Binary numeral system
The binary number system, also referred to as base-2, or radix-2, represents numbers using only the digits 0 and 1. This is in contrast with the more familiar decimal number system (a.k.a. base-10, radix-10) which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In the binary system each digit position represents a power of two. The numeral "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10} " in binary represents the value consisting of one set of twos (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^1} ) and no sets of ones (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^0} ), which is represented by the number 2 in the decimal system. This is analogous to the decimal system, where each digit position represents a power of ten: the numeral "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10} ", for example, represents the value consisting of one set of tens (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^1} ), and no sets of ones (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^0} ). When the numbering system used for a number is in question, one can write the radix as a subscript to the number as done in the following table.
| Binary | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 100_2 = (1 \times 2^2)+(0 \times 2^1)+(0 \times 2^0)= 4_{10} + 0 + 0 = 4_{10}} |
|---|---|
| Decimal | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 100_{10} = (1 \times 10^2) + (0 \times 10^1) + (0 \times 10^0)} |
Binary arithmetic
Arithmetic with binary numbers is similar to arithmetic with decimal numbers, except that the addition and multiplication tables are much simpler:
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Division and subtraction are performed in the same way as for decimal numbers, but using the corresponding rules for binary addition and multiplication. Non-integer quantities can be represented as binary digits to the right of the binary point. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3/16 = 0.1875_{10} = 0.0011_2 = (0 \times 2^0)+(0 \times 2^{-1})+(0 \times 2^{-2})+(1 \times 2^{-3})+(1 \times 2^{-4})}
Repeating binary expansions also occur, for any fraction where the denominator is not a power of 2. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/5 = 0.001100110011_2} (with 0011 repeating).
Irrational numbers can also be expressed, and will have irregular distributions of digits. For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi = 11.001001000011111..._2}
Use in computing
The binary system is used is most electronic computers, as the values of 0 and 1 can be easily represented by a low and a high voltage in a circuit. A single digit of a binary number is referred to as a bit, short for binary digit. (The term bit was coined in 1947 at Bell Labs.) A bit can be a measure of data size, or a measure of information entropy, which are often not equal in size.
Other representations
Because the number of digits in the binary representation of a value can grow quickly, when human readability is desired, binary values are often represented in the octal number system (base 8) or the hexadecimal number system (base 16). Octal uses the digits 0 through 7; hexadecimal uses the digits 0 through 9, followed by the letters A through F to represent the values ten, eleven, twelve, thirteen, fourteen, and fifteen.
Binary numbers can be converted to octal by grouping the binary digits in groups of three beginning at the ones place; each group of three binary digits converts to a single octal digit. Similarly, binary numbers can be converted to hexadecimal by grouping the binary digits in groups of four beginning at the ones place; each group of four binary digits converts to a single hexadecimal digit.
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 3 | 3 |
| 4 | 100 | 4 | 4 |
| 5 | 101 | 5 | 5 |
| 6 | 110 | 6 | 6 |
| 7 | 111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 17 | 10001 | 21 | 11 |
| 20 | 10100 | 24 | 14 |
| 25 | 11001 | 31 | 19 |
| 32 | 100000 | 40 | 20 |
| 40 | 101000 | 50 | 28 |
| 49 | 110001 | 61 | 31 |
| 63 | 111111 | 77 | 3F |
| 99 | 1100011 | 143 | 63 |