Binary number

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Decimal number	Binary number
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

1⁄8, 1⁄64. Early forms of this system can be found in documents from the Fifth Dynasty of Egypt, approximately 2400 BC, and its fully developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt, approximately 1200 BC.

The method used for ancient Egyptian multiplication is also closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it; the order in which these steps are to be performed is given by the binary representation of the second number. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC.

Daoist Bagua

The I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary technique of divination.

It is based on taoistic duality of yin and yang. Eight trigrams (Bagua) and a set of 64 hexagrams ("sixty-four" gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou dynasty of ancient China.

The Song dynasty scholar Shao Yong (1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.

Etruscans divided the outer edge of divination livers into sixteen parts, each inscribed with the name of a divinity and its region of the sky. Each liver region produced a binary reading which was combined into a final binary for divination.

Divination at Ancient Greek Dodona oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy.

Metrical combinations generated via Prastāra for length 2n)

Combinatorial sequence (Prasatāra order)
1	2	G L
2	4	GG LG GL LL
3	8	GGG LGG GLG LLG GGL LGL GLL LLL

The Indian scholar Pingala (c. 2nd century BC) developed a binary system for describing prosody. He described meters in the form of short and long syllables (the latter equal in length to two short syllables). They were known as laghu (light) and guru (heavy) syllables.

Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to science of meters in Sanskrit. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern positional notation. In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values.

The Ifá is a West African divination system popular among the Yoruba tribe of the Old Oyo Empire. Similar to the I Ching, but has up to 256 binary signs, unlike the I Ching which has 64. The number comes from squaring 16 which also matches the total possibilities in an 8-bit sequence. In Ifá divination, this reflects the possible outcomes called Odú. These Odú are determined using an Ọpẹlẹ chain, which has 8 seeds. Each seed can land in one of two positions (open or closed) creating all the possible combinations. The Ifá originated in 15th century West Africa among Yoruba people. In 2008, UNESCO added Ifá to its list of the "Masterpieces of the Oral and Intangible Heritage of Humanity".

The residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa and Asia. Sets of binary combinations similar to the I Ching have also been used in traditional African divination systems, such as Ifá among others, as well as in medieval Western geomancy. The majority of Indigenous Australian languages use a base-2 system.

In 1605, Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text. Importantly for the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature".

","txt":"//upload.wikimedia.org/wikipedia/commons/4/4a/Binary_guess_number_trick_SMIL.svg?utm_source=en.wikipedia.org&utm_campaign=index&utm_content=original"}]]}'>A party trick to guess a number from which cards it is printed on uses the bits of the binary representation of the number. In the SVG file, click a card to toggle it

Binary counting follows the exact same procedure, and again the incremental substitution begins with the least significant binary digit, or bit (the rightmost one, also called the first bit), except that only the two symbols 0 and 1 are available. Thus, after a bit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next bit to the left:

0000,

0001, (rightmost bit starts over, and the next bit is incremented)

0010, 0011, (rightmost two bits start over, and the next bit is incremented)

0100, 0101, 0110, 0111, (rightmost three bits start over, and the next bit is incremented)

1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...

In the binary system, each bit represents an increasing power of 2, with the rightmost bit representing 2, the next representing 2, then 2, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" bit. For example, the binary number 100101 is converted to decimal form as follows:

1001012 = [ ( 1 ) × 2 ] + [ ( 0 ) × 2 ] + [ ( 0 ) × 2 ] + [ ( 1 ) × 2 ] + [ ( 0 ) × 2 ] + [ ( 1 ) × 2 ]

1001012 = [ 1 × 32 ] + [ 0 × 16 ] + [ 0 × 8 ] + [ 1 × 4 ] + [ 0 × 2 ] + [ 1 × 1 ]

1001012 = 3710

Binary arithmetic

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Arithmetic in binary is much like arithmetic in other numeral systems using positional notation. Addition, subtraction, multiplication, and division can be performed on binary numerals.

Addition

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1 1 1 1 1 (carried digits) 0 1 1 0 1 + 1 0 1 1 1 ------------- = 1 0 0 1 0 0 = 36

In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).

When computers must add two numbers, the rule that: x xor y = (x + y) mod 2 for any two bits x and y allows for very fast calculation, as well.

Long carry method

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A simplification for many binary addition problems is the "long carry method" or "Brookhouse Method of Binary Addition". This method is particularly useful when one of the numbers contains a long stretch of ones. It is based on the simple premise that under the binary system, when given a stretch of digits composed entirely of n ones (where n 9s will result in the number 1 followed by a string of ∨. The difference is that