6815:: "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, 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 science which exists in the world, this restitution of their meaning, after such a great interval of time, will seem all the more curious."
1165:
1911:
5945:
4442:
852:
907:
4181:
1305:
1560:
2478:
2526:. 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:
1180:, 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..
3245:. 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)
4437:{\displaystyle {\begin{aligned}x&=&1100&.1{\overline {01110}}\ldots \\x\times 2^{6}&=&1100101110&.{\overline {01110}}\ldots \\x\times 2&=&11001&.{\overline {01110}}\ldots \\x\times (2^{6}-2)&=&1100010101\\x&=&1100010101/111110\\x&=&(789/62)_{10}\end{aligned}}}
3260:
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
2661:
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
1172:
Leibniz wrote in excess of a hundred manuscripts on binary, most of them remaining unpublished. 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
1131:
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
2590:
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
1969:
In the binary system, each bit represents an increasing power of 2, with the rightmost bit representing 2, the next representing 2, then 2, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" bit. For example, the binary number 100101 is converted to decimal
893:
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
1123:
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
3164:
1 0 0 1 --------- √ 1010001 1 --------- 101 01 0 -------- 1001 100 0 -------- 10001 10001 10001 ------- 0
2510:
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:
1583:
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. In a modern computer, the numeric values may be represented by two different
965:
on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.
4533:
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, where
4175:
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:
2848:. 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.
5874:
2618:
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
5780:
1903:
1834:
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
4168:... . 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 binary
1364:
itchen", where he had assembled it), which calculated using binary addition. 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
3148:
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.
3112:
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:
1280:
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
3203:
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
2662:
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 1
4542:. 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 and added to the second converted piece, where
1027:. In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of
2614:
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
5254:
of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 2, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.
5784:
1424:(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:
1228:
he admired. Of this parallel invention, Liebniz 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."
1151:
investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers. Possibly the first publication of the system in Europe was by
2670:). 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.
3675:, 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.
3994:
3875:
3933:
3814:
1064:
4186:
3646:
5690:
1132:
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
981:
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.
6281:
Leibniz' Binary System and Shao Yong's "Xiantian Tu" in :Das
Neueste über China: G.W. Leibnizens Novissima Sinica von 1697 : Internationales Symposium, Berlin 4. bis 7. Oktober 1997
5878:
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
1224:
was an independent, parallel invention of binary notation. Leibniz & Bouvet concluded that this mapping was evidence of major
Chinese accomplishments in the sort of philosophical
5316:
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:
2057:+ ... = 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.
5350:(namely, 2, 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
2658:
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
817:, 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.
5671:
3716:
2877:
notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation, subtraction can be summarized by the following formula:
1674:, 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
874:, although this has been disputed). Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a
1870:), 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:
1341:
1189:"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
5926:
3756:
3673:
3593:
961:(1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the
2772:
1266:
of his own religious beliefs as a
Christian. Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of
1914:
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
6624:
3062:
2804:
5283:
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
2985:+ 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)
2740:
1287:
4528:
4508:
4488:
4468:
3222:
1666:
When spoken, binary numerals are usually read digit-by-digit, to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced
3122:
1 0 1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 ----- 1 1 1 − 1 0 1 ----- 0 1 0
1023:
in
Sanskrit. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern
723:
2555:. 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 = 10
1381:. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were
3561:
The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.
7061:
988:
oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy.
443:
878:
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
2493:
The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:
1678:(a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct
870:
fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of
7197:
7285:
7151:
6918:
6891:
6864:
6765:
6473:
6439:
6370:
6345:
6237:
6207:
6179:
1934:
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:
1918:
Binary counting follows the exact same procedure, and again the incremental substitution begins with the least significant binary digit, or
6266:
You could say is a more sensible way of rendering hexagram as binary numbers ... The reasoning, if any, that informs sequence is unknown.
4172:. 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.
2844:
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
6789:
Leibniz G., Explication de l'Arithmétique
Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.
2989:
7280:
7181:
1004:. He described meters in the form of short and long syllables (the latter equal in length to two short syllables). They were known as
276:
1195:. Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows:
6834:
6394:
6149:
6122:
6096:
3158:
1339:
that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled
837:. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India.
5882:. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that
1019:(8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to
6408:
1164:
67:
7078:
6018:
4538:
is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are
1858:. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the
1108:
890:
5869:{\displaystyle {\frac {12_{10}}{17_{10}}}={\frac {1100_{2}}{10001_{2}}}=0.1011010010110100{\overline {10110100}}\ldots \,_{2}}
3944:
3825:
7275:
6289:
5998:
5901:
0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational
510:
3886:
3767:
7245:
6622:
3179:
Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using
2870:
2815:
1370:
716:
291:
6957:
1220:
while a missionary in China, Leibniz explained his binary notation, and Bouvet demonstrated in his 1701 letters that the
7270:
846:
636:
3119:
The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:
1377:
on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a
646:
5354:
in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.
1233:
883:
463:
3598:
2563:. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100
1603:", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.
3157:
The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained
7265:
3234:
2854:* (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
895:
806:
523:
2977:(6.25 in decimal) ------------------- 1 . 0 1 1 0 1 ← Corresponds to a 'one' in
6910:
An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities
4169:
2911:
is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in
2851:* * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1
2038:
1409:
1263:
1153:
1091:
with binary tones are used to encode messages across Africa and Asia. Sets of binary combinations similar to the
925:
894:
is given by the binary representation of the second number. This method can be seen in use, for instance, in the
830:
619:
388:
5525:
Non-integers can be represented by using negative powers, which are set off from the other digits by means of a
2918:
Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:
5258:
To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:
709:
36:
6745:
1874:
000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
6913:(Macmillan, Dover Publications, reprinted with corrections ed.). New York: Cambridge University Press.
1016:
1325:
699:
483:
80:
6698:
1887:
090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
6454:
B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
5883:
5288:
3093:
1245:
879:
383:
299:
6013:
5775:{\displaystyle {\frac {1_{10}}{3_{10}}}={\frac {1_{2}}{11_{2}}}=0.01010101{\overline {01}}\ldots \,_{2}}
5642:
3242:
3196:
2642:
carry the 1 until it is one digit past the "string" below 1 1 1 0 1 1 1 1 1 0
1618:. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or
974:
962:
501:
6468:. Contemporary ethnography (1st ed.). Philadelphia: University of Pennsylvania Press. p. 25.
6053:
5944:
3681:
2638:
Traditional Carry Method Long Carry Method vs.
7135:
Arbeitsbuch Informatik – eine praxisorientierte Einführung in die Datenverarbeitung mit Projektaufgabe
5324:= (12 × 16) + (0 × 16) + (14 × 16) + (7 × 16) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,383
3469:. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the
1862:. When the available symbols for this position are exhausted, the least significant digit is reset to
6671:
6614:
5978:
2603:
zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of
1572:
1568:
596:
457:
450:
331:
1910:
7201:
6028:
5677:
binary numeral—the binary representation has a finite number of terms after the radix point. Other
3192:
3188:
3041:
3037:
2874:
2819:
2719:
2715:
2486:
2472:
2457:
1650:
1144:
1024:
790:
678:
543:
494:
306:
238:
93:
54:
3208:
left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.
6721:
6594:
5950:
5907:
3180:
3073:
1328:. His logical calculus was to become instrumental in the design of digital electronic circuitry.
943:
928:
591:
344:
181:
176:
123:
3734:
3651:
3571:
2041:. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 ×
6854:
6648:
3221:
2745:
1659:
6b100101 (a prefix indicating number of bits in binary format, common in programming languages)
946:, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the
7226:
7177:
7173:
Informatik für Ingenieure und Naturwissenschaftler: PC- und Mikrocomputertechnik, Rechnernetze
7147:
6914:
6887:
6881:
6860:
6830:
6761:
6586:
6555:
6469:
6435:
6390:
6366:
6341:
6318:
6285:
6279:
6233:
6203:
6175:
6145:
6139:
6118:
6092:
6086:
6023:
5968:
5894:
3184:
3174:
2014:
1596:
1374:
1268:
1133:
863:
834:
673:
663:
651:
631:
586:
581:
517:
349:
321:
228:
161:
151:
138:
103:
98:
6227:
6169:
6112:
3199:
may be performed on corresponding bits in two binary numerals provided as input. The logical
3145:. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2.
7139:
7070:
6939:
6753:
6713:
6679:
6545:
6535:
5963:
5929:
5887:
5678:
3648:, etc. So if there is a 1 in the first place after the decimal, then the number is at least
3205:
3047:
2777:
2579:
1630:
1382:
1080:
1001:
576:
470:
223:
211:
156:
146:
113:
88:
6501:
4446:
Another way of converting from binary to decimal, often quicker for a person familiar with
2725:
2635:), using the traditional carry method on the left, and the long carry method on the right:
2463:. Addition, subtraction, multiplication, and division can be performed on binary numerals.
1187:(published in 1703). The full title of Leibniz's article is translated into English as the
782:
that has a finite representation in the binary numeral system, that is, the quotient of an
7249:
6628:
6618:
6412:
5637:
4546:
is the number of decimal digits in the second, least-significant piece before conversion.
3238:
3116:
1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 ----- 0 0 1
2862:
2482:
1607:
1366:
1346:
779:
688:
658:
601:
571:
556:
316:
284:
256:
233:
216:
75:
7104:
2559:
again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11
668:
6675:
6431:
The mathematics of harmony: from Euclid to contemporary mathematics and computer science
1959:
00, 0101, 0110, 0111, (rightmost three bits start over, and the next bit is incremented)
825:
The modern binary number system was studied in Europe in the 16th and 17th centuries by
6550:
6523:
6425:
6082:
6078:
4513:
4493:
4473:
4453:
2892:
2866:
2460:
1866:, and the next digit of higher significance (one position to the left) is incremented (
1589:
1390:
1353:
1332:
1255:
1213:
1148:
826:
763:
750:
683:
626:
606:
561:
434:
166:
133:
118:
44:
6405:
6229:
Groundbreaking Scientific Experiments, Inventions, and Discoveries of the 18th Century
2551:). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10
851:
7259:
7049:
7006:"Pioneers – The people and ideas that made a difference – George Stibitz (1904–1995)"
6800:
6725:
6644:
6003:
5983:
5530:
4539:
3470:
3101:
3078:
2599:
is any integer length), adding 1 will result in the number 1 followed by a string of
1405:
1397:
1237:
1128:
1056:
489:
311:
251:
186:
128:
108:
7053:
7239:
7235:
7230:
6757:
5347:
2963:
2951:) --------- 0 0 0 0 ← Corresponds to the rightmost 'zero' in
2575:
1592:
1564:
1386:
1313:
1241:
954:
947:
935:
867:
856:
641:
566:
6812:
6340:. Blackwell ancient religions (1. publ ed.). Malden, Mass.: Wiley-Blackwell.
1662:#b100101 (a prefix indicating binary format, common in Lisp programming languages)
1304:
906:
7171:
7133:
6790:
6429:
5250:
Binary may be converted to and from hexadecimal more easily. This is because the
5973:
5526:
5434:
5351:
5218:
5176:
5134:
5092:
5046:
4555:
4447:
3141:, as shown on the top line, while the remainder, shown on the bottom line, is 10
2824:
2018:
1576:
1401:
1378:
1225:
1147:
for doing binary calculations using a non-positional representation by letters.
1140:
1120:
1100:
1028:
882:, approximately 2400 BC, and its fully developed hieroglyphic form dates to the
611:
476:
428:
418:
1906:
This counter shows how to count in binary from numbers zero through thirty-one.
7143:
6961:
5940:
2453:
2022:
1600:
810:
413:
171:
17:
7005:
6590:
6322:
3183:. When a string of binary symbols is manipulated in this way, it is called a
2915:
that was used. The sum of all these partial products gives the final result.
