# Binary numeral system

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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 "${\displaystyle 10}$" in binary represents the value consisting of one set of twos (${\displaystyle 2^{1}}$) and no sets of ones (${\displaystyle 2^{0}}$), 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 "${\displaystyle 10}$", for example, represents the value consisting of one set of tens (${\displaystyle 10^{1}}$), and no sets of ones (${\displaystyle 10^{0}}$). 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 ${\displaystyle 100_{2}=(1\times 2^{2})+(0\times 2^{1})+(0\times 2^{0})=4_{10}+0+0=4_{10}}$ ${\displaystyle 100_{10}=(1\times 10^{2})+(0\times 10^{1})+(0\times 10^{0})}$

## Binary arithmetic

Arithmetic with binary numerals is similar to arithmetic with decimal numerals, except that the addition and multiplication tables are much simpler:

 ${\displaystyle +}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 10}$
 ${\displaystyle \times }$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1}$

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, ${\displaystyle 3/16=0.1875_{10}=0.0011_{2}=(0\times 2^{0})+(0\times 2^{-1})+(0\times 2^{-2})+(1\times 2^{-3})+(1\times 2^{-4})}$

Repeating binary expansions also occur, for any fraction where the denominator is not a power of 2. For example, ${\displaystyle 1/5=0.001100110011_{2}}$ (with 0011 repeating).

Irrational numbers can also be expressed, and will have irregular distributions of digits. For example, ${\displaystyle \pi =11.001001000011111..._{2}}$

## 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.

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