Binary Numbers
In mathematics and digital electronics, abinary number is a number expressed in thebase-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).
The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all moderncomputers and computer-based devices.
History
The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. Leibniz was specifically inspired by the Chinese I Ching.
Egypt
Arithmetic values represented by parts of the Eye of Horus
The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions (not related to the binary number system) and Horus-Eye fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of Horus, although this has been disputed).[1] Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 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 theNineteenth Dynasty of Egypt, approximately 1200 BC.[2]
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.[3]
China
Daoist Bagua
The I Ching dates from the 9th century BC in China.[4] The binary notation in the I Ching is used to interpret its quaternary divinationtechnique.[5]
It is based on taoistic duality of yin and yang.[6] 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.[4]
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.[5] Viewing the least significant bit on top of single hexagrams inShao 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. [7]
India
The Indian scholar Pingala (c. 2nd century BC) developed a binary system for describingprosody.[8][9] He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.[10][11] Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern, Westernpositional notation.[12][13]
Other cultures
The residents of the island of Mangareva inFrench Polynesia were using a hybrid binary-decimal system before 1450.[14] Slit drumswith binary tones are used to encode messages across Africa and Asia.[6] Sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as inmedieval Western geomancy. The base-2 system utilized in geomancy had long been widely applied in sub-Saharan Africa.
Western predecessors to Leibniz
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.[15]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".[15] (See Bacon's cipher.)
John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriotinvestigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers.[16] Possibly the first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700.[17]
Leibniz and the I Ching
Gottfried Leibniz
Leibniz studied binary numbering in 1679; his work appears in his article Explication de l'Arithmétique Binaire (published in 1703) The full title of Leibniz's article is translated into English as the "Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi".[18] (1703). Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows:[18]
- 0 0 0 1 numerical value 20
- 0 0 1 0 numerical value 21
- 0 1 0 0 numerical value 22
- 1 0 0 0 numerical value 23
Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus.[19] As aSinophile, Leibniz was aware of the I Ching, noted with fascination how its hexagrams correspond to the binary numbers from 0 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.[20] Leibniz was first introduced to theI Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Chinghexagrams as an affirmation of theuniversality of his own religious beliefs as a Christian.[19] Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing.[21]
Later developments
George Boole
In 1854, British mathematician George Boolepublished a landmark paper detailing analgebraic system of logic that would become known as Boolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry.[22]
In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuitdesign.[23]
In November 1937, George Stibitz, then working at Bell Labs, completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition.[24] Bell Labs authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed 8 January 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference atDartmouth College on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a teletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John von Neumann, John Mauchly and Norbert Wiener, who wrote about it in his memoirs.[25][26][27]
The Z1 computer, which was designed and built by Konrad Zuse between 1935 and 1938, used Boolean logic and binary floating point numbers.[28]
Representation
Any number can be represented by a sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667:
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimalnumeral of the traditional sexagesimaltime.
The numeric value represented in each case is dependent upon the value assigned to each symbol. In a computer, the numeric values may be represented by two different voltages; on a magnetic disk, magnetic polarities may be used. A "positive", "yes", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.
In keeping with customary representation of numerals using Arabic numerals, binary numbers are commonly written using the symbols 0 and 1. When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix. The following notations are equivalent:
- 100101 binary (explicit statement of format)
- 100101b (a suffix indicating binary format; also known as Intel convention[29][30])
- 100101B (a suffix indicating binary format)
- bin 100101 (a prefix indicating binary format)
- 1001012 (a subscript indicating base-2 (binary) notation)
- %100101 (a prefix indicating binary format; also known as Motorola convention[29][30])
- 0b100101 (a prefix indicating binary format, common in programming languages)
- 6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced one zero zero, rather than one hundred, to make its binary nature explicit, and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as one hundred (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct value), but this does not make its binary nature explicit.
Counting in binary
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.
Decimal counting
Decimal counting uses the ten symbols 0through 9. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the first digit. When the available symbols for this position are exhausted, the least significant digit is reset to 0, and the next digit of higher significance (one position to the left) is incremented (overflow), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:
- 000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
- 010, 011, 012, ...
- ...
- 090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
- 100, 101, 102, ...
Binary counting
This counter shows how to count in binary from numbers zero through thirty-one.
Binary counting follows the same procedure, except that only the two symbols 0 and 1 are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:
- 0000,
- 0001, (rightmost digit starts over, and next digit is incremented)
- 0010, 0011, (rightmost two digits start over, and next digit is incremented)
- 0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)
- 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...