2650:
0 cross out the "string", + 1 0 1 0 1 1 0 0 1 1 + 1 0
1656:
0b100101 (a prefix indicating binary format, common in programming languages)
862:
The scribes of ancient Egypt used two different systems for their fractions,
7027:
6540:
6254:
6008:
5988:
5433:
Converting from octal to binary proceeds in the same fashion as it does for
2010:
1902:
1357:
1088:
1076:
958:
423:
6986:. Math & Computer Science Department, Denison University. 30 April 2004
6717:
6559:
5904:
1.0110101000001001111001100110011111110... is the binary representation of
1559:
1236:, a popular idea that would be followed closely by his successors such as
1232:
The relation was a central idea to his universal concept of a language or
7176:(in German). Vol. 2 (5 ed.). Vieweg, reprint: Springer-Verlag.
6908:
5993:
3200:
3126:
2940:
For example, the binary numbers 1011 and 1010 are multiplied as follows:
1952:
0, 0011, (rightmost two bits start over, and the next bit is incremented)
1104:
978:
814:
6943:
6598:
6574:
6306:
5685:, with a finite sequence of digits repeating indefinitely. For instance
6033:
5879:
3085:
2477:
2066:
1847:
1836:
1585:
1317:
1250:
1095:
have also been used in traditional African divination systems, such as
1084:
997:
916:
783:
408:
393:
7074:
6983:
6683:
3261:
organized in a multi-column table. For example, to convert 10010101101
2582:
2 for any two bits x and y allows for very fast calculation, as well.
1571:
to express binary values. In this clock, each column of LEDs shows a
1060:
985:
939:
759:
741:
398:
3074:
Division algorithm § Integer division (unsigned) with remainder
1096:
1040:
805:, or binary digit. Because of its straightforward implementation in
1055:
which has 64. The Ifá originated in 15th century West Africa among
920:
dates from the 9th century BC in China. The binary notation in the
855:
Arithmetic values thought to have been represented by parts of the
5958:
5343:
5337:
5251:
2714:
The binary addition table is similar to, but not the same as, the
2476:
2025:. As a result, 1/10 does not have a finite binary representation (
1909:
1901:
1619:
1558:
1321:
1303:
1190:
1163:
875:
871:
850:
794:
746:
403:
365:
326:
6255:"Yijing hexagram sequences: The Shao Yong square (Fuxi sequence)"
5513:= (1 × 8) + (2 × 8) + (7 × 8) = (1 × 64) + (2 × 8) + (7 × 1) = 87
3100:, or 27 in decimal. The procedure is the same as that of decimal
1360:, completed a relay-based computer he dubbed the "Model K" (for "
767:
6750:
Handbook of the History and Philosophy of Mathematical Practice
2959:+ 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0
1606:
In keeping with the customary representation of numerals using
1945:, (rightmost bit starts over, and the next bit is incremented)
1421:
1336:
802:
771:
7030:. Computer History Association of California. 6 February 1995
6856:
Leibniz: What Kind of Rationalist?: What Kind of Rationalist?
6662:
Shirley, John W. (1951). "Binary numeration before Leibniz".
6524:"Mangarevan invention of binary steps for easier calculation"
5681:
have binary representation, but instead of terminating, they
2895:
in binary is similar to its decimal counterpart. Two numbers
2535:
In this example, two numerals are being added together: 01101
1258:, who visited China in 1685 as a missionary. Leibniz saw the
1000:(c. 2nd century BC) developed a binary system for describing
6938:(Thesis). Cambridge: Massachusetts Institute of Technology.
5533:
in the decimal system). For example, the binary number 11.01
2489:, which adds two bits together, producing sum and carry bits
2037:). This causes 10 × 1/10 not to precisely equal 1 in binary
1065:
Masterpieces of the Oral and Intangible Heritage of Humanity
5893:
Binary numerals that neither terminate nor recur represent
5004:
2981:+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in
6752:, Cham: Springer International Publishing, pp. 1–31,
6612:
6575:"Diversity in the Numeral Systems of Australian Languages"
5346:
numeral system, since octal uses a radix of 8, which is a
4559:
1649:%100101 (a prefix indicating binary format; also known as
1629:
100101b (a suffix indicating binary format; also known as
7105:"Introducing binary – Revision 1 – GCSE Computer Science"
7054:"Konrad Zuse's Legacy: The Architecture of the Z1 and Z3"
6307:"Mapping the Entrails: The Practice of Greek Hepatoscopy"
4962:
4920:
4874:
4832:
4790:
4748:
4702:
4660:
4618:
4574:
3989:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1}
3870:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1}
2903:
can be multiplied by partial products: for each digit in
2532:
0 1 1 0 1 + 1 0 1 1 1 ------------- = 1 0 0 1 0 0 = 36
798:
7165:
7163:
7127:
7125:
2962:
Binary numbers can also be multiplied with bits after a
1610:, binary numbers are commonly written using the symbols
3928:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1}
3809:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1}
3081:
in binary is again similar to its decimal counterpart.
2607:
9s will result in the number 1 followed by a string of
7138:(in German). Vieweg-Verlag, reprint: Springer-Verlag.
6085:, eds. (2009), "Myth No. 2: the Horus eye fractions",
3947:
3889:
3828:
3770:
3737:
3684:
3654:
3601:
3574:
2574:
When computers must add two numbers, the rule that: x
2518:
7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 10) )
2515:
5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 10) )
1682:), but this does not make its binary nature explicit.
6958:"National Inventors Hall of Fame – George R. Stibitz"
6522:
Bender, Andrea; Beller, Sieghard (16 December 2013).
6284:. Stuttgart: Franz Steiner Verlag. pp. 165–170.
5910:
5787:
5693:
5645:
4516:
4496:
4476:
4456:
4184:
4008:... is equivalent to the repeating binary fraction 0.
3050:
2955:+ 1 0 1 1 ← Corresponds to the next 'one' in
2780:
2748:
2728:
2640:
1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ←
1276:
is not easy to impart to the pagans, is the creation
6168:
Edward Hacker; Steve Moore; Lorraine Patsco (2002).
5932:, another irrational. It has no discernible pattern.
2973:(5.625 in decimal) × 1 1 0 . 0 1
2506:
1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 2) )
7170:Küveler, Gerd; Schwoch, Dietrich (4 October 2007).
6936:
A symbolic analysis of relay and switching circuits
6744:Strickland, Lloyd (2020), Sriraman, Bharath (ed.),
6706:
Mitteilungen der deutschen Mathematiker-Vereinigung
3564:
In a fractional binary number such as 0.11010110101
3036:The binary multiplication table is the same as the
1342:
A Symbolic Analysis of Relay and Switching Circuits
5920:
5868:
5774:
5665:
4522:
4502:
4482:
4462:
4436:
3988:
3927:
3869:
3808:
3750:
3710:
3667:
3640:
3587:
3056:
2798:
2766:
2734:
6278:Zhonglian, Shi; Wenzhao, Li; Poser, Hans (2000).
6163:
6161:
6088:The Oxford Handbook of the History of Mathematics
3237:to its base-2 (binary) equivalent, the number is
1965:000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...
1646:(a subscript indicating base-2 (binary) notation)
1345:, Shannon's thesis essentially founded practical
813:, the binary system is used by almost all modern
6801:"Bouvet and Leibniz: A Scholarly Correspondence"
6141:How Mathematics Happened: The First 50,000 Years
3641:{\textstyle ({\frac {1}{2}})^{2}={\frac {1}{4}}}
1183:His most well known work appears in his article
6699:"Leibniz, Caramuel, Harriot und das Dualsystem"
6528:Proceedings of the National Academy of Sciences
6054:"3.3. Binary and Its Advantages — CS160 Reader"
2857:Subtracting a positive number is equivalent to
1420:Any number can be represented by a sequence of
1274:
1216:in 1700, who had made himself an expert on the
6466:Vodún: secrecy and the search for divine power
6365:. Boca Raton, Florida: CRC Press. p. 37.
6363:Microcontroller programming: the microchip PIC
6117:, Cambridge University Press, pp. 42–43,
1639:bin 100101 (a prefix indicating binary format)
866:(not related to the binary number system) and
6193:
6191:
2873:to handle negative numbers—most commonly the
1051:, but has up to 256 binary signs, unlike the
717:
27:Number expressed in the base-2 numeral system
8:
7244:Sir Francis Bacon's BiLiteral Cypher system
6639:
6637:
6221:
6219:
5503:= (6 × 8) + (5 × 8) = (6 × 8) + (5 × 1) = 53
4164:This is also a repeating binary fraction 0.0
3212:Conversion to and from other numeral systems
1626:100101 binary (explicit statement of format)
4450:, is to do so indirectly—first converting (
2456:in binary is much like arithmetic in other
1636:100101B (a suffix indicating binary format)
1254:through his contact with the French Jesuit
1212:While corresponding with the Jesuit priest
7132:Küveler, Gerd; Schwoch, Dietrich (2013) .
1622:. The following notations are equivalent:
724:
710:
60:
31:
6549:
6539:
6361:Sanchez, Julio; Canton, Maria P. (2007).
6114:Numerical Notation: A Comparative History
5911:
5909:
5860:
5859:
5845:
5831:
5821:
5815:
5804:
5794:
5788:
5786:
5766:
5765:
5751:
5737:
5727:
5721:
5710:
5700:
5694:
5692:
5655:
5646:
5644:
4515:
4495:
4475:
4455:
4424:
4412:
4382:
4336:
4306:
4261:
4236:
4209:
4185:
4183:
3970:
3948:
3946:
3909:
3890:
3888:
3851:
3829:
3827:
3790:
3771:
3769:
3738:
3736:
3702:
3688:
3683:
3655:
3653:
3628:
3619:
3605:
3600:
3575:
3573:
3229:to binary notation results in (101100101)
3049:
2779:
2747:
2727:
1839:counting system as a frame of reference.
6848:
6846:
6829:. Taylor & Francis. pp. 245–8.
6202:. Oxford University Press. p. 227.
6091:, Oxford University Press, p. 790,
5356:
4021:
3720:
3220:
2999:
2677:
2059:
1689:
1316:published a landmark paper detailing an
905:
7062:IEEE Annals of the History of Computing
6198:Redmond, Geoffrey; Hon, Tze-Ki (2014).
6045:
5342:Binary is also easily converted to the
43:
6785:
6783:
6488:
6144:, Prometheus Books, pp. 135–136,
4004:Thus the repeating decimal fraction 0.
2982:
2978:
2974:
2970:
2956:
2952:
2948:
2944:
2934:
2933:is 1, the partial product is equal to
2930:
2923:
2912:
2908:
2904:
2900:
2896:
2608:
2604:
2600:
2596:
2592:
1248:. Leibniz was first introduced to the
6739:
6737:
6735:
6573:Bowern, Claire; Zentz, Jason (2012).
5299:= 0101 0010 grouped with padding = 52
4490:in hexadecimal) and then converting (
1393:, who wrote about it in his memoirs.
1185:Explication de l'Arithmétique Binaire
1124:science and artificial intelligence.
7:
6232:. Greenwood Publishing. p. 29.
3108:goes into the first three digits 110
1926:), except that only the two symbols
1922:(the rightmost one, also called the
815:computers and computer-based devices
2926:is 0, the partial product is also 0
1262:hexagrams as an affirmation of the
934:It is based on taoistic duality of
762:that uses only two symbols for the
7028:"George Robert Stibitz – Obituary"
6171:I Ching: An Annotated Bibliography
5666:{\displaystyle {\frac {p}{2^{a}}}}
3711:{\textstyle ({\frac {1}{3}})_{10}}
1400:, which was designed and built by
25:
7008:. Kerry Redshaw. 20 February 2006
6311:The American Journal of Philology
898:, which dates to around 1650 BC.
801:. Each digit is referred to as a
7252:, predates binary number system.
7084:from the original on 3 July 2022
6960:. 20 August 2008. Archived from
5943:
2990:Booth's multiplication algorithm
1335:produced his master's thesis at
1208:1 0 0 0 numerical value 2
1205:0 1 0 0 numerical value 2
1202:0 0 1 0 numerical value 2
1199:0 0 0 1 numerical value 2
1176:His first known work on binary,
6934:Shannon, Claude Elwood (1940).
