Binary numeral system
Information about Binary numeral system
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Chinese counting rods
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HistoryThe ancient Indian mathematician Pingala presented the first known description of a binary numeral system around 800 BC written in Hindu numerals. The numeration system was based on the Eye of Horus Old Kingdom numeration system. 
A full set of 8 trigrams and 64 hexagrams, analogous to the 3-bit and 6-bit binary numerals, were known to the ancient Chinese in the classic text I Ching. Similar sets of binary combinations have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.
An ordered binary arrangement of the hexagrams of the I Ching, representing the decimal sequence from 0 to 63, and a method for generating the same, was developed by the Chinese scholar and philosopher Shao Yong in the 11th century. However, there is no evidence that Shao understood binary computation.
In 1605 Francis Bacon discussed a system by which 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 onely; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature." (See Bacon's cipher.)
The modern binary number system was fully documented by Gottfried Leibniz in the 17th century in his article Explication de l'Arithmétique Binaire. Leibniz's system used 0 and 1, like the modern binary numeral system.
In 1854, British mathematician George Boole published a landmark paper detailing a system of logic that would become known as Boolean algebra. His logical system proved instrumental in the development of the binary system, particularly in its implementation in electronic circuitry.
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 circuit design.
In November of 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. Bell Labs thus authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11, 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.
RepresentationA binary number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. The following sequences of symbols could all be interpreted as the same binary numeric value of 666:
1 0 1 0 0 1 1 0 1 0 | - | - - | | - | - x o x o o x x o x o y n y n n y y n y n
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)
- 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)
- 0b100101 (a prefix indicating binary format, common in programming languages)
When spoken, binary numerals are usually pronounced by pronouncing each individual digit, in order to distinguish them from decimal numbers. 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" is equal to the decimal value four, it would be confusing, and numerically incorrect, to refer to the numeral as "one hundred" so to speak.
Counting in binary
When the symbols for the first digit are exhausted, the next-higher digit (to the left) is incremented, and counting starts over at 0. In decimal, counting proceeds like so:
- 000, 001, 002, ... 007, 008, 009, (rightmost digit starts over, and next digit is incremented)
- 010, 011, 012, ...
- 090, 091, 092, ... 097, 098, 099, (rightmost two digits start over, and next digit is incremented)
- 100, 101, 102, ...
After a digit reaches 9, an increment resets it to 0 but also causes an increment of the next digit to the left. In binary, counting is the same except that only the two symbols 0 and 1 are used. 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:
- 000, 001, (rightmost digit starts over, and next digit is incremented)
- 010, 011, (rightmost two digits start over, and next digit is incremented)
- 100, 101, ...
Binary simplifiedOne can think about binary by comparing it with our usual numbers. We use a base ten system. This means that the value of each position in a numerical value can be represented by one of ten possible symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. We are all familiar with these and how the decimal system works using these ten symbols. When we begin counting values, we should start with the symbol 0, and proceed to 9 when counting. We call this the "ones", or "units" place.
The "ones" place, with those digits, might be thought of as a multiplication problem. 5 can be thought of as 5 × 100 (10 to the zeroeth power, which equals 5 × 1, since any nonzero number to the zero power is one). As we move to the left of the ones place, we increase the power of 10 by one. Thus, to represent 50 in this same manner, it can be thought of as 5 × 101, or 5 × 10.
When we run out of symbols in the decimal numeral system, we "move to the left" one place and use a "1" to represent the "tens" place. Then we reset the symbol in the "ones" place back to the first symbol, zero.
Binary is a base two system which works just like our decimal system, however with only two symbols which can be used to represent numerical values: 0 and 1. We begin in the "ones" place with 0, then go up to 1. Now we are out of symbols, so to represent a higher value, we must place a "1" in the "twos" place, since we don't have a symbol we can use in the binary system for 2, like we do in the decimal system.
In the binary numeral system, the value represented as 10 is (1 × 21) + (0 × 20). Thus, it equals "2" in our decimal system.
To see the actual algorithm used in computing the conversion, see the conversion guide below.
Here is another way of thinking about it: When you run out of symbols, for example 11111, add a "1" on the left end and reset all the numerals on the right to "0", producing 100000. This also works for symbols in the middle. Say the number is 100111. If you add one to it, you move the leftmost repeating "1" one space to the left (from the "fours" place to the "eights" place) and reset all the numerals on the right to "0", producing 101000.
