Abinary number is anumber expressed in thebase-2numeral system orbinary numeral system, a method for representingnumbers that uses only two symbols for thenatural numbers: typically "0" (zero) and "1" (one). Abinary number may also refer to arational number that has a finite representation in the binary numeral system, that is, the quotient of aninteger by a power of two.
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
The base-2 numeral system is apositional notation with aradix of2. Each digit is referred to as abit, or binary digit. Because of its straightforward implementation indigital electronic circuitry usinglogic gates, the binary system is used by almost all moderncomputers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation.[1]
The modern binary number system was studied in Europe in the 16th and 17th centuries byThomas Harriot, andGottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Europe and India.
Arithmetic values thought to have been represented by parts of theEye of Horus
The scribes of ancient Egypt used two different systems for their fractions,Egyptian fractions (not related to the binary number system) andHorus-Eye fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye ofHorus, although this has been disputed).[2] Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of ahekat 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 theFifth Dynasty of Egypt, approximately 2400 BC, and its fully developed hieroglyphic form dates to theNineteenth Dynasty of Egypt, approximately 1200 BC.[3]
The method used forancient 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 theRhind Mathematical Papyrus, which dates to around 1650 BC.[4]
TheSong dynasty scholarShao 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.[6] Viewing theleast significant bit on top of single hexagrams in Shao Yong's square[8]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.[9]
Etruscans divided the outer edge ofdivination 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.[10]
Divination at Ancient GreekDodona oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy.[11]
The Indian scholarPingala (c. 2nd century BC) developed a binary system for describingprosody.[12][13] He described meters in the form of short and long syllables (the latter equal in length to two short syllables).[14] They were known aslaghu (light) andguru (heavy) syllables.
Pingala's Hindu classic titledChandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates toscience 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 modernpositional notation.[15] 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 ofplace values.[16]
The residents of the island ofMangareva inFrench Polynesia were using a hybrid binary-decimal system before 1450.[20]Slit drums with binary tones are used to encode messages across Africa and Asia.[7]Sets of binary combinations similar to theI Ching have also been used in traditional African divination systems, such asIfá among others, as well as inmedieval Westerngeomancy. The majority ofIndigenous Australian languages use a base-2 system.[21]
In the late 13th centuryRamon 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.[22]
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.[23] 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".[23] (SeeBacon's cipher.)
In 1617,John Napier described a system he calledlocation arithmetic for doing binary calculations using a non-positional representation by letters.Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers.[24]Possibly the first publication of the system in Europe was byJuan Caramuel y Lobkowitz, in 1700.[25]
Leibniz wrote in excess of a hundred manuscripts on binary, most of them remaining unpublished.[26] Before his first dedicated work in 1679, numerous manuscripts feature early attempts to explore binary concepts, including tables of numbers and basic calculations, often scribbled in the margins of works unrelated to mathematics.[26]
His first known work on binary,“On the Binary Progression", in 1679, Leibniz introduced conversion between decimal and binary, along with algorithms for performing basic arithmetic operations such as addition, subtraction, multiplication, and division using binary numbers. He also developed a form of binary algebra to calculate the square of a six-digit number and to extract square roots.[26]
His most well known work appears in his articleExplication 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 ofFu Xi".[27] Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows:[27]
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
While corresponding with the Jesuit priestJoachim Bouvet in 1700, who had made himself an expert on theI Ching while a missionary in China, Leibniz explained his binary notation, and Bouvet demonstrated in his 1701 letters that theI Ching was an independent, parallel invention of binary notation.Leibniz & Bouvet concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophicalmathematics he admired.[28] Of this parallel invention, Leibniz wrote in his "Explanation Of Binary Arithmetic" that "this restitution of their meaning, after such a great interval of time, will seem all the more curious."[29]
The relation was a central idea to his universal concept of a language orcharacteristica universalis, a popular idea that would be followed closely by his successors such asGottlob Frege andGeorge Boole in formingmodern symbolic logic.[30]Leibniz was first introduced to theI Ching through his contact with the French JesuitJoachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw theI Ching hexagrams as an affirmation of theuniversality of his own religious beliefs as a Christian.[31] Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea ofcreatio ex nihilo or creation out of nothing.[32]
[A concept that] is not easy to impart to the pagans, is the creationex nihilo through God's almighty power. Now one can say that nothing in the world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing.