6883:Leibniz, Mysticism and Religion
6428:; Olsen, Scott Anthony (2009).
6019:Redundant binary representation
2907:, the product of that digit in
1312:In 1854, British mathematician
1115:Western predecessors to Leibniz
1109:Indigenous Australian languages
1075:The residents of the island of
1043:is an African divination system
1015:Pingala's Hindu classic titled
944:64 hexagrams ("sixty-four" gua)
891:ancient Egyptian multiplication
789:The base-2 numeral system is a
6886:. Springer. pp. 149–150.
6758:10.1007/978-3-030-19071-2_90-1
5999:Linear-feedback shift register
4421:
4406:
4348:
4329:
3699:
3685:
3616:
3602:
1850:counting uses the ten symbols
1063:added Ifá to its list of the "
1:
6651:. London. pp. Chapter 1.
6649:"The Advancement of Learning"
6336:Johnston, Sarah Iles (2008).
6111:Chrisomalis, Stephen (2010),
6058:computerscience.chemeketa.edu
5633:For a total of 3.25 decimal.
3092:, or 5 in decimal, while the
2871:signed number representations
2816:signed number representations
2530:1 1 1 1 1 (carried digits)
1371:American Mathematical Society
1143:described a system he called
7286:Power-of-two numeral systems
6385:W. S. Anglin and J. Lambek,
6138:Rudman, Peter Strom (2007),
5890:2 + 2 + 2 + ... which is 1.
5850:
5756:
4311:
4266:
4214:
2827:works in much the same way:
2591:digits composed entirely of
2392:1/16 + 1/128 + 1/1024 . . .
1599:may be used. A "positive", "
1404:between 1935 and 1938, used
1369:. In a demonstration to the
1272:or creation out of nothing.
1099:among others, as well as in
984:Divination at Ancient Greek
847:Ancient Egyptian mathematics
807:digital electronic circuitry
758:, a method for representing
6984:"George Stibitz : Bio"
6746:"Leibniz on Number Systems"
6664:American Journal of Physics
6579:Anthropological Linguistics
6464:Landry, Timothy R. (2019).
6406:Math for Poets and Drummers
5921:{\displaystyle {\sqrt {2}}}
5495:And from octal to decimal:
3751:{\textstyle {\frac {1}{3}}}
3668:{\textstyle {\frac {1}{2}}}
3588:{\textstyle {\frac {1}{2}}}
3444:
3428:
3412:
3396:
3380:
3364:
3348:
3332:
3316:
3300:
3284:
3249:is expressed as (101100101)
2375:1/16 + 1/128 + 1/256 . . .
1575:numeral of the traditional
1324:that would become known as
1234:characteristica universalis
1178:“On the Binary Progression"
1083:were using a hybrid binary-
884:Nineteenth Dynasty of Egypt
7302:
6226:Jonathan Shectman (2003).
5462:And from binary to octal:
5335:
5076:
4904:
4732:
4590:
4553:
3233:To convert from a base-10
3172:
3084:In the example below, the
3071:
2813:
2470:
2442:1/32 + 1/64 + 1/128 . . .
2358:1/16 + 1/64 + 1/256 . . .
2341:1/16 + 1/64 + 1/128 . . .
2324:1/16 + 1/32 + 1/256 . . .
1173:unrelated to mathematics.
977:divided the outer edge of
896:Rhind Mathematical Papyrus
844:
444:Non-standard radices/bases
7281:Gottfried Wilhelm Leibniz
7248:23 September 2016 at the
7144:10.1007/978-3-322-92907-5
6859:. Springer. p. 415.
6174:. Routledge. p. 13.
5521:Representing real numbers
5488:grouped with padding = 23
5287:bits at the left (called
4170:floating point arithmetic
3181:Boolean logical operators
2767:{\displaystyle 1\lor 1=1}
2742:. The difference is that
2627:) and 1 0 1 0 1 1 0 0 1 1
2299:1/16 + 1/32 + 1/64 . . .
2282:1/16 + 1/32 + 1/64 . . .
2249:1/8 + 1/64 + 1/512 . . .
2232:1/8 + 1/32 + 1/128 . . .
2215:1/8 + 1/16 + 1/128 . . .
2074:Fractional approximation
2039:floating-point arithmetic
1697:
1692:
1154:Juan Caramuel y Lobkowitz
1119:In the late 13th century
924:is used to interpret its
886:, approximately 1200 BC.
831:Juan Caramuel y Lobkowitz
6338:Ancient Greek divination
5309:= 1101 1101 grouped = DD
3187:; the logical operators
2947:) × 1 0 1 0 (
2190:1/8 + 1/16 + 1/32 . . .
2157:1/4 + 1/16 + 1/64 . . .
1286:Leibniz's letter to the
7236:Conversion of Fractions
6907:Boole, George (2009) .
6825:Aiton, Eric J. (1985).
6541:10.1073/pnas.1309160110
6502:"Ifa Divination System"
6305:Collins, Derek (2008).
5638:dyadic rational numbers
3241:. The remainder is the
3104:; here, the divisor 101
2140:1/4 + 1/8 + 1/16 . . .
700:List of numeral systems
6880:Yuen-Ting Lai (1998).
6718:10.1515/dmvm-2008-0009
6387:The Heritage of Thales
5922:
5870:
5776:
5667:
4524:
4510:in hexadecimal) into (
4504:
4484:
4464:
4438:
3990:
3929:
3871:
3810:
3752:
3712:
3669:
3642:
3589:
3230:
3058:
3057:{\displaystyle \land }
2800:
2799:{\displaystyle 1+1=10}
2768:
2736:
2490:
1915:
1907:
1580:
1410:floating-point numbers
1309:
1297:
1169:
940:Eight trigrams (Bagua)
911:
880:Fifth Dynasty of Egypt
859:
7276:Elementary arithmetic
6853:J.E.H. Smith (2008).
6697:Ineichen, R. (2008).
6621:, Fidora et al. 2011
6014:Reduction of summands
5923:
5871:
5777:
5668:
4525:
4505:
4485:
4465:
4439:
3991:
3930:
3872:
3811:
3753:
3713:
3670:
3643:
3590:
3568:, the first digit is
3243:least-significant bit
3224:
3059:
2882:A − B = A + not B + 1
2814:Further information:
2801:
2769:
2737:
2735:{\displaystyle \lor }
2480:
2013:in binary arithmetic
1913:
1905:
1884: ...
1562:
1307:
1246:modern symbolic logic
1167:
1111:use a base-2 system.
963:least significant bit
909:
854:
756:binary numeral system
68:Hindu–Arabic numerals
6827:Leibniz: A Biography
6627:8 April 2019 at the
6617:3 April 2014 at the
6434:. World Scientific.
6411:16 June 2012 at the
6200:Teaching the I Ching
5979:Binary-coded decimal
5908:
5785:
5691:
5643:
4514:
4494:
4474:
4454:
4182:
3945:
3887:
3826:
3768:
3735:
3682:
3652:
3599:
3572:
3048:
2996:Multiplication table
2778:
2746:
2726:
1573:binary-coded decimal
1087:system before 1450.
889:The method used for
778:may also refer to a
597:Prehistoric counting
373:Common radices/bases
55:Place-value notation
7271:Computer arithmetic
7052:(April–June 1997).
6676:1951AmJPh..19..452S
6083:Stedall, Jacqueline
4015:Or for example, 0.1
3225:Conversion of (357)
3042:logical conjunction
2834:0 − 1 → 1, borrow 1
2720:logical disjunction
2473:Adder (electronics)
2458:positional notation
2409:1/16 + 1/256 . . .
2383:0.0714285714285...
2107:1/2 + 1/4 + 1/8...
1651:Motorola convention
1597:magnetic polarities
1145:location arithmetic
1025:positional notation
1012:(heavy) syllables.
996:The Indian scholar
970:Classical antiquity
791:positional notation
786:by a power of two.
544:Sign-value notation
7204:on 23 October 2017
6389:, Springer, 1995,
5951:Mathematics portal
5918:
5895:irrational numbers
5886:is the sum of the
5866:
5843:0.1011010010110100
5772:
5663:
4520:
4500:
4480:
4460:
4434:
4432:
3986:
3925:
3867:
3806:
3748:
3708:
3665:
3638:
3585:
3465:The result is 1197
3231:
3169:Bitwise operations
3054:
2969:1 0 1 . 1 0 1
2796:
2764:
2732:
2491:
2366:0.076923076923...
2240:0.142857142857...
2029:has prime factors
1916:
1908:
1686:Counting in binary
1581:
1356:, then working at
1352:In November 1937,
1310:
1300:Later developments
1290:attached with the
1170:
1107:. The majority of
950:of ancient China.
912:
864:Egyptian fractions
860:
200:East Asian systems
7266:Binary arithmetic
7153:978-3-528-04952-2
7075:10.1109/85.586067
6920:978-1-108-00153-3
6893:978-0-7923-5223-5
6866:978-1-4020-8668-7
6767:978-3-030-19071-2
6684:10.1119/1.1933042
6611:(see Bonner 2007
6475:978-0-8122-5074-9
6441:978-981-277-582-5
6372:978-0-8493-7189-9
6347:978-1-4051-1573-5
6253:Marshall, Steve.
6239:978-0-313-32015-6
6209:978-0-19-976681-9
6181:978-0-415-93969-0
6024:Repeating decimal
5969:Bitwise operation
5916:
5853:
5837:
5810:
5759:
5743:
5716:
5661:
5631:
5630:
5431:
5430:
5248:
5247:
4523:{\displaystyle x}
4503:{\displaystyle x}
4483:{\displaystyle x}
4470:in binary) into (
4463:{\displaystyle x}
4314:
4269:
4217:
4162:
4161:
4019:, in binary, is:
4002:
4001:
3978:
3956:
3917:
3898:
3859:
3837:
3798:
3779:
3746:
3718:, in binary, is:
3696:
3663:
3636:
3613:
3583:
3559:
3558:
3463:
3462:
3256:Binary to decimal
3217:Decimal to binary
3185:bitwise operation
3175:Bitwise operation
3161:. An example is:
3034:
3033:
2712:
2711:
2586:Long carry method
2522:This is known as
2449:Binary arithmetic
2446:
2445:
1987:= + + + + +
1978:= + + + + +
1970:form as follows:
1893:00, 101, 102, ...
1832:
1831:
1557:
1556:
1375:Dartmouth College
1288:Duke of Brunswick
1269:creatio ex nihilo
1168:Gottfried Leibniz
1021:science of meters
979:divination livers
835:Gottfried Leibniz
766:: typically "0" (
744:expressed in the
734:
733:
533:
532:
16:(Redirected from
7293:
7214:
7213:
7211:
7209:
7200:. Archived from
7194:
7188:
7187:
7186:. 9783834891914.
7167:
7158:
7157:
7156:. 9783322929075.
7129:
7120:
7119:
7117:
7115:
7101:
7095:
7093:
7091:
7089:
7083:
7058:
7046:
7040:
7039:
7037:
7035:
7024:
7018:
7017:
7015:
7013:
7002:
6996:
6995:
6993:
6991:
6980:
6974:
6973:
6971:
6969:
6954:
6948:
6947:
6931:
6925:
6924:
6904:
6898:
6897:
6877:
6871:
6870:
6850:
6841:
6840:
6822:
6816:
6810:
6804:
6803:, Swiderski 1980
6798:
6792:
6787:
6778:
6777:
6776:
6774:
6741:
6730:
6729:
6703:
6694:
6688:
6687:
6659:
6653:
6652:
6641:
6632:
6609:
6603:
6602:
6570:
6564:
6563:
6553:
6543:
6534:(4): 1322–1327.