Binary arithmeticArithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.
The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (carry:1)
- 5 + 5 = 10
- 7 + 9 = 16
This is known as carrying in most numeral systems. When the result of an addition exceeds the value of the radix, the procedure is to "carry the one" to the left, adding it to the next positional value. 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
In this example, two numerals are being added together: 011012 (13 decimal) and 101112 (23 decimal). 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 ^ y = x + y % 2 for any two bits x and y allows for very fast calculation, as well.
SubtractionSubtraction works in much the same way:
- 0 − 0 = 0
- 0 − 1 = 1 (with borrow)
- 1 − 0 = 1
- 1 − 1 = 0
One binary numeral can be subtracted from another as follows:
- * * * (starred columns are borrowed from)
= 1 0 1 0 1 1 1
Subtracting a positive number is equivalent to adding a negative number of equal absolute value; computers typically use two's complement notation to represent negative values. This notation eliminates the need for a separate "subtract" operation. The subtraction can be summarized with this formula:
A - B = A + not B + 1
For further details, see two's complement.
MultiplicationMultiplication in binary is similar to its decimal counterpart. Two numbers A and B can 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
1 0 1 1 (A) × 1 0 1 0 (B)
0 0 0 0 ← Corresponds to a zero in B + 1 0 1 1 ← Corresponds to a one in B + 0 0 0 0 + 1 0 1 1
= 1 1 0 1 1 1 0
See also Booth's multiplication algorithm.
DivisionBinary division is again similar to its decimal counterpart:
__________ 1 0 1 | 1 1 0 1 1
Here, 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 1012 goes 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 1 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
0 1 1 − 0 0 0
1 1 1 − 1 0 1
Thus, the quotient of 110112 divided by 1012 is 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.
Conversion to and from other numeral systems
DecimalTo convert from a base-10 integer numeral to its base-2 (binary) equivalent, the number is divided by two, and the remainder is the least-significant bit. The (integer) result is again divided by two, its remainder is the next most significant bit. This process repeats until the result of further division becomes zero.
For example, 11810, in binary, is:
Operation Remainder 118 ÷ 2 = 59 0 59 ÷ 2 = 29 1 29 ÷ 2 = 14 1 14 ÷ 2 = 7 0 7 ÷ 2 = 3 1 3 ÷ 2 = 1 1 1 ÷ 2 = 0 1
Reading the sequence of remainders from the bottom up gives the binary numeral .
This method works for conversion from any base, but there are better methods for bases which are powers of two, such as octal and hexadecimal given below.
To convert from base-2 to base-10 is the reverse algorithm. Starting from the left, double the result and add the next digit until there are no more. For example to convert 1100101011012 to decimal:
Result Remaining digits 0 110010101101 0 × 2 + 1 = 1 10010101101 1 × 2 + 1 = 3 0010101101 3 × 2 + 0 = 6 010101101 6 × 2 + 0 = 12 10101101 12 × 2 + 1 = 25 0101101 25 × 2 + 0 = 50 101101 50 × 2 + 1 = 101 01101 101 × 2 + 0 = 202 1101 202 × 2 + 1 = 405 101 405 × 2 + 1 = 811 01 811 × 2 + 0 = 1622 1 1622 × 2 + 1 = 3245
The result is 324510.
The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.
In a fractional binary number such as .110101101012, the first digit is , the second , etc. So if there is a 1 in the first place after the decimal, then the number is at least , and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.
For example, 10, in binary, is:
Converting Result 0. 0.0 0.01 0.010 0.0101
Thus the repeating decimal fraction 0.3... is equivalent to the repeating binary fraction 0.01... .
Or for example, 0.110, in binary, is:
Converting Result 0.1 0. 0.1 × 2 = 0.2 < 1 0.0 0.2 × 2 = 0.4 < 1 0.00 0.4 × 2 = 0.8 < 1 0.000 0.8 × 2 = 1.6 ≥ 1 0.0001 0.6 × 2 = 1.2 ≥ 1 0.00011 0.2 × 2 = 0.4 < 1 0.000110 0.4 × 2 = 0.8 < 1 0.0001100 0.8 × 2 = 1.6 ≥ 1 0.00011001 0.6 × 2 = 1.2 ≥ 1 0.000110011 0.2 × 2 = 0.4 < 1 0.0001100110
This is also a repeating binary fraction 0.000110011... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 0.1 + ... + 0.1, (10 additions) differs from 1 in floating point arithmetic. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not.