In 1854, British mathematicianGeorge Boole published a landmark paper detailing analgebraic system oflogic that would become known asBoolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry.[33]
In November 1937,George Stibitz, then working atBell Labs, completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition.[35] 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 calculatecomplex numbers. In a demonstration to theAmerican Mathematical Society conference atDartmouth College on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by ateletype. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration wereJohn von Neumann,John Mauchly andNorbert Wiener, who wrote about it in his memoirs.[36][37][38]
Any number can be represented by a sequence ofbits (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:
The numeric value represented in each case depends on the value assigned to each symbol. In the earlier days of computing, switches, punched holes, and punched paper tapes were used to represent binary values.[40] In a modern computer, the numeric values may be represented by two differentvoltages; on amagneticdisk,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 the customary representation of numerals usingArabic numerals, binary numbers are commonly written using the symbols0 and1. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, orradix. The following notations are equivalent:
0b100101 (a prefix indicating binary format, common in programming languages)
6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
#b100101 (a prefix indicating binary format, common in Lisp programming languages)
When spoken, binary numerals are usually read digit-by-digit, to distinguish them from decimal numerals. For example, the binary numeral 100 is pronouncedone zero zero, rather thanone 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 asone hundred (a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correctvalue), but this does not make its binary nature explicit.
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 familiardecimal counting system as a frame of reference.
Decimal counting uses the ten symbols0 through9. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called thefirst digit. When the available symbols for this position are exhausted, the least significant digit is reset to0, 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)
This counter shows how to count in binary from numbers zero through thirty-one.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, orbit (the rightmost one, also called thefirst bit), except that only the two symbols0 and1 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)
In the binary system, each bit represents an increasing power of 2, with the rightmost bit representing 20, the next representing 21, then 22, 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:
Fractions in binary arithmeticterminate only if thedenominator is apower of 2. As a result, 1/10 does not have a finite binary representation (10 has prime factors2 and5). This causes 10 × 1/10 not to precisely equal 1 in binaryfloating-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.
Arithmetic in binary is much like arithmetic in otherpositional notationnumeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.
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:
This is known ascarrying. 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:
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:xxor y = (x + y)mod 2for any two bits x and y allows for very fast calculation, as well.
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 ofn ones (wheren is any integer length), adding 1 will result in the number 1 followed by a string ofn zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string ofn 9s will result in the number 1 followed by a string ofn 0s:
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 01 1 1 01 1 1 1 1 0 cross out the "string",+ 1 0 1 0 1 1 0 0 1 1 + 1 01 0 1 1 0 01 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.
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 asborrowing. 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.
Subtracting a positive number is equivalent toadding anegative number of equalabsolute value. Computers usesigned number representations to handle negative numbers—most commonly thetwo'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 in binary is similar to its decimal counterpart. Two numbersA andB can be multiplied by partial products: for each digit inB, the product of that digit inA is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit inB 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 inB is 0, the partial product is also 0
If the digit inB is 1, the partial product is equal toA
For example, the binary numbers 1011 and 1010 are multiplied as follows:
Long division in binary is again similar to its decimal counterpart.
In the example below, thedivisor is 1012, or 5 in decimal, while thedividend is 110112, or 27 in decimal. The procedure is the same as that of decimallong 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:
Thus, thequotient 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.
The process oftaking a binary square root digit by digit is essentially the same as for a decimal square root but much simpler, due to the binary nature. First group the digits in pairs, using a leading 0 if necessary so there are an even number of digits. Now at each step, consider the answer so far, extended with the digits 01. If this can be subtracted from the current remainder, do so. Then extend the remainder with the next pair of digits. If you subtracted, the next digit of the answer is 1, otherwise it's 0.
1 1 1 1 1 0 1 1 0 1 ------------- ------------- ------------- ------------- -------------√ 10 10 10 01 √ 10 10 10 01 √ 10 10 10 01 √ 10 10 10 01 √ 10 10 10 01 - 1 - 1 - 1 - 1 Answer so far is 0, ---- ---- ---- ---- extended by 01 is 001, 1 10 1 10 1 10 1 10this CAN be subtracted - 1 01 - 1 01 - 1 01from first pair 10, Answer so far is 1, ------- ------- -------so first digit of extended by 01 is 101, 1 10 1 10 01 1 10 01answer is 1. this CAN be subtracted - 1 10 01 from remainder 110, so Answer so far is 11, Answer so far is 110, ---------- next answer digit is 1. extended by 01 is 1101, extended by 01 is 11001, 0 this is TOO BIG to this CAN be subtracted subtract from remainder from remainder 11001, so Done! 110, so next digit of next digit of answer is 1. answer is 0.
Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated usingBoolean logical operators. When a string of binary symbols is manipulated in this way, it is called abitwise operation; the logical operatorsAND,OR, andXOR may be performed on corresponding bits in two binary numerals provided as input. The logicalNOT 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, anarithmetic shift left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.
