ,d_{b-1}$ digits.

Theorem: Representing Integers

Let $b>1$ be a natural number.

For every $N\in {\mathbb{N}}_0$, there exists a finite sequence of [extended natural numbers] $a_0,a_1,\dots ,a_n$ such that $0\le a_k<b$ and

$$N=\sum _{k=0}^n{a}_k{b}^k$$

PROOF

TODO

The above theorem tells us that we can use any positional numeral system to represent all non-negative integers by identifying the digits $D=\{d_0,d_1,\dots ,d_{b-1}\}$ with the first $b$ numbers from ${\mathbb{N}}_0$. Then, each finite string $a_n{a}_{n-1}\cdots a_1{a}_0$ represents a number $N\in {\mathbb{N}}_0$.

NOTATION

When multiple positional numeral systems are used, it can be difficult to know in which system a particular number is given, especially when the systems use the same symbols for some digits. This is why we usually write the radix $b$ as a subscript at the end:

$$(a_n{a}_{n-1}\cdots a_1{a}_0{)}_b$$

Example: Base 2

Consider the positional numeral system with radix $b=2$ and digits $D=\{0,1\}$.

The string $11011_2$ represents the number

$$11011_2=1\cdot 2^4+1\cdot 2^3+0\cdot 2^2+1\cdot 2^1+1\cdots 2^0=16+8+0+2+1=27$$

Example: Base 5

Consider the positional numeral system with radix $b=5$ and digits $D=\{0,1,2,3,4\}$.

The string $3402_5$ represents the number

$$\begin{array}{rl}3402& =3\cdot 5^3+4\cdot 5^2+0\cdot 5^1+2\cdot 5^0\\ & =375+100+0+2=477\end{array}$$

Example: Base 16

Consider the positional numeral system with radix $b=16$ and digits $D=\{0,1,2,3,4,5,6,7,8,9,\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D},\mathrm{E},\mathrm{F}\}$.

The string ${2AF70}_{16}$ represents the number

$$\begin{array}{rl}{2AF70}_{16}& =2\cdot 16^4+\mathrm{A}\cdot 16^3+\mathrm{F}\cdot 16^2+7\cdot 16^1+0\cdots 16^0\\ & =2\cdot 16^4+10\cdot 16^3+15\cdot 16^2+7\cdot 16^1+0\cdot 16^0\\ & =131072+40960+3840+112+0\\ & =175984\end{array}$$

Theorem: Representing Real Numbers

Let $b>1$ be a natural number.

For every real number $r\ge 0$ there exists a finite sequence of [extended natural numbers] $a_0,a_1,\dots ,a_n$ and an infinite series of extended natural numbers $(c_k{)}_{k\in \mathbb{N}}$ such that $0\le a_k<b$ and $0\le c_k<b$ and

$$r=\sum _{k=0}^n{a}_k{b}^k+\sum _{k=1}^{\infty }{c}_k{b}^{-k}$$

PROOF

TODO

According to this theorem, we can use any positional numeral system to represent all real numbers by again identifying the digits $D=\{d_0,\dots ,d_{n-1}\}$ with the first $b$ numbers from ${\mathbb{N}}_0$.

NOTATION

Each each string of the form $a_n{a}_{n-1}\cdots a_1{a}_0.c_1{c}_2\cdots$ represents a real number $r\ge 0$:

$$r=\sum _{k=0}^n{a}_k{b}^k+\sum _{k=1}^{\infty }{c}_k{b}^{-k}$$

If there is some index $p$ such that $c_k=0$ for all $k>p$, then we can also omit the trailing zeros and just write $a_n{a}_{n-1}\cdots a_1{a}_0.c_1{c}_2\cdots c_p$.

When multiple positional numeral systems are used, it can be difficult to know in which system a particular number is given, especially when the systems use the same symbols for some digits. This is why we usually write the radix $b$ as a subscript at the end:

$$(a_n{a}_{n-1}\cdots a_1{a}_0.c_1{c}_2\cdots {)}_b$$