History
The modern binary number system was studied in Europe in the 16th and 17th centuries byEgypt
China
India
The Indian scholar Pingala (c. 2nd century BC) developed a binary system for describing prosody (poetry), prosody. 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.Binary Numbers in Ancient IndiaOther cultures
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 as well as in Middle Ages, medieval Western geomancy.Western predecessors to Leibniz
In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or 'Ars generalis' based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. 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". (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.Leibniz and the ''I Ching''
Later developments
Representation
Any number can be represented by a sequence ofCounting 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 ''0'' through ''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
Fractions
Fractions in binary arithmetic Repeating decimal, terminate only if 2 (number), 2 is the only prime factor in the denominator. As a result, 1/10 does not have a finite binary representation (10 has prime factors 2 and 5). This causes 10 × 0.1 not to precisely equal 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 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: 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 Exclusive or, xor y = (x + y) Modulo operation, 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" or "Brookhouse Method of Binary Addition". This method is particularly 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 ones (where is any integer length), adding 1 will result in the number 1 followed by a string of zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of 9s will result in the number 1 followed by a string of 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. carry the 1 until it is one digit past the "string" below 1 1 1 0 1 1 1 1 1 0Addition table
The binary addition table is similar, but not the same, as the Logical disjunction#Truth table, 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 to ''adding'' a negative number of equal absolute value. Computers use signed number representations to handle negative numbers—most commonly the two's complement notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula: :Multiplication
Multiplication in binary is similar to its decimal counterpart. Two numbers and can be multiplied by partial products: for each digit in , the product of that digit in is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in 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 is 0, the partial product is also 0 * If the digit in is 1, the partial product is equal to For example, the binary numbers 1011 and 1010 are multiplied as follows: 1 0 1 1 () × 1 0 1 0 () --------- 0 0 0 0 ← Corresponds to the rightmost 'zero' in + 1 0 1 1 ← Corresponds to the next 'one' in + 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 (5.625 in decimal) × 1 1 0 . 0 1 (6.25 in decimal) ------------------- 1 . 0 1 1 0 1 ← Corresponds to a 'one' in + 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in + 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 Logical conjunction#Truth table, truth table of the logical conjunction operation .Division
Long division in binary is again similar to its decimal counterpart. In the example below, the divisor is 1012, or 5 in decimal, while the Division (mathematics), dividend is 110112, or 27 in 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 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 ----- 0 1 0 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, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.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 Methods of computing square roots#Binary numeral system (base 2), here. An example is: 1 0 0 1 --------- √ 1010001 1 --------- 101 01 0 -------- 1001 100 0 -------- 10001 10001 10001 ------- 0Bitwise operations
Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using logical connective, Boolean logical operators. When a string of binary symbols is manipulated in this way, it is called a bitwise operation; the logical operators Logical conjunction, AND, Logical disjunction, OR, and Exclusive disjunction, XOR may be performed on corresponding bits in two binary numerals provided as input. The logical Negation, 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.Conversion to and from other numeral systems
Decimal to Binary
Binary to Decimal
Conversion from base-2 to base-10 simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be organized in a multi-column table. For example, to convert 100101011012 to decimal: The result is 119710. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the Horner scheme. 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 0.110101101012, 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, , in binary, is: Thus the repeating decimal fraction 0.... is equivalent to the repeating binary fraction 0.... . Or for example, 0.110, in binary, is: This is also a repeating binary fraction 0.0... . 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: 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). For very large numbers, these simple methods are inefficient because they perform a large number of multiplications or divisions where one operand is very large. A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10''k'', where ''k'' is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are Concatenation, concatenated. Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10''k'' and added to the second converted piece, where ''k'' is the number of decimal digits in the second, least-significant piece before conversion.Hexadecimal
Binary may be converted to and from hexadecimal more easily. This is because 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, as shown in the adjacent table. 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 (cryptography)#Bit padding, padding). For example: :10100102 = 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) = 49,38310Octal
Binary 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. 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) = 8710Representing real numbers
Non-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 means: For a total of 3.25 decimal. All dyadic fraction, 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 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 0.111... = 1 (binary), 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.See also
* Balanced ternary * Binary code * Binary-coded decimal * Finger binary * Gray code * IEEE 754 * Linear-feedback shift register * Offset binary * Quibinary * Reduction of summands * Redundant binary representation * Repeating decimal * Two's complementReferences
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