About

A numeral system (or system of numeration) is a mathematical notation system for expressing numbers using digits (or other symbols). The numeral system gives the context that allows the (digits|symbols) to be interpreted.

This article talk about the positional notation. For other, see Non-positional positions

A positional notation has the following properties:

Example, the symbols 11 is interpreted

For systems for classifying numbers according to their type, see Number System (Classification|Type).

List

The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations.

Numeral System Base Best known prefix
Number, Numeric, Quantity 2 %
Number - Octal Number System 8 0
Number, Numeric, Quantity 10 Default
Number - Hexadecimal notation (0x) 16 0x
Number - Babylonian numeral system, base-60 60 ?

Number translation

2 8 10 16
00000 0 0 0
00001 1 1 1
00010 2 2 2
00011 3 3 3
00100 4 4 4
00101 5 5 5
00110 6 6 6
00111 7 7 7
01000 10 8 8
01001 11 9 9
01010 12 10 A
01011 13 11 B
01100 14 12 C
01101 15 13 D
01110 16 14 E
01111 17 15 F
10000 20 16 10
10001 21 17 11
10010 22 18 12
10011 23 19 13
10100 24 20 14
10101 25 21 15

Machine Data Representation

Most computers store data by block of bytes (of eight bits)

Representation of a full bytes 1111111 in:

Base Representation Note
2 11111111 Eight Full Digit
8 377 Digits are not used completely (Not 777)
10 255
16 FF Digits are used completely

As the octal digit are not used completely, the Octal system is not the most practical number system to store a byte.

Example

Python with an set expression in order to calculate the set of numbers that are made with:

  • at-most-three-digit numbers.
base = 2
digits = {0, 1}
{(base**2)*x + base*y + z for x in digits for y in digits for z in digits}

{0, 1, 2, 3, 4, 5, 6, 7}
  • at-most-four-digit numbers.
{(base**3)*w + (base**2)*x + base*y + z for x in digits for y in digits for z in digits for w in digits}

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

Documentation / Reference