6519:
6513:
6512:
6510:
6508:
6498:
6492:
6486:
6480:
6479:
6461:
6455:
6452:
6446:
6445:
6422:
6416:
6403:
6397:
6383:
6377:
6376:
6358:
6352:
6351:
6333:
6327:
6326:
6302:
6296:
6295:
6275:
6269:
6268:
6263:
6261:
6250:
6244:
6243:
6223:
6214:
6213:
6195:
6186:
6185:
6165:
6156:
6154:
6135:
6129:
6127:
6108:
6102:
6101:
6075:
6069:
6068:
6066:
6064:
6050:
6029:Two's complement
5964:Balanced ternary
5953:
5948:
5947:
5930:square root of 2
5927:
5925:
5924:
5919:
5917:
5912:
5897:. For instance,
5888:geometric series
5875:
5873:
5872:
5867:
5865:
5864:
5854:
5846:
5838:
5836:
5835:
5826:
5825:
5816:
5811:
5809:
5808:
5799:
5798:
5789:
5781:
5779:
5778:
5773:
5771:
5770:
5760:
5752:
5744:
5742:
5741:
5732:
5731:
5722:
5717:
5715:
5714:
5705:
5704:
5695:
5679:rational numbers
5672:
5670:
5669:
5664:
5662:
5660:
5659:
5647:
5622:
5621:
5617:
5594:
5593:
5589:
5540:
5539:
5357:
5291:). For example:
4560:
4529:
4527:
4526:
4521:
4509:
4507:
4506:
4501:
4489:
4487:
4486:
4481:
4469:
4467:
4466:
4461:
4443:
4441:
4440:
4435:
4433:
4429:
4428:
4416:
4386:
4341:
4340:
4315:
4307:
4270:
4262:
4241:
4240:
4218:
4210:
4167:
4022:
4011:
4007:
3995:
3993:
3992:
3987:
3979:
3971:
3957:
3949:
3934:
3932:
3931:
3926:
3918:
3910:
3899:
3891:
3876:
3874:
3873:
3868:
3860:
3852:
3838:
3830:
3815:
3813:
3812:
3807:
3799:
3791:
3780:
3772:
3757:
3755:
3754:
3749:
3747:
3739:
3721:
3717:
3715:
3714:
3709:
3707:
3706:
3697:
3689:
3674:
3672:
3671:
3666:
3664:
3656:
3647:
3645:
3644:
3639:
3637:
3629:
3624:
3623:
3614:
3606:
3594:
3592:
3591:
3586:
3584:
3576:
3476:
3475:
3268:
3267:
3206:arithmetic shift
3063:
3061:
3060:
3055:
3000:
2929:If the digit in
2922:If the digit in
2883:
2875:two's complement
2869:. Computers use
2820:two's complement
2805:
2803:
2802:
2797:
2773:
2771:
2770:
2765:
2741:
2739:
2738:
2733:
2678:
2641:
2531:
2439:
2435:
2431:
2424:
2420:
2406:
2389:
2372:
2355:
2338:
2321:
2314:
2310:
2296:
2279:
2275:
2271:
2264:
2260:
2246:
2229:
2212:
2205:
2201:
2187:
2183:
2179:
2172:
2168:
2154:
2137:
2133:
2129:
2122:
2118:
2104:
2100:
2096:
2089:
2085:
2060:
1881:0, 011, 012, ...
1843:Decimal counting
1690:
1631:Intel convention
1427:
1426:
1383:John von Neumann
1295:
1081:French Polynesia
726:
719:
712:
515:
499:
481:
471:balanced ternary
468:
455:
61:
32:
21:
7301:
7300:
7296:
7295:
7294:
7292:
7291:
7290:
7256:
7255:
7250:Wayback Machine
7223:
7218:
7217:
7207:
7205:
7196:
7195:
7191:
7184:
7169:
7168:
7161:
7154:
7131:
7130:
7123:
7113:
7111:
7103:
7102:
7098:
7087:
7085:
7081:
7056:
7048:
7047:
7043:
7033:
7031:
7026:
7025:
7021:
7011:
7009:
7004:
7003:
6999:
6989:
6987:
6982:
6981:
6977:
6967:
6965:
6956:
6955:
6951:
6933:
6932:
6928:
6921:
6906:
6905:
6901:
6894:
6879:
6878:
6874:
6867:
6852:
6851:
6844:
6837:
6824:
6823:
6819:
6811:
6807:
6799:
6795:
6788:
6781:
6772:
6770:
6768:
6743:
6742:
6733:
6701:
6696:
6695:
6691:
6661:
6660:
6656:
6643:
6642:
6635:
6629:Wayback Machine
6619:Wayback Machine
6610:
6606:
6572:
6571:
6567:
6521:
6520:
6516:
6506:
6504:
6500:
6499:
6495:
6487:
6483:
6476:
6463:
6462:
6458:
6453:
6449:
6442:
6426:Stakhov, Alexey
6424:
6423:
6419:
6413:Wayback Machine
6404:
6400:
6384:
6380:
6373:
6360:
6359:
6355:
6348:
6335:
6334:
6330:
6304:
6303:
6299:
6292:
6277:
6276:
6272:
6259:
6257:
6252:
6251:
6247:
6240:
6225:
6224:
6217:
6210:
6197:
6196:
6189:
6182:
6167:
6166:
6159:
6152:
6137:
6136:
6132:
6125:
6110:
6109:
6105:
6099:
6079:Robson, Eleanor
6077:
6076:
6072:
6062:
6060:
6052:
6051:
6047:
6042:
5949:
5942:
5939:
5906:
5905:
5858:
5827:
5817:
5800:
5790:
5783:
5782:
5764:
5733:
5723:
5706:
5696:
5689:
5688:
5651:
5641:
5640:
5619:
5615:
5614:
5591:
5587:
5586:
5536:
5523:
5516:
5512:
5506:
5502:
5491:
5487:
5483:
5477:
5473:
5469:
5458:
5454:
5448:
5444:
5340:
5334:
5327:
5323:
5312:
5308:
5302:
5298:
5279:
5275:
5269:
5265:
5232:
5222:
5212:
5190:
5180:
5170:
5148:
5138:
5128:
5106:
5096:
5086:
5060:
5050:
5040:
5018:
5008:
4998:
4976:
4966:
4956:
4934:
4924:
4914:
4888:
4878:
4868:
4846:
4836:
4826:
4804:
4794:
4784:
4762:
4752:
4742:
4716:
4706:
4696:
4674:
4664:
4654:
4632:
4622:
4612:
4588:
4578:
4568:
4558:
4552:
4512:
4511:
4492:
4491:
4472:
4471:
4452:
4451:
4431:
4430:
4420:
4404:
4397:
4391:
4390:
4377:
4370:
4364:
4363:
4358:
4351:
4332:
4320:
4319:
4299:
4294:
4287:
4275:
4274:
4254:
4249:
4242:
4232:
4223:
4222:
4204:
4199:
4192:
4180:
4179:
4165:
4018:
4009:
4005:
3943:
3942:
3885:
3884:
3824:
3823:
3766:
3765:
3733:
3732:
3698:
3680:
3679:
3650:
3649:
3615:
3597:
3596:
3570:
3569:
3567:
3468:
3264:
3258:
3252:
3248:
3228:
3219:
3214:
3177:
3171:
3166:
3155:
3144:
3140:
3136:
3132:
3123:
3117:
3111:
3107:
3099:
3091:
3076:
3070:
3046:
3045:
2998:
2986:
2984:
2980:
2976:
2972:
2960:
2958:
2954:
2950:
2946:
2936:
2932:
2925:
2914:
2910:
2906:
2902:
2898:
2890:
2881:
2863:negative number
2855:
2852:
2822:
2812:
2776:
2775:
2744:
2743:
2724:
2723:
2676:
2669:
2665:
2659:
2639:
2634:
2630:
2626:
2622:
2616:
2610:
2606:
2602:
2598:
2594:
2588:
2570:
2566:
2562:
2558:
2554:
2550:
2546:
2542:
2538:
2533:
2529:
2483:circuit diagram
2475:
2469:
2461:numeral systems
2451:
2437:
2433:
2429:
2422:
2418:
2404:
2387:
2370:
2353:
2336:
2319:
2312:
2308:
2294:
2277:
2273:
2269:
2262:
2258:
2244:
2227:
2210:
2203:
2199:
2185:
2181:
2177:
2170:
2166:
2152:
2135:
2131:
2127:
2120:
2116:
2102:
2098:
2094:
2087:
2083:
2008:
2000:
1996:
1986:
1977:
1900:
1898:Binary counting
1845:
1699:
1694:
1688:
1645:
1608:Arabic numerals
1418:
1367:complex numbers
1347:digital circuit
1326:Boolean algebra
1302:
1296:
1285:
1162:
1117:
1073:
1047:Similar to the
1037:
994:
972:
904:
849:
843:
823:
780:rational number
764:natural numbers
730:
694:
693:
616:
602:Proto-cuneiform
547:
546:
535:
534:
529:
528:
513:
497:
479:
466:
453:
440:
369:
368:
356:
355:
336:
296:
281:
272:
271:
262:
261:
243:
202:
201:
192:
191:
143:
85:
71:
70:
58:
57:
45:Numeral systems
28:
23:
22:
15:
12:
11:
5:
7299:
7297:
7289:
7288:
7283:
7278:
7273:
7268:
7258:
7257:
7254:
7253:
7242:
7233:
7222:
7221:External links
7219:
7216:
7215:
7189:
7183:978-3834891914
7182:
7159:
7152:
7121:
7096:
7041:
7019:
6997:
6975:
6964:on 9 July 2010
6949:
6926:
6919:
6899:
6892:
6872:
6865:
6842:
6835:
6817:
6805:
6793:
6779:
6766:
6731:
6689:
6670:(8): 452–454.
6654:
6645:Bacon, Francis
6633:
6604:
6585:(2): 133–160.
6565:
6514:
6493:
6491:, p. 154.
6481:
6474:
6456:
6447:
6440:
6417:
6398:
6378:
6371:
6353:
6346:
6328:
6317:(3): 319–345.
