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
- as three in the binary system
- as eleven in the decimal system.
For systems for classifying numbers according to their type, see Number System (Classification|Type).
Articles Related
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}