The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example:
= 1100 .101110011100... = 1100101110 .0111001110... = 11001 .0111001110... = 1100010101 = (789/62)10
Another way of converting from binary to decimal, often quicker for a person familiar with hexadecimal, is to do so indirectly—first converting ( in binary) into ( in hexadecimal) and then converting ( in hexadecimal) into ( in decimal).
HexadecimalBinary may be converted to and from hexadecimal somewhat more easily. This is due to the fact that the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, so it takes four digits of binary to represent one digit of hexadecimal.
The following table shows each hexadecimal digit along with the equivalent decimal value and four-digit binary sequence:
Hex Dec Binary 0 0 0000 1 1 0001 2 2 0010 3 3 0011 Hex Dec Binary 4 4 0100 5 5 0101 6 6 0110 7 7 0111 Hex Dec Binary 8 8 1000 9 9 1001 A 10 1010 B 11 1011 Hex Dec Binary C 12 1100 D 13 1101 E 14 1110 F 15 1111
To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:
- 3A16 = 0011 10102
- E716 = 1110 01112
To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra 0 bits at the left (called padding). For example:
- 010100102 = 0101 0010 grouped with padding = 5216
- 110111012 = 1101 1101 grouped = DD16
To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:
- C0E716 = (12 × 163) + (0 × 162) + (14 × 161) + (7 × 160) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,38310
OctalBinary is also easily converted to the octal numeral system, since octal uses a radix of 8, which is a power of two (namely, 23, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of hexadecimal in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.
Octal Binary 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111
Converting from octal to binary proceeds in the same fashion as it does for hexadecimal:
- 658 = 110 1012
- 178 = 001 1112
And from binary to octal:
- 1011002 = 101 1002 grouped = 548
- 100112 = 010 0112 grouped with padding = 238
And from octal to decimal:
- 658 = (6 × 81) + (5 × 80) = (6 × 8) + (5 × 1) = 5310
- 1278 = (1 × 82) + (2 × 81) + (7 × 80) = (1 × 64) + (2 × 8) + (7 × 1) = 8710
Representing real numbersNon-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in the decimal system). For example, the binary number 11.012 thus means:
1 × 21 (1 × 2 = 2) plus 1 × 20 (1 × 1 = 1) plus 0 × 2-1 (0 × ½ = 0) plus 1 × 2-2 (1 × ¼ = 0.25)
For a total of 3.25 decimal.
All dyadic rational numbers have a terminating binary numeral—the binary representation has a finite number of terms after the radix point. Other rational numbers have binary representation, but instead of terminating, they recur, with a finite sequence of digits repeating indefinitely. For instance
- = = 0.0101010101...2
- = = 0.10110100 10110100 10110100...2
The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in decimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the geometric series 2-1 + 2-2 + 2-3 + ... which is 1.
Binary numerals which neither terminate nor recur represent irrational numbers. For instance,
- 0.10100100010000100000100.... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
- 1.0110101000001001111001100110011111110... is the binary representation of , the square root of 2, another irrational. It has no discernible pattern, although a proof that is irrational requires more than this. See irrational number.
References1. ^ W. S. Anglin and J. Lambek, The Heritage of Thales, Springer, 1995, ISBN 038794544X online
2. ^ Bacon, Francis The Advancement of Learning, Book 6, Chapter 1, 1605. Online here.
- Two's complement
- Finger binary
- Binary-coded decimal
- reflected binary code, also known as Gray code
- linear feedback shift register
numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three
- BinaryMath.info - arithmetic examples and conversion methods
- Introduction to Binary Numbers
- A brief overview of Leibniz and the connection to binary numbers
- Simple Conversion Methods
- Binary System at cut-the-knot
- Conversion of Fractions at cut-the-knot
- Binary Digits at Math Is Fun
- at wikiHow
- Binary and Hexadecimal Tutorial
- Binary numbers conversion software Software for conversion of Numbers from Binary to other number systems and vice versa.