6297:
6290:
6270:
6245:
6238:
6215:
6208:
6187:
6180:
6157:
6150:
6130:
6123:
6103:
6097:
6070:
6044:
6043:
6041:
6038:
6037:
6036:
6031:
6026:
6021:
6016:
6011:
6006:
6001:
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5961:
5955:
5954:
5938:
5935:
5934:
5933:
5915:
5902:
5863:
5857:
5852:
5849:
5844:
5841:
5834:
5830:
5824:
5820:
5814:
5807:
5803:
5797:
5793:
5769:
5763:
5758:
5755:
5750:
5747:
5740:
5736:
5730:
5726:
5720:
5713:
5709:
5703:
5699:
5658:
5654:
5650:
5629:
5628:
5611:
5604:
5603:
5600:
5583:
5576:
5575:
5572:
5565:
5558:
5557:
5554:
5547:
5534:
5522:
5519:
5518:
5517:
5514:
5510:
5507:
5504:
5500:
5493:
5492:
5489:
5485:
5481:
5478:
5475:
5471:
5467:
5460:
5459:
5456:
5452:
5449:
5446:
5442:
5429:
5428:
5425:
5421:
5420:
5417:
5413:
5412:
5409:
5405:
5404:
5401:
5397:
5396:
5393:
5389:
5388:
5385:
5381:
5380:
5377:
5373:
5372:
5369:
5365:
5364:
5361:
5336:Main article:
5333:
5330:
5329:
5328:
5325:
5321:
5314:
5313:
5310:
5306:
5303:
5300:
5296:
5281:
5280:
5277:
5273:
5270:
5267:
5263:
5246:
5245:
5242:
5239:
5236:
5233:
5230:
5227:
5224:
5220:
5216:
5213:
5210:
5204:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5178:
5174:
5171:
5168:
5162:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5136:
5132:
5129:
5126:
5120:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5094:
5090:
5087:
5084:
5078:
5077:
5074:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5048:
5044:
5041:
5038:
5032:
5031:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
5006:
5002:
4999:
4996:
4990:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4964:
4960:
4957:
4954:
4948:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4922:
4918:
4915:
4912:
4906:
4905:
4902:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4876:
4872:
4869:
4866:
4860:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4834:
4830:
4827:
4824:
4818:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4792:
4788:
4785:
4782:
4776:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4750:
4746:
4743:
4740:
4734:
4733:
4730:
4729:
4726:
4723:
4720:
4717:
4714:
4711:
4708:
4704:
4700:
4697:
4694:
4688:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4662:
4658:
4655:
4652:
4646:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4620:
4616:
4613:
4610:
4604:
4603:
4600:
4597:
4594:
4591:
4589:
4586:
4583:
4580:
4576:
4572:
4569:
4566:
4554:Main article:
4551:
4548:
4519:
4499:
4479:
4459:
4427:
4423:
4419:
4415:
4411:
4408:
4405:
4403:
4400:
4398:
4396:
4393:
4392:
4389:
4385:
4381:
4378:
4376:
4373:
4371:
4369:
4366:
4365:
4362:
4359:
4357:
4354:
4352:
4350:
4347:
4344:
4339:
4335:
4331:
4328:
4325:
4322:
4321:
4318:
4313:
4310:
4305:
4302:
4300:
4298:
4295:
4293:
4290:
4288:
4286:
4283:
4280:
4277:
4276:
4273:
4268:
4265:
4260:
4257:
4255:
4253:
4250:
4248:
4245:
4243:
4239:
4235:
4231:
4228:
4225:
4224:
4221:
4216:
4213:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4191:
4188:
4187:
4160:
4159:
4156:
4148:
4147:
4144:
4136:
4135:
4132:
4124:
4123:
4120:
4112:
4111:
4108:
4100:
4099:
4096:
4088:
4087:
4084:
4076:
4075:
4072:
4064:
4063:
4060:
4052:
4051:
4048:
4040:
4039:
4036:
4030:
4029:
4026:
4016:
4000:
3999:
3996:
3985:
3982:
3977:
3974:
3969:
3966:
3963:
3960:
3955:
3952:
3939:
3938:
3935:
3924:
3921:
3916:
3913:
3908:
3905:
3902:
3897:
3894:
3881:
3880:
3877:
3866:
3863:
3858:
3855:
3850:
3847:
3844:
3841:
3836:
3833:
3820:
3819:
3816:
3805:
3802:
3797:
3794:
3789:
3786:
3783:
3778:
3775:
3762:
3761:
3758:
3745:
3742:
3729:
3728:
3725:
3705:
3701:
3695:
3692:
3687:
3662:
3659:
3635:
3632:
3627:
3622:
3618:
3612:
3609:
3604:
3582:
3579:
3565:
3557:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3519:Decimal
3516:
3515:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3466:
3461:
3460:
3454:
3449:
3446:
3442:
3441:
3438:
3433:
3430:
3426:
3425:
3422:
3417:
3414:
3410:
3409:
3406:
3401:
3398:
3394:
3393:
3390:
3385:
3382:
3378:
3377:
3374:
3369:
3366:
3362:
3361:
3358:
3353:
3350:
3346:
3345:
3342:
3337:
3334:
3330:
3329:
3326:
3321:
3318:
3314:
3313:
3310:
3305:
3302:
3298:
3297:
3294:
3289:
3286:
3282:
3281:
3278:
3275:
3272:
3262:
3257:
3254:
3250:
3246:
3239:divided by two
3226:
3218:
3215:
3213:
3210:
3173:Main article:
3170:
3167:
3163:
3154:
3151:
3142:
3138:
3134:
3133:divided by 101
3130:
3121:
3115:
3109:
3105:
3097:
3089:
3069:
3066:
3053:
3032:
3031:
3028:
3025:
3021:
3020:
3017:
3014:
3010:
3009:
3006:
3003:
2997:
2994:
2968:
2942:
2938:
2937:
2927:
2893:Multiplication
2889:
2888:Multiplication
2886:
2885:
2884:
2867:absolute value
2853:
2850:
2842:
2841:
2838:
2835:
2832:
2811:
2808:
2795:
2792:
2789:
2786:
2783:
2763:
2760:
2757:
2754:
2751:
2731:
2710:
2709:
2706:
2703:
2699:
2698:
2695:
2692:
2688:
2687:
2684:
2681:
2675:
2674:Addition table
2672:
2667:
2663:
2637:
2632:
2628:
2624:
2620:
2613:
2587:
2584:
2568:
2564:
2560:
2556:
2552:
2548:
2544:
2540:
2536:
2528:
2520:
2519:
2516:
2508:
2507:
2504:
2501:
2498:
2471:Main article:
2468:
2465:
2450:
2447:
2444:
2443:
2440:
2426:
2415:
2411:
2410:
2407:
2401:
2398:
2394:
2393:
2390:
2384:
2381:
2377:
2376:
2373:
2367:
2364:
2360:
2359:
2356:
2350:
2347:
2343:
2342:
2339:
2333:
2330:
2326:
2325:
2322:
2316:
2305:
2301:
2300:
2297:
2291:
2288:
2284:
2283:
2280:
2266:
2255:
2251:
2250:
2247:
2241:
2238:
2234:
2233:
2230:
2224:
2221:
2217:
2216:
2213:
2207:
2196:
2192:
2191:
2188:
2174:
2163:
2159:
2158:
2155:
2149:
2146:
2142:
2141:
2138:
2124:
2113:
2109:
2108:
2105:
2091:
2080:
2076:
2075:
2072:
2069:
2064:
2007:
2004:
2003:
2002:
1998:
1994:
1989:
1988:
1984:
1980:
1979:
1975:
1967:
1966:
1960:
1953:
1946:
1939:
1899:
1896:
1895:
1894:
1888:
1885:
1882:
1875:
1844:
1841:
1830:
1829:
1826:
1822:
1821:
1818:
1814:
1813:
1810:
1806:
1805:
1802:
1798:
1797:
1794:
1790:
1789:
1786:
1782:
1781:
1778:
1774:
1773:
1770:
1766:
1765:
1762:
1758:
1757:
1754:
1750:
1749:
1746:
1742:
1741:
1738:
1734:
1733:
1730:
1726:
1725:
1722:
1718:
1717:
1714:
1710:
1709:
1706:
1702:
1701:
1696:
1687:
1684:
1670:, rather than
1664:
1663:
1660:
1657:
1654:
1647:
1643:
1640:
1637:
1634:
1627:
1555:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1523:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1491:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1459:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1417:
1416:Representation
1414:
1391:Norbert Wiener
1373:conference at
1354:George Stibitz
1333:Claude Shannon
1301:
1298:
1283:
1256:Joachim Bouvet
1214:Joachim Bouvet
1210:
1209:
1206:
1203:
1200:
1161:
1158:
1149:Thomas Harriot
1134:Bacon's cipher
1116:
1113:
1072:
1071:Other cultures
1069:
1036:
1033:
993:
990:
971:
968:
903:
900:
842:
839:
827:Thomas Harriot
822:
819:
751:numeral system
732:
731:
729:
728:
721:
714:
706:
703:
702:
696:
695:
692:
691:
686:
681:
676:
671:
666:
661:
656:
655:
654:
649:
644:
634:
629:
623:
622:
615:
614:
609:
604:
599:
594:
589:
584:
579:
574:
569:
564:
559:
553:
552:
551:Non-alphabetic
548:
542:
541:
540:
537:
536:
531:
530:
527:
526:
521:
508:
492:
487:
474:
461:
447:
446:
439:
438:
431:
426:
421:
416:
411:
406:
401:
396:
391:
386:
381:
375:
374:
370:
363:
362:
361:
358:
357:
354:
353:
347:
341:
340:
335:
334:
329:
324:
319:
314:
309:
303:
302:
300:Post-classical
295:
294:
288:
287:
280:
279:
273:
269:
268:
267:
264:
263:
260:
259:
254:
248:
247:
242:
241:
236:
231:
226:
221:
220:
219:
208:
207:
203:
199:
198:
197:
194:
193:
190:
189:
184:
179:
174:
169:
164:
159:
154:
149:
142:
141:
136:
131:
126:
121:
116:
111:
106:
101:
96:
91:
84:
83:
81:Eastern Arabic
78:
76:Western Arabic
72:
66:
65:
64:
59:
53:
52:
51:
48:
47:
41:
40:
26:
24:
18:Binary numbers
14:
13:
10:
9:
6:
4:
3:
2:
7298:
7287:
7284:
7282:
7279:
7277:
7274:
7272:
7269:
7267:
7264:
7263:
7261:
7251:
7247:
7243:
7241:
7237:
7234:
7232:
7228:
7227:Binary System
7225:
7224:
7220:
7203:
7199:
7198:"Base System"
7193:
7190:
7185:
7179:
7175:
7174:
7166:
7164:
7160:
7155:
7149:
7145:
7141:
7137:
7136:
7128:
7126:
7122:
7110:
7106:
7100:
7097:
7080:
7076:
7072:
7068:
7064:
7063:
7055:
7051:
7045:
7042:
7029:
7023:
7020:
7007:
7001:
6998:
6985:
6979:
6976:
6963:
6959:
6953:
6950:
6945:
6941:
6937:
6930:
6927:
6922:
6916:
6912:
6911:
6903:
6900:
6895:
6889:
6885:
6884:
6876:
6873:
6868:
6862:
6858:
6857:
6849:
6847:
6843:
6838:
6836:0-85274-470-6
6832:
6828:
6821:
6818:
6814:
6809:
6806:
6802:
6797:
6794:
6791:
6786:
6784:
6780:
6769:
6763:
6759:
6755:
6751:
6747:
6740:
6738:
6736:
6732:
6727:
6723:
6719:
6715:
6711:
6708:(in German).
