About
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1.
Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers.
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Counting in binary
It is possible to do arithmetic in base two, e.g. 3+5 is written:
<math> \begin{array}{cccc} & 0 & 0 & 1 & 1 \\ + & 0 & 1 & 0 & 1 \\ \hline & 1 & 0 & 0 & 0 \\ \end{array} </math>
The addition works like normal (base-10) arithmetic, where:
<math> \begin{array}{cccc} & & 1 \\ + & & 1 \\ \hline & 1 & 0 \\ \end{array} </math> where 1 + 1 = 0 with a carry of 1 and 1 + 0 = 1
Subtraction, multiplication, etc. work this way:
<math> \begin{array}{cccc} & & \textit{1} & \textit{1} & & \textit{carried digits} \\ \\ & 0 & 0 & 1 & 1 & \\ + & 0 & 1 & 0 & 1 &\\ \hline & 1 & 0 & 0 & 0 & \\ \end{array} </math>
Translation in
Decimal
Integer
Using Arabic numerals, binary numbers are commonly written using the symbols 0 and 1.
| Number | Binary coding |
|---|---|
| 0 | 0000 |
| 1 | 0001 |
| 2 | 0010 |
| 3 | 0011 |
| 4 | 0100 |
| 5 | 0101 |
| 6 | 0110 |
| 7 | 0111 |
| 8 | 1000 |
| ……. | |
| 14 | 1110 |
| 15 | 1111 |
| ……. | |
Floating point
1.01 in base-2 notation is 1 + 0/2 + 1/4, or 1.25 in decimal notation.
Hexadecimal
| Binary | Hexadecimal |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 10 | 2 |
| 11 | 3 |
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
| 1000 | 8 |
| 1001 | 9 |
| 1010 | A |
| 1011 | B |
| 1100 | C |
| 1101 | D |
| 1110 | E |
| 1111 | F |
Hexadecimal is then to easier write. For example, the binary number “100110110100” is “9B4” in hexadecimal.