In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. The equivalent decimal representation of a binary number is sum of the powers of 2 which each digit represents. For example, the binary number 100101 is converted to decimal form as follows:
- 1001012 = [ ( 1 ) × 25 ] + [ ( 0 ) × 24 ] + [ ( 0 ) × 23 ] + [ ( 1 ) × 22 ] + [ ( 0 ) × 21 ] + [ ( 1 ) × 20]
- 1001012 = [ 1 × 32 ] + [ 0 × 16 ] + [ 0 × 8 ] + [1 × 4 ] + [ 0 × 2 ] + [ 1 × 1 ]
- 1001012 = 3710
Fractions
Fractions in binary only terminate if the denominator has 2 as the only prime factor. As a result, 1/10 does not have a finite binary representation, and this causes 10 × 0.1 not to be precisely equal to 1 in floating point arithmetic. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 2−1 + 1 × 2−2 + 0 × 2−3 + 1 ×2−4 + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever.
Binary arithmetic
Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.
Addition
The circuit diagram for a binaryhalf adder, which adds two bits together, producing sum and carry bits.
The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:
- 0 + 0 → 0
- 0 + 1 → 1
- 1 + 0 → 1
- 1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
- 5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101) )
- 7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101) )
This is known as carrying. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:
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 (36 decimal).
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
A simplification for many binary addition problems is the Long Carry Method orBrookhouse Method of Binary Addition. This method is generally useful in any binary addition where one of the numbers contains a long "string" of ones. It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely ofn ones (where: n is any integer length), adding 1 will result in the number 1 followed by a string of n zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of n 9s will result in the number 1 followed by a string of n 0s:
Binary Decimal
1 1 1 1 1 likewise 9 9 9 9 9
+ 1 + 1
——————————— ———————————
1 0 0 0 0 0 1 0 0 0 0 0
Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 02 (95810) and 1 0 1 0 1 1 0 0 1 12(69110), using the traditional carry method on the left, and the long carry method on the right:
Traditional Carry Method Long Carry Method
vs.
1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ← carry the 1 until it is one digit past the "string" below
1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 0 cross out the "string",
+ 1 0 1 0 1 1 0 0 1 1 + 1 0 1 0 1 1 0 0 1 1 and cross out the digit that was added to it
——————————————————————— ——————————————————————
= 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1
The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique. Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 12 (164910). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.
Addition table
The binary addition table is similar, but not the same, as the truth table of the logical disjunction operation . The difference is that , while .
Subtraction
Subtraction works in much the same way:
- 0 − 0 → 0
- 0 − 1 → 1, borrow 1
- 1 − 0 → 1
- 1 − 1 → 0
Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as borrowing. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value.
* * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1
* (starred columns are borrowed from) 1 0 1 1 1 1 1 - 1 0 1 0 1 1 ---------------- = 0 1 1 0 1 0 0
Subtracting a positive number is equivalent toadding a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complementnotation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:
A − B = A + not B + 1
Multiplication
Multiplication in binary is similar to its decimal counterpart. Two numbers A and Bcan be multiplied by partial products: for each digit in B, the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used. The sum of all these partial products gives the final result.
Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
- If the digit in B is 0, the partial product is also 0
- If the digit in B is 1, the partial product is equal to A
For example, the binary numbers 1011 and 1010 are multiplied as follows:
1 0 1 1 (A)
× 1 0 1 0 (B)
---------
0 0 0 0 ← Corresponds to the rightmost 'zero' in B
+ 1 0 1 1 ← Corresponds to the next 'one' in B
+ 0 0 0 0
+ 1 0 1 1
---------------
= 1 1 0 1 1 1 0
Binary numbers can also be multiplied with bits after a binary point:
1 0 1 . 1 0 1 A (5.625 in decimal)
× 1 1 0 . 0 1 B (6.25 in decimal)
-------------------
1 . 0 1 1 0 1 ← Corresponds to a 'one' in B
+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in B
+ 0 0 0 . 0 0 0
+ 1 0 1 1 . 0 1
+ 1 0 1 1 0 . 1
---------------------------
= 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal)
See also Booth's multiplication algorithm.
Multiplication table
The binary multiplication table is the same as the truth table of the logical conjunctionoperation .
Division
Long division in binary is again similar to its decimal counterpart.
In the example below, the divisor is 1012, or 5 decimal, while the dividend is 110112, or 27 decimal. The procedure is the same as that of decimal long division; here, the divisor 1012goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence:
1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
0 0 1
The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:
1 0 1
___________
1 0 1 ) 1 1 0 1 1
− 1 0 1
-----
1 1 1
− 1 0 1
-----
1 0
Thus, the quotient of 110112 divided by 1012is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102. In decimal, 27 divided by 5 is 5, with a remainder of 2.
Square root
The process of taking a binary square root digit by digit is the same as for a decimal square root, and is explained here. An example is:
1 0 0 1
---------
√ 1010001
1
---------
101 01
0
--------
1001 100
0
--------
10001 10001
10001
-------
0
Bitwise operations
Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators. When a string of binary symbols is manipulated in this way, it is called a bitwise operation; the logical operators AND,OR, and XOR may be performed on corresponding bits in two binary numerals provided as input. The logical NOT operation may be performed on individual bits in a single binary numeral provided as input. Sometimes, such operations may be used as arithmetic short-cuts, and may have other computational benefits as well. For example, an arithmetic shift left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.
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