6707:
6700:
6693:
6690:
6685:
6681:
6677:
6673:
6669:
6665:
6658:
6655:
6650:
6646:
6640:
6638:
6634:
6630:
6626:
6623:
6620:
6616:
6613:
6608:
6605:
6600:
6596:
6592:
6588:
6584:
6580:
6576:
6569:
6566:
6561:
6557:
6552:
6547:
6542:
6537:
6533:
6529:
6525:
6518:
6515:
6503:
6497:
6494:
6490:
6485:
6482:
6477:
6471:
6467:
6460:
6457:
6451:
6448:
6443:
6437:
6433:
6432:
6427:
6421:
6418:
6414:
6410:
6407:
6402:
6399:
6396:
6395:0-387-94544-X
6392:
6388:
6382:
6379:
6374:
6368:
6364:
6357:
6354:
6349:
6343:
6339:
6332:
6329:
6324:
6320:
6316:
6312:
6308:
6301:
6298:
6293:
6287:
6283:
6282:
6274:
6271:
6267:
6256:
6249:
6246:
6241:
6235:
6231:
6230:
6222:
6220:
6216:
6211:
6205:
6201:
6194:
6192:
6188:
6183:
6177:
6173:
6172:
6164:
6162:
6158:
6153:
6151:9781615921768
6147:
6143:
6142:
6134:
6131:
6126:
6124:9780521878180
6120:
6116:
6115:
6107:
6104:
6100:
6098:9780199213122
6094:
6090:
6089:
6084:
6080:
6074:
6071:
6059:
6055:
6049:
6046:
6039:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6004:Offset binary
6002:
6000:
5997:
5995:
5992:
5990:
5987:
5985:
5984:Finger binary
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5956:
5952:
5946:
5941:
5936:
5931:
5913:
5903:
5900:
5899:
5898:
5896:
5891:
5889:
5885:
5881:
5876:
5861:
5855:
5847:
5842:
5839:
5832:
5828:
5822:
5818:
5812:
5805:
5801:
5795:
5791:
5767:
5761:
5753:
5748:
5745:
5738:
5734:
5728:
5724:
5718:
5711:
5707:
5701:
5697:
5686:
5684:
5680:
5676:
5656:
5652:
5648:
5639:
5634:
5626:
5612:
5609:
5606:
5605:
5601:
5598:
5584:
5581:
5578:
5577:
5573:
5570:
5566:
5563:
5560:
5559:
5555:
5552:
5548:
5545:
5542:
5541:
5538:
5532:
5531:decimal point
5528:
5520:
5508:
5498:
5497:
5496:
5479:
5465:
5464:
5463:
5450:
5440:
5439:
5438:
5436:
5426:
5423:
5422:
5418:
5415:
5414:
5410:
5407:
5406:
5402:
5399:
5398:
5394:
5391:
5390:
5386:
5383:
5382:
5378:
5375:
5374:
5370:
5367:
5366:
5362:
5359:
5358:
5355:
5353:
5349:
5345:
5339:
5331:
5319:
5318:
5317:
5304:
5294:
5293:
5292:
5290:
5286:
5271:
5261:
5260:
5259:
5256:
5253:
5243:
5240:
5237:
5234:
5228:
5225:
5223:
5217:
5214:
5209:
5206:
5205:
5201:
5198:
5195:
5192:
5186:
5183:
5181:
5175:
5172:
5167:
5164:
5163:
5159:
5156:
5153:
5150:
5144:
5141:
5139:
5133:
5130:
5125:
5122:
5121:
5117:
5114:
5111:
5108:
5102:
5099:
5097:
5091:
5088:
5083:
5080:
5079:
5075:
5071:
5068:
5065:
5062:
5056:
5053:
5051:
5045:
5042:
5037:
5034:
5033:
5029:
5026:
5023:
5020:
5014:
5011:
5009:
5003:
5000:
4995:
4992:
4991:
4987:
4984:
4981:
4978:
4972:
4969:
4967:
4961:
4958:
4953:
4950:
4949:
4945:
4942:
4939:
4936:
4930:
4927:
4925:
4919:
4916:
4911:
4908:
4907:
4903:
4899:
4896:
4893:
4890:
4884:
4881:
4879:
4873:
4870:
4865:
4862:
4861:
4857:
4854:
4851:
4848:
4842:
4839:
4837:
4831:
4828:
4823:
4820:
4819:
4815:
4812:
4809:
4806:
4800:
4797:
4795:
4789:
4786:
4781:
4778:
4777:
4773:
4770:
4767:
4764:
4758:
4755:
4753:
4747:
4744:
4739:
4736:
4735:
4731:
4727:
4724:
4721:
4718:
4712:
4709:
4707:
4701:
4698:
4693:
4690:
4689:
4685:
4682:
4679:
4676:
4670:
4667:
4665:
4659:
4656:
4651:
4648:
4647:
4643:
4640:
4637:
4634:
4628:
4625:
4623:
4617:
4614:
4609:
4606:
4605:
4601:
4598:
4595:
4592:
4584:
4581:
4579:
4573:
4570:
4565:
4562:
4561:
4557:
4549:
4547:
4545:
4541:
4537:
4531:
4530:in decimal).
4517:
4497:
4477:
4457:
4449:
4444:
4425:
4417:
4413:
4409:
4401:
4399:
4394:
4387:
4383:
4379:
4374:
4372:
4367:
4360:
4355:
4353:
4345:
4342:
4337:
4333:
4326:
4323:
4316:
4308:
4303:
4301:
4296:
4291:
4289:
4284:
4281:
4278:
4271:
4263:
4258:
4256:
4251:
4246:
4244:
4237:
4233:
4229:
4226:
4219:
4211:
4206:
4201:
4196:
4194:
4189:
4177:
4173:
4171:
4158:0.0001100110
4157:
4154:
4150:
4149:
4145:
4142:
4138:
4137:
4133:
4130:
4126:
4125:
4121:
4118:
4114:
4113:
4109:
4106:
4102:
4101:
4097:
4094:
4090:
4089:
4085:
4082:
4078:
4077:
4073:
4070:
4066:
4065:
4061:
4058:
4054:
4053:
4049:
4046:
4042:
4041:
4037:
4035:
4032:
4031:
4027:
4024:
4023:
4020:
4013:
3997:
3983:
3980:
3975:
3972:
3967:
3964:
3961:
3958:
3953:
3950:
3941:
3940:
3936:
3922:
3919:
3914:
3911:
3906:
3903:
3900:
3895:
3892:
3883:
3882:
3878:
3864:
3861:
3856:
3853:
3848:
3845:
3842:
3839:
3834:
3831:
3822:
3821:
3817:
3803:
3800:
3795:
3792:
3787:
3784:
3781:
3776:
3773:
3764:
3763:
3759:
3743:
3740:
3731:
3730:
3726:
3723:
3722:
3719:
3703:
3693:
3690:
3678:For example,
3676:
3660:
3657:
3633:
3630:
3625:
3620:
3610:
3607:
3595:, the second
3580:
3577:
3562:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3479:Binary
3478:
3477:
3474:
3472:
3471:Horner scheme
3459:
3455:
3453:
3450:
3447:
3443:
3439:
3437:
3434:
3431:
3427:
3423:
3421:
3418:
3415:
3411:
3407:
3405:
3402:
3399:
3395:
3391:
3389:
3386:
3383:
3379:
3375:
3373:
3370:
3367:
3363:
3359:
3357:
3354:
3351:
3347:
3343:
3341:
3338:
3335:
3331:
3327:
3325:
3322:
3319:
3315:
3311:
3309:
3306:
3303:
3299:
3295:
3293:
3290:
3287:
3283:
3280:= Next value
3279:
3276:
3273:
3270:
3269:
3266:
3255:
3253:
3244:
3240:
3236:
3223:
3216:
3211:
3209:
3207:
3202:
3198:
3194:
3190:
3186:
3182:
3176:
3168:
3162:
3160:
3152:
3150:
3146:
3128:
3120:
3114:
3103:
3102:long division
3095:
3087:
3082:
3080:
3079:Long division
3075:
3067:
3065:
3051:
3043:
3039:
3029:
3026:
3023:
3022:
3018:
3015:
3012:
3011:
3007:
3004:
3002:
3001:
2995:
2993:
2991:
2967:
2965:
2941:
2928:
2921:
2920:
2919:
2916:
2894:
2887:
2880:
2879:
2878:
2876:
2872:
2868:
2864:
2860:
2849:
2847:
2839:
2836:
2833:
2830:
2829:
2828:
2826:
2821:
2817:
2809:
2807:
2793:
2790:
2787:
2784:
2781:
2761:
2758:
2755:
2752:
2749:
2729:
2721:
2717:
2707:
2704:
2701:
2700:
2696:
2693:
2690:
2689:
2685:
2682:
2680:
2679:
2673:
2671:
2657:
2653:
2649:
2645:
2636:
2612:
2585:
2583:
2581:
2577:
2572:
2527:
2525:
2517:
2514:
2513:
2512:
2505:
2502:
2499:
2496:
2495:
2494:
2488:
2485:for a binary
2484:
2479:
2474:
2466:
2464:
2462:
2459:
2455:
2448:
2441:
2427:
2425:0.0624999...
2416:
2413:
2412:
2408:
2402:
2399:
2396:
2395:
2391:
2385:
2382:
2379:
2378:
2374:
2368:
2365:
2362:
2361:
2357:
2351:
2348:
2345:
2344:
2340:
2334:
2331:
2328:
2327:
2323:
2317:
2306:
2303:
2302:
2298:
2292:
2289:
2286:
2285:
2281:
2267:
2256:
2253:
2252:
2248:
2242:
2239:
2236:
2235:
2231:
2225:
2222:
2219:
2218:
2214:
2208:
2197:
2194:
2193:
2189:
2175:
2164:
2161:
2160:
2156:
2150:
2147:
2144:
2143:
2139:
2125:
2114:
2111:
2110:
2106:
2092:
2081:
2078:
2077:
2073:
2070:
2068:
2065:
2062:
2061:
2058:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2005:
2001:
1991:
1990:
1982:
1981:
1973:
1972:
1971:
1964:
1961:
1958:
1954:
1951:
1947:
1944:
1940:
1937:
1936:
1935:
1933:
1929:
1925:
1921:
1912:
1904:
1897:
1892:
1889:
1886:
1883:
1880:
1876:
1873:
1872:
1871:
1869:
1865:
1861:
1857:
1853:
1849:
1842:
1840:
1838:
1827:
1824:
1823:
1819:
1816:
1815:
1811:
1808:
1807:
1803:
1800:
1799:
1795:
1792:
1791:
1787:
1784:
1783:
1779:
1776:
1775:
1771:
1768:
1767:
1763:
1760:
1759:
1755:
1752:
1751:
1747:
1744:
1743:
1739:
1736:
1735:
1731:
1728:
1727:
1723:
1720:
1719:
1715:
1712:
1711:
1707:
1704:
1703:
1691:
1685:
1683:
1681:
1677:
1673:
1669:
1668:one zero zero
1661:
1658:
1655:
1652:
1648:
1641:
1638:
1635:
1632:
1628:
1625:
1624:
1623:
1621:
1617:
1613:
1609:
1604:
1602:
1598:
1594:
1591:
1587:
1578:
1574:
1570:
1566:
1561:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1524:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1492:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1460:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1428:
1425:
1423:
1415:
1413:
1411:
1407:
1406:Boolean logic
1403:
1399:
1394:
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1363:
1359:
1355:
1350:
1348:
1344:
1343:
1338:
1334:
1329:
1327:
1323:
1319:
1315:
1306:
1299:
1293:
1289:
1282:
1279:
1273:
1271:
1270:
1265:
1261:
1257:
1253:
1252:
1247:
1243:
1239:
1238:Gottlob Frege
1235:
1230:
1227:
1223:
1219:
1215:
1207:
1204:
1201:
1198:
1197:
1196:
1194:
1192:
1186:
1181:
1179:
1174:
1166:
1159:
1157:
1155:
1150:
1146:
1142:
1137:
1135:
1130:
1129:Francis Bacon
1125:
1122:
1114:
1112:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1070:
1068:
1066:
1062:
1058:
1057:Yoruba people
1054:
1050:
1046:
1042:
1034:
1032:
1030:
1026:
1022:
1018:
1017:Chandaḥśāstra
1013:
1011:
1007:
1003:
999:
991:
989:
987:
982:
980:
976:
969:
967:
964:
960:
956:
951:
949:
945:
942:and a set of
941:
937:
932:
930:
927:
923:
919:
918:
908:
901:
899:
897:
892:
887:
885:
881:
877:
873:
869:
865:
858:
853:
848:
840:
838:
836:
832:
828:
820:
818:
816:
812:
808:
804:
800:
796:
792:
787:
785:
781:
777:
776:binary number
773:
769:
765:
761:
757:
753:
752:
748:
743:
739:
738:binary number
727:
722:
720:
715:
713:
708:
707:
705:
704:
701:
698:
697:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
653:
650:
648:
645:
643:
640:
639:
638:
637:Alphasyllabic
635:
633:
630:
628:
625:
624:
621:
618:
617:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
554:
550:
549:
545:
539:
538:
525:
522:
519:
512:
509:
506:
505:
496:
493:
491:
488:
485:
478:
475:
472:
465:
462:
459:
452:
449:
448:
445:
442:
441:
436:
432:
430:
427:
425:
422:
420:
417:
415:
412:
410:
407:
405:
402:
400:
397:
395:
392:
390:
387:
385:
382:
380:
377:
376:
372:
371:
367:
360:
359:
351:
348:
346:
343:
342:
338:
337:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
304:
301:
298:
297:
293:
290:
289:
286:
283:
282:
278:
275:
274:
270:Other systems
266:
265:
258:
255:
253:
252:Counting rods
250:
249:
245:
244:
240:
237:
235:
232:
230:
227:
225:
222:
218:
215:
214:
213:
210:
209:
205:
204:
196:
195:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
144:
140:
137:
135:
132:
130:
127:
125:
122:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
86:
82:
79:
77:
74:
73:
69:
63:
62:
56:
50:
49:
46:
42:
38:
34:
33:
30:
19:
7240:cut-the-knot
7231:cut-the-knot
7206:. Retrieved
7202:the original
7192:
7172:
7134:
7112:. Retrieved
7108:
7099:
7086:. Retrieved
7066:
7060:
7044:
7032:. Retrieved
7022:
7010:. Retrieved
7000:
6988:. Retrieved
6978:
6966:. Retrieved
6962:the original
6952:
6944:1721.1/11173
6935:
6929:
6909:
6902:
6882:
6875:
6855:
6826:
6820:
6808:
6796:
6771:, retrieved
6749:
6712:(1): 12–15.