Conversion of (357)10 to binary notation results in (101100101)
To convert from a base-10integer to its base-2 (binary) equivalent, the number isdivided by two. The remainder is theleast-significant bit. The quotient is again divided by two; its remainder becomes the next least significant bit. This process repeats until a quotient of one is reached. The sequence of remainders (including the final quotient of one) forms the binary value, as each remainder must be either zero or one when dividing by two. For example, (357)10 is expressed as (101100101)2.[43]
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:
Prior value
× 2 +
Next bit
= Next value
0
× 2 +
1
= 1
1
× 2 +
0
= 2
2
× 2 +
0
= 4
4
× 2 +
1
= 9
9
× 2 +
0
= 18
18
× 2 +
1
= 37
37
× 2 +
0
= 74
74
× 2 +
1
= 149
149
× 2 +
1
= 299
299
× 2 +
0
= 598
598
× 2 +
1
=1197
The result is 119710. The first Prior Value of 0 is simply an initial decimal value. This method is an application of theHorner scheme.
Binary
1
0
0
1
0
1
0
1
1
0
1
Decimal
1×210 +
0×29 +
0×28 +
1×27 +
0×26 +
1×25 +
0×24 +
1×23 +
1×22 +
0×21 +
1×20 =
1197
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:
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 0.00011... . 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 1/10 + ... + 1/10 (addition of 10 numbers) differs from 1 in binaryfloating 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 withhexadecimal, 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 10k, wherek is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two areconcatenated. 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 10k and added to the second converted piece, wherek is the number of decimal digits in the second, least-significant piece before conversion.
Binary may be converted to and from hexadecimal more easily. This is because theradix 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 extra0 bits at the left (calledpadding). 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:
Binary is also easily converted to theoctal numeral system, since octal uses a radix of 8, which is apower 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 ofhexadecimal 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 forhexadecimal:
Non-integers can be represented by using negative powers, which are set off from the other digits by means of aradix point (called adecimal point in the decimal system). For example, the binary number 11.012 means:
1 × 21
(1 × 2 =2)
plus
1 × 20
(1 × 1 =1)
plus
0 × 2−1
(0 ×1⁄2 =0)
plus
1 × 2−2
(1 ×1⁄4 =0.25)
For a total of 3.25 decimal.
Alldyadic rational numbers have aterminating binary numeral—the binary representation has a finite number of terms after the radix point. Otherrational numbers have binary representation, but instead of terminating, theyrecur, 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 indecimal. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that0.111111... is the sum of thegeometric series 2−1 + 2−2 + 2−3 + ... which is 1.
Binary numerals that neither terminate nor recur representirrational 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, thesquare root of 2, another irrational. It has no discernible pattern.
^Marshall, Steve."Yijing hexagram sequences: The Shao Yong square (Fuxi sequence)". Retrieved15 September 2022.You could say [the Fuxi binary sequence] is a more sensible way of rendering hexagram as binary numbers ... The reasoning, if any, that informs [the King Wen] sequence is unknown.
^B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
^Landry, Timothy R. (2019).Vodún: secrecy and the search for divine power. Contemporary ethnography (1st ed.). Philadelphia: University of Pennsylvania Press. p. 25.ISBN978-0-8122-5074-9.
^Leibniz: "The Chinese lost the meaning of the Cova or Lineations of Fuxi, perhaps more than a thousand years ago, and they have written commentaries on the subject in which they have sought I know not what far out meanings, so that their true explanation now has to come from Europeans. Here is how: It was scarcely more than two years ago that I sent to Reverend Father Bouvet,3 the celebrated French Jesuit who lives in Peking, my method of counting by 0 and 1, and nothing more was required to make him recognize that this was the key to the figures of Fuxi. Writing to me on 14 November 1701, he sent me this philosophical prince's grand figure, which goes up to 64, and leaves no further room to doubt the truth of our interpretation, such that it can be said that this Father has deciphered the enigma of Fuxi, with the help of what I had communicated to him. And as these figures are perhaps the most ancient monument of [GM VII, p227] science which exists in the world, this restitution of their meaning, after such a great interval of time, will seem all the more curious."
^Aiton, Eric J. (1985).Leibniz: A Biography. Taylor & Francis. pp. 245–8.ISBN0-85274-470-6.
^Shannon, Claude Elwood (1940).A symbolic analysis of relay and switching circuits (Thesis). Cambridge: Massachusetts Institute of Technology.hdl:1721.1/11173.
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