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 numeral 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 we are accustomed to seeing represented as "2". 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 numeral system used for a number is in question, one can write the radix as a subscript to the number, as is 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 numerals is similar to arithmetic with decimal numerals, except that the addition and multiplication tables are much simpler:

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 +} 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 0} 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} 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 0} 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 0} 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} 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} 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} 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}

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 \times} 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 0} 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} 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 0} 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 0} 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 0} 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} 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 0} 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}

Division and subtraction are performed in the same way as for decimal numerals, 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 in most electronic computers, as the values of 0 and 1 can be easily represented by a low and a high voltage in a circuit (i.e., by an "on/off" switch). A single digit of a binary numeral is referred to as a bit, short for binary digit. (The term bit was coined in 1947 at Bell Laboratories.) 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 numeral system (base 8) or the hexadecimal numeral system (base 16). Octal uses the digits 0 through 7, while 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 numerals can be converted to octal by grouping the binary digits in groups of three beginning at the ones place, with each group of three binary digits converting to a single octal digit. Similarly, binary numerals can be converted to hexadecimal by grouping the binary digits in groups of four beginning at the ones place, with each group of four binary digits converting 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