6709:
6705:
6692:
6667:
6663:
6657:
6607:
6582:
6578:
6568:
6531:
6527:
6517:
6505:. Retrieved
6496:
6484:
6465:
6459:
6450:
6430:
6420:
6415:(pdf, 145KB)
6401:
6386:
6381:
6362:
6356:
6337:
6331:
6314:
6310:
6300:
6280:
6273:
6265:
6260:15 September
6258:. Retrieved
6248:
6228:
6199:
6170:
6140:
6133:
6113:
6106:
6087:
6073:
6061:. Retrieved
6057:
6048:
5892:
5877:
5687:
5682:
5674:
5635:
5632:
5624:
5607:
5596:
5579:
5568:
5561:
5550:
5543:
5524:
5494:
5474:grouped = 54
5461:
5432:
5348:power of two
5341:
5315:
5284:
5282:
5257:
5249:
5207:
5165:
5123:
5081:
5035:
4993:
4951:
4909:
4863:
4821:
4779:
4737:
4691:
4649:
4607:
4563:
4543:
4540:concatenated
4535:
4532:
4445:
4178:
4174:
4163:
4152:
4146:0.000110011
4140:
4128:
4116:
4104:
4092:
4080:
4068:
4056:
4044:
4033:
4014:
4003:
3677:
3563:
3560:
3464:
3457:
3451:
3435:
3419:
3403:
3387:
3371:
3355:
3339:
3323:
3307:
3291:
3271:Prior value
3265:to decimal:
3259:
3232:
3178:
3156:
3147:
3124:
3118:
3083:
3077:
3035:
2987:
2964:binary point
2961:
2939:
2917:
2891:
2858:
2856:
2845:
2843:
2823:
2713:
2660:
2655:
2651:
2647:
2643:
2617:
2595:ones (where
2589:
2578:y = (x + y)
2573:
2534:
2523:
2521:
2509:
2492:
2452:
2371:000100111011
2332:0.090909...
2265:0.124999...
2054:
2050:
2046:
2042:
2034:
2030:
2026:
2017:only if the
2009:
1992:
1968:
1962:
1956:
1949:
1942:
1931:
1927:
1923:
1919:
1917:
1890:
1878:
1867:
1863:
1859:
1855:
1851:
1846:
1833:
1679:
1675:
1671:
1667:
1665:
1615:
1611:
1605:
1582:
1565:binary clock
1419:
1395:
1387:John Mauchly
1361:
1351:
1340:
1330:
1314:George Boole
1311:
1308:George Boole
1291:
1277:
1275:
1267:
1264:universality
1259:
1249:
1242:George Boole
1231:
1221:
1217:
1211:
1188:
1184:
1182:
1177:
1175:
1171:
1138:
1126:
1118:
1092:
1074:
1052:
1048:
1044:
1038:
1029:place values
1020:
1014:
1009:
1008:(light) and
1005:
995:
983:
973:
955:Song dynasty
952:
948:Zhou dynasty
936:yin and yang
933:
921:
915:
913:
910:Daoist Bagua
888:
861:
857:Eye of Horus
824:
788:
775:
755:
745:
737:
735:
503:
464:Signed-digit
378:
339:Contemporary
206:Contemporary
29:
7069:(2): 5–16.
7050:Rojas, Raúl
6489:Landry 2019
5974:Binary code
5884:0.111111...
5675:terminating
5527:radix point
5435:hexadecimal
5352:hexadecimal
5276:= 1110 0111
5266:= 0011 1010
4556:Hexadecimal
4550:Hexadecimal
4448:hexadecimal
4134:0.00011001
3153:Square root
3038:truth table
2943:1 0 1 1 (
2825:Subtraction
2810:Subtraction
2716:truth table
2543:) and 10111
2349:0.08333...
2173:0.24999...
2019:denominator
1860:first digit
1676:one hundred
1672:one hundred
1577:sexagesimal
1408:and binary
1402:Konrad Zuse
1398:Z1 computer
1244:in forming
1226:mathematics
1156:, in 1700.
1141:John Napier
1121:Ramon Llull
1059:. In 2008,
931:technique.
811:logic gates
770:) and "1" (
642:Akṣarapallī
612:Tally marks
511:Non-integer
7260:Categories
7094:(12 pages)
6291:3515074481
6040:References
5749:0.01010101
5529:(called a
4380:1100010101
4361:1100010101
4252:1100101110
4151:0.2 × 2 =
4139:0.6 × 2 =
4127:0.8 × 2 =
4122:0.0001100
4115:0.4 × 2 =
4103:0.2 × 2 =
4091:0.6 × 2 =
4079:0.8 × 2 =
4067:0.4 × 2 =
4055:0.2 × 2 =
4043:0.1 × 2 =
4025:Converting
3724:Converting
3125:Thus, the
3072:See also:
3044:operation
2722:operation
2654:0 1 1 0 0
2487:half adder
2454:Arithmetic
2400:0.0666...
2337:0001011101
2315:0.0999...
2223:0.1666...
2206:0.1999...
2123:0.4999...
2023:power of 2
1567:might use
1320:system of
1089:Slit drums
929:divination
926:quaternary
845:See also:
679:Glagolitic
652:Kaṭapayādi
620:Alphabetic
524:Asymmetric
366:radix/base
307:Cistercian
292:Babylonian
239:Vietnamese
94:Devanagari
7208:31 August
6773:20 August
6726:179000299
6591:0003-5483
6323:0002-9475
6009:Quibinary
5989:Gray code
5856:…
5851:¯
5762:…
5757:¯
5567:(1 × 1 =
5549:(1 × 2 =
5484:= 010 011
5470:= 101 100
5455:= 001 111
5445:= 110 101
4343:−
4327:×
4317:…
4312:¯
4282:×
4272:…
4267:¯
4230:×
4220:…
4215:¯
4110:0.000110
3981:≥
3959:×
3901:×
3862:≥
3840:×
3782:×
3277:Next bit
3052:∧
2988:See also
2865:of equal
2846:borrowing
2840:1 − 1 → 0
2837:1 − 0 → 1
2831:0 − 0 → 0
2753:∨
2730:∨
2648:1 1 1 1 1
2503:1 + 0 → 1
2500:0 + 1 → 1
2497:0 + 0 → 0
2290:0.111...
2148:0.333...
2090:0.999...
2063:Fraction
2015:terminate
2011:Fractions
2006:Fractions
1924:first bit
1358:Bell Labs
1331:In 1937,
1318:algebraic
1294:hexagrams
1278:ex nihilo
1139:In 1617,
1127:In 1605,
1077:Mangareva
975:Etruscans
959:Shao Yong
868:Horus-Eye
647:Āryabhaṭa
592:Kharosthi
484:factorial
451:Bijective
352:(Iñupiaq)
182:Sundanese
177:Mongolian
124:Malayalam
7246:Archived
7079:Archived
6647:(1605).
6625:Archived
6615:Archived
6599:23621076
6560:24344278
6409:Archived
5994:IEEE 754
5937:See also
5848:10110100
5305:11011101
4098:0.00011
3129:of 11011
3127:quotient
3096:is 11011
3094:dividend
3068:Division
2774:, while
2524:carrying
2467:Addition
1868:overflow
1854:through
1590:magnetic
1586:voltages
1379:teletype
1349:design.
1284:—
1281:Nothing.
1105:geomancy
1103:Western
1101:medieval
957:scholar
674:Georgian
664:Cyrillic
632:Armenian
587:Etruscan
582:Egyptian
490:Negative
350:Kaktovik
345:Cherokee
322:Pentadic
246:Historic
229:Japanese
162:Javanese
152:Balinese
139:Dzongkha
104:Gurmukhi
99:Gujarati
37:a series
35:Part of
7114:26 June
6813:Leibniz
6672:Bibcode
6551:3910603
6034:Unicode
5880:decimal
5673:have a
5618:⁄
5590:⁄
5537:means:
5363:Binary
5295:1010010
5289:padding
4086:0.0001
4028:Result
3998:0.0101
3727:Result
3235:integer
3086:divisor
3040:of the
2718:of the
2071:Binary
2067:Decimal
1848:Decimal
1837:decimal
1693:Decimal
1588:; on a
1292:I Ching
1260:I Ching
1251:I Ching
1222:I Ching
1218:I Ching
1160:Leibniz
1093:I Ching
1085:decimal
1053:I Ching
1049:I Ching
1002:prosody
998:Pingala
922:I Ching
917:I Ching
821:History
793:with a
784:integer
760:numbers
577:Chuvash
495:Complex
285:Ancient
277:History
224:Hokkien
212:Chinese
157:Burmese
147:Tibetan
134:Kannada
114:Sinhala
89:Bengali
7180:
7150:
7088:3 July
7034:5 July
7012:5 July
6990:5 July
6968:5 July
6917:
6890:
6863:
6833:
6764:
6724:
6597:
6589:
6558:
6548:
6507:5 July
6472:
6438:
6393:
6369:
6344:
6321:
6288:
6236:
6206:
6178:
6148:
6121:
6095:
6063:22 May
5928:, the
5466:101100
4388:111110
4155:< 1
4119:< 1
4107:< 1
4074:0.000
4071:< 1
4059:< 1
4047:< 1
4012:... .
3937:0.010
3440:= 598
3424:= 299
3408:= 149
3274:× 2 +
3195:, and
3137:is 101
3088:is 101
2859:adding
2436:0.0000
2434:
2430:
2428:0.0001
2423:
2419:
2417:0.0625
2352:0.0001
2313:
2309:
2295:000111
2274:
2270:
2263:
2259:
2204:
2200:
2182:
2178:
2171:
2167:
2132:
2128:
2121:
2117:
2099:
2095:
2088:
2084:
2053:+ 1 ×
2049:+ 0 ×
2045:+ 1 ×
1993:100101
1983:100101
1974:100101
1700:number
1698:Binary
1695:number
1642:100101
1061:UNESCO
1035:Africa
986:Dodona
833:, and
809:using
742:number
689:Hebrew
659:Coptic
572:Brahmi
557:Aegean
514:
498:
480:
467:
454:
317:Muisca
257:Tangut
234:Korean
217:Suzhou
129:Telugu
7082:(PDF)
7057:(PDF)
6722:S2CID
6702:(PDF)
6595:JSTOR
5959:ASCII
5829:10001
5683:recur
5613:(1 ×
5602:plus
5585:(0 ×
5574:plus
5556:plus
5480:10011
5360:Octal
5344:octal
5338:Octal
5332:Octal
5252:radix
4309:01110
4297:11001
4264:01110
4212:01110
4062:0.00
3879:0.01
3555:1197
3552:1×2 =
3549:0×2 +
3546:1×2 +
3543:1×2 +
3540:0×2 +
3537:1×2 +
3534:0×2 +
3531:1×2 +
3528:0×2 +
3525:0×2 +
3522:1×2 +
3448:× 2 +
3432:× 2 +
3416:× 2 +
3400:× 2 +
3392:= 74
3384:× 2 +
3376:= 37
3368:× 2 +
3360:= 18
3352:× 2 +
3336:× 2 +
3320:× 2 +
3304:× 2 +
3288:× 2 +
2666:(1649
2644:1 1 1
2414:1/16
2397:1/15
2380:1/14
2363:1/13
2346:1/12
2329:1/11
2304:1/10
2276:0.000
2268:0.001
2257:0.125
2226:0.001
2021:is a
1938:0000,
1680:value
1620:radix
1579:time.