..... Click the link for more information.Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century.
The symbols (glyphs) used to represent the system are in principle independent of the system itself.
..... Click the link for more information.Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world.
..... Click the link for more information.The Eastern Arabic numerals (also called Arabic-Indic numerals, Arabic Eastern Numerals) are the symbols (glyphs) used to represent the Hindu-Arabic numeral system in conjunction with the Arabic alphabet in Egypt, Iran, Afghanistan, Pakistan and parts of India, and also in
..... Click the link for more information.Khmer numerals are the numerals used in the Khmer language of Cambodia. In informal spoken language one can ignore the last "sep" (30 to 90) and it is still understood.
..... Click the link for more information.symbols used in various modern Indian scripts for the numbers from zero to nine:
Variant 0 1 2 3 4 5 6 7 8 9 Used in
Eastern Nagari numerals ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ? Bengali language
..... Click the link for more information.Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). They are the direct graphic ancestors of the modern Indic and Hindu-Arabic numerals.
..... Click the link for more information.Thai numerals (ตัวเลขไทย) are traditionally used in Thailand, although the Arabic numerals (also known as Western numerals) are more common.
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Without proper rendering support, you may see question marks, boxes, or other symbols instead of Chinese characters.
Numeral systems by culture
Khmer Indian family
..... Click the link for more information.Counting rods (Traditional Chinese: 籌; Simplified Chinese: 筹; Pinyin: chou2
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- 여덟 권 yeodeolgwon (eight (books)) is pronounced like [여덜꿘] yeodeolkkwon
..... Click the link for more information.Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are entirely based on the Chinese numerals and the grouping of large numbers follow the Chinese tradition of grouping by 10,000.
..... Click the link for more information.Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times.
..... Click the link for more information.Armenian numerals is a historic numeral system created using the majuscules (uppercase letters) of the Armenian alphabet.
There was no notation for zero in the old system, and the numeric values for individual letters were added together.
..... Click the link for more information.Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. The system was used in Russia as late as the 1700s when Peter the Great replaced it with the Hindu-Arabic numeral system.
..... Click the link for more information.Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet.
In this system, there is no notation for zero, and the numeric values for individual letters are added together. Each unit (1, 2, ...
..... Click the link for more information.Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Milesian numerals, Alexandrian numerals, or alphabetic numerals.
..... Click the link for more information.Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodian.
..... Click the link for more information.Etruscan numerals were used by the ancient Etruscans. The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman numerals.
Etruscan Decimal Symbol *
θu 1 I
ma? 5 ?
śar 10 X
..... Click the link for more information./» and the fifths place with a stroke from the top-left to the bottom-right «\». The numbers from 1 = / to 29 = ////\\\\\ have been found.
InterpretationThese embossed marks, unique in objects from the Bronze Age, were introduced in cast-iron molds and were not
..... Click the link for more information.Roman numerals is a numeral system originating in ancient Rome, adapted from Etruscan numerals. The system used in classical antiquity was slightly modified in the Middle Ages to produce the system we use today. It is based on certain letters which are given values as numerals.
..... Click the link for more information.Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.
..... Click the link for more information.Egyptian numerals was a numeral system used in ancient Egypt. It was a decimal system, often rounded off to the higher power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, being ciphered one:one onto the Egyptian alphabet.
..... Click the link for more information.Maya numerals is very simple. 
Addition is performed by combining the numeric symbols at each level:
If five or more dots result from the combination, five dots are removed and replaced by a bar.
..... Click the link for more information.This is a list of numeral system topics (and "numeric representations"), by Wikipedia page. It does not systematically list computer formats for storing numbers (computer numbering formats). See also number names.
..... Click the link for more information.A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system.
..... Click the link for more information.base or radix is usually the number of various unique digits, including zero, that a positional numeral system uses to represent numbers. For example, the decimal system, the most common system in use today, uses base ten, hence the maximum number a single digit will ever
..... Click the link for more information.decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have four fingers and a thumb on each hand, giving a total of ten digits over both hands.
..... Click the link for more information.Quaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.
It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the
..... Click the link for more information.octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7.
Octal numerals can be made from binary numerals by grouping consecutive digits into groups of three (starting from the right).
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