1322:logic
1191:Fu Xi
1006:laghu
992:India
902:China
876:hekat
872:Horus
841:Egypt
795:radix
774:). A
740:is a
684:Greek
669:Geʽez
627:Abjad
607:Roman
567:Aztec
562:Attic
477:Mixed
435:table
327:Quipu
312:Mayan
167:Khmer
119:Tamil
7210:2016
7178:ISBN
7148:ISBN
7116:2019
7090:2022
7036:2010
7014:2010
6992:2010
6970:2010
6915:ISBN
6888:ISBN
6861:ISBN
6831:ISBN
6775:2024
6762:ISBN
6587:ISSN
6556:PMID
6509:2017
6470:ISBN
6436:ISBN
6391:ISBN
6367:ISBN
6342:ISBN
6319:ISSN
6286:ISBN
6262:2022
6234:ISBN
6204:ISBN
6176:ISBN
6146:ISBN
6119:ISBN
6093:ISBN
6065:2022
5819:1100
5636:All
5625:0.25
5427:111
5419:110
5411:101
5403:100
5395:011
5387:010
5379:001
5371:000
5320:C0E7
4202:1100
4166:0011
4050:0.0
3920:<
3818:0.0
3801:<
3458:1197
3344:= 9
3328:= 4
3312:= 2
3296:= 1
3159:here
2899:and
2818:and
2631:(691
2623:(958
2611:0s:
2481:The
2405:0001
2320:0011
2287:1/9
2254:1/8
2237:1/7
2220:1/6
2211:0011
2195:1/5
2184:0.00
2176:0.01
2165:0.25
2162:1/4
2145:1/3
2112:1/2
2079:1/1
2033:and
1997:= 37
1930:and
1828:1111
1820:1110
1812:1101
1804:1100
1796:1011
1788:1010
1780:1001
1772:1000
1614:and
1593:disk
1569:LEDs
1422:bits
1396:The
1389:and
1240:and
1039:The
1010:guru
953:The
914:The
768:zero
747:base
332:Rumi
187:Thai
109:Odia
7238:at
7229:at
7140:doi
7109:BBC
7071:doi
6940:hdl
6754:doi
6714:doi
6680:doi
6546:PMC
6536:doi
6532:111
6315:129
5610:× 2
5582:× 2
5564:× 2
5546:× 2
5509:127
5231:oct
5221:dec
5211:hex
5189:oct
5179:dec
5169:hex
5147:oct
5137:dec
5127:hex
5105:oct
5095:dec
5085:hex
5059:oct
5049:dec
5039:hex
5017:oct
5007:dec
4997:hex
4975:oct
4965:dec
4955:hex
4933:oct
4923:dec
4913:hex
4887:oct
4877:dec
4867:hex
4845:oct
4835:dec
4825:hex
4803:oct
4793:dec
4783:hex
4761:oct
4751:dec
4741:hex
4715:oct
4705:dec
4695:hex
4673:oct
4663:dec
4653:hex
4631:oct
4621:dec
4611:hex
4587:oct
4577:dec
4567:hex
4410:789
4153:0.4
4143:≥ 1
4141:1.2
4131:≥ 1
4129:1.6
4117:0.8
4105:0.4
4095:≥ 1
4093:1.2
4083:≥ 1
4081:1.6
4069:0.8
4057:0.4
4045:0.2
4038:0.
4034:0.1
3760:0.
3445:598
3429:299
3413:149
3201:NOT
3197:XOR
3189:AND
2708:10
2580:mod
2576:xor
2571:).
2567:(36
2547:(23
2539:(13
2388:001
2386:0.0
2318:0.0
2307:0.1
2245:001
2198:0.2
2134:0.0
2126:0.1
2115:0.5
1941:000
1920:bit
1764:111
1756:110
1748:101
1740:100
1601:yes
1337:MIT
1136:.)
1097:Ifá
1079:in
1067:".
1041:Ifá
803:bit
797:of
772:one
754:or
749:-2
364:By
172:Lao
7262::
7162:^
7146:.
7124:^
7107:.
7077:.
7067:19
7065:.
7059:.
6845:^
6782:^
6760:,
6748:,
6734:^
6720:.
6710:16
6704:.
6678:.
6668:19
6666:.
6636:^
6593:.
6583:54
6581:.
6577:.
6554:.
6544:.
6530:.
6526:.
6313:.
6309:.
6264:.
6218:^
6190:^
6160:^
6081:;
6056:.
5806:10
5802:17
5796:10
5792:12
5754:01
5735:11
5712:10
5702:10
5627:)
5623:=
5595:=
5515:10
5505:10
5499:65
5451:17
5441:65
5437::
5326:10
5322:16
5311:16
5301:16
5274:16
5272:E7
5264:16
5262:3A
5244:1
5229:17
5219:15
5202:0
5187:16
5177:14
5160:1
5145:15
5135:13
5118:0
5103:14
5093:12
5072:1
5057:13
5047:11
5030:0
5015:12
5005:10
4988:1
4973:11
4946:0
4931:10
4900:1
4858:0
4816:1
4774:0
4728:1
4686:0
4644:1
4602:0
4426:10
4418:62
4207:.1
4017:10
4010:01
3704:10
3473:.
3467:10
3456:=
3397:74
3381:37
3365:18
3251:2.
3247:10
3227:10
3193:OR
3191:,
3064:.
3030:1
3027:0
3024:1
3019:0
3016:0
3013:0
3008:1
3005:0
2992:.
2966::
2861:a
2806:.
2794:10
2705:1
2702:1
2697:1
2694:0
2691:0
2686:1
2683:0
2668:10
2646:0
2633:10
2625:10
2569:10
2549:10
2541:10
2432:or
2421:or
2403:0.
2369:0.
2354:01
2335:0.
2311:or
2293:0.
2272:or
2261:or
2243:0.
2228:01
2209:0.
2202:or
2180:or
2169:or
2153:01
2151:0.
2130:or
2119:or
2101:0.
2097:or
2086:or
2027:10
1999:10
1948:00
1825:15
1817:14
1809:13
1801:12
1793:11
1785:10
1732:11
1724:10
1595:,
1563:A
1553:y
1521:☒
1489:|
1457:1
1412:.
1385:,
1031:.
938:.
829:,
736:A
429:60
424:20
419:16
414:12
409:10
39:on
7212:.
7142::
7118:.
7092:.
7073::
7038:.
7016:.
6994:.
6972:.
6946:.
6942::
6923:.
6896:.
6869:.
6839:.
6756::
6728:.
6716::
6686:.
6682::
6674::
6631:)
6601:.
6562:.
6538::
6511:.
6478:.
6444:.
6375:.
6350:.
6325:.
6294:.
6242:.
6212:.
6184:.
6155:.
6128:.
6067:.
5914:2
5862:2
5840:=
5833:2
5823:2
5813:=
5768:2
5746:=
5739:2
5729:2
5725:1
5719:=
5708:3
5698:1
5657:a
5653:2
5649:p
5620:4
5616:1
5608:1
5599:)
5597:0
5592:2
5588:1
5580:0
5571:)
5569:1
5562:1
5553:)
5551:2
5544:1
5535:2
5511:8
5501:8
5490:8
5486:2
5482:2
5476:8
5472:2
5468:2
5457:2
5453:8
5447:2
5443:8
5424:7
5416:6
5408:5
5400:4
5392:3
5384:2
5376:1
5368:0
5307:2
5297:2
5285:0
5278:2
5268:2
5241:1
5238:1
5235:1
5226:=
5215:=
5208:F
5199:1
5196:1
5193:1
5184:=
5173:=
5166:E
5157:0
5154:1
5151:1
5142:=
5131:=
5124:D
5115:0
5112:1
5109:1
5100:=
5089:=
5082:C
5069:1
5066:0
5063:1
5054:=
5043:=
5036:B
5027:1
5024:0
5021:1
5012:=
5001:=
4994:A
4985:0
4982:0
4979:1
4970:=
4963:9
4959:=
4952:9
4943:0
4940:0
4937:1
4928:=
4921:8
4917:=
4910:8
4897:1
4894:1
4891:0
4885:7
4882:=
4875:7
4871:=
4864:7
4855:1
4852:1
4849:0
4843:6
4840:=
4833:6
4829:=
4822:6
4813:0
4810:1
4807:0
4801:5
4798:=
4791:5
4787:=
4780:5
4771:0
4768:1
4765:0
4759:4
4756:=
4749:4
4745:=
4738:4
4725:1
4722:0
4719:0
4713:3
4710:=
4703:3
4699:=
4692:3
4683:1
4680:0
4677:0
4671:2
4668:=
4661:2
4657:=
4650:2
4641:0
4638:0
4635:0
4629:1
4626:=
4619:1
4615:=
4608:1
4599:0
4596:0
4593:0
4585:0
4582:=
4575:0
4571:=
4564:0
4544:k
4536:k
4518:x
4498:x
4478:x
4458:x
4422:)
4414:/
4407:(
4402:=
4395:x
4384:/
4375:=
4368:x
4356:=
4349:)
4346:2
4338:6
4334:2
4330:(
4324:x
4304:.
4292:=
4285:2
4279:x
4259:.
4247:=
4238:6
4234:2
4227:x
4197:=
4190:x
4006:3
3984:1
3976:3
3973:1
3968:1
3965:=
3962:2
3954:3
3951:2
3923:1
3915:3
3912:2
3907:=
3904:2
3896:3
3893:1
3865:1
3857:3
3854:1
3849:1
3846:=
3843:2
3835:3
3832:2
3804:1
3796:3
3793:2
3788:=
3785:2
3777:3
3774:1
3744:3
3741:1
3700:)
3694:3
3691:1
3686:(
3661:2
3658:1
3634:4
3631:1
3626:=
3621:2
3617:)
3611:2
3608:1
3603:(
3581:2
3578:1
3566:2
3512:1
3509:0
3506:1
3503:1
3500:0
3497:1
3494:0
3491:1
3488:0
3485:0
3482:1
3452:1
3436:0
3420:1
3404:1
3388:0
3372:1
3356:0
3349:9
3340:1
3333:4
3324:0
3317:2
3308:0
3301:1
3292:1
3285:0
3263:2
3143:2
3139:2
3135:2
3131:2
3110:2
3106:2
3098:2
3090:2
2983:B
2979:B
2975:B
2971:A
2957:B
2953:B
2949:B
2945:A
2935:A
2931:B
2924:B
2913:B
2909:A
2905:B
2901:B
2897:A
2791:=
2788:1
2785:+
2782:1
2762:1
2759:=
2756:1
2750:1
2664:2
2656:1
2652:1
2629:2
2621:2
2619:0
2609:n
2605:n
2601:n
2597:n
2593:n
2565:2
2561:2
2557:2
2553:2
2545:2
2537:2
2438:1
2278:1
2186:1
2136:1
2103:1
2093:1
2082:1
2055:2
2051:2
2047:2
2043:2
2035:5
2031:2
1995:2
1985:2
1976:2
1963:1
1957:1
1955:0
1950:1
1943:1
1932:1
1928:0
1891:1
1879:1
1877:0
1864:0
1856:9
1852:0
1777:9
1769:8
1761:7
1753:6
1745:5
1737:4
1729:3
1721:2
1716:1
1713:1
1708:0
1705:0
1653:)
1644:2
1633:)
1616:1
1612:0
1550:y
1547:n
1544:y
1541:y
1538:n
1535:n
1532:y
1529:n
1526:y
1518:☒
1515:☐
1512:☒
1509:☒
1506:☐
1503:☐
1500:☒
1497:☐
1494:☒
1486:|
1483:―
1480:|
1477:|
1474:―
1471:―
1468:|
1465:―
1462:|
1454:1
1451:0
1448:1
1445:1
1442:0
1439:0
1436:1
1433:0
1430:1
1362:K
1193:"
1045:.
799:2
725:e
718:t
711:v
520:)
518:φ
516:(
507:)
504:i
502:2
500:(
486:)
482:(
473:)
469:(
460:)
458:1
456:(
437:)
433:(
404:8
399:6
394:5
389:4
384:3
379:2
20:)
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