Binary number - Knowledge

6808:: "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." 1158: 1904: 5938: 4435: 845: 900: 4174: 1298: 1553: 2471: 2519:. 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: 1173:, 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.. 3238:. 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) 4430:{\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}}} 3253:

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

2654:

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

1165:

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

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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

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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

1962:

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

886:

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

1116:

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

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1 0 0 1 --------- √ 1010001 1 --------- 101 01 0 -------- 1001 100 0 -------- 10001 10001 10001 ------- 0

2503:

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:

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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

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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.

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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

4168:

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:

2841:. 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. 5867: 2611:

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

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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

4161:... . 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 1357:

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

3141:

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.

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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:

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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

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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

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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

4535:. 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 1020:. 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 2607:

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

5247:

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.

5777: 1417:(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: 1221:

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."

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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

2663:). 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. 3668:, 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. 3987: 3868: 3926: 3807: 1057: 4179: 3639: 5683: 1125:

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

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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.

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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

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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

1217:

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

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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:

2050:+ ... = 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. 5343:(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 2651:

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

810:, 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. 5664: 3709: 2870:

notation. Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation, subtraction can be summarized by the following formula:

1667:, 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 867:, 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 1863:), 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: 1334: 1182:"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 5919: 3749: 3666: 3586: 954:(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 2765: 1259:

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

1907:

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

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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

2978:+ 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) 2733: 1280: 4521: 4501: 4481: 4461: 3215: 1659:

When spoken, binary numerals are usually read digit-by-digit, to distinguish them from decimal numerals. For example, the binary numeral 100 is pronounced

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1 0 1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 ----- 1 1 1 − 1 0 1 ----- 0 1 0

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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

716: 2548:. 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 1374:. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were 3554:

The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving.

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oracle worked by drawing from separate jars, questions tablets and "yes" and "no" pellets. The result was then combined to make a final prophecy.

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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

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The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

1671:(a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct 863:

fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of

7190: 7278: 7144: 6911: 6884: 6857: 6758: 6466: 6432: 6363: 6338: 6230: 6200: 6172: 1927:

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:

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Binary counting follows the exact same procedure, and again the incremental substitution begins with the least significant binary digit, or

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You could say is a more sensible way of rendering hexagram as binary numbers ... The reasoning, if any, that informs sequence is unknown.

4165:. 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. 2837:

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

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Leibniz G., Explication de l'Arithmétique Binaire, Die Mathematische Schriften, ed. C. Gerhardt, Berlin 1879, vol.7, p.223; Engl. transl.

2982: 7273: 7174: 997:. 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: 1188:. Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 6827: 6387: 6142: 6115: 6089: 3151: 1332:

that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled

830:. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. 5875:. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 1012:(8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to 6401: 1157: 67: 7071: 6011: 4531:

is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are

1851:. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the 1101: 883: 5862:{\displaystyle {\frac {12_{10}}{17_{10}}}={\frac {1100_{2}}{10001_{2}}}=0.1011010010110100{\overline {10110100}}\ldots \,_{2}} 3937: 3818: 7268: 6282: 5991: 5894:

0.10100100010000100000100... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational

503: 3879: 3760: 7238: 6615: 3172:

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using

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while a missionary in China, Leibniz explained his binary notation, and Bouvet demonstrated in his 1701 letters that the

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The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:

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on 11 September 1940, Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a

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in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.

1226: 876: 456: 3591: 2556:. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100 1596:", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use. 3150:

The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained

7258: 3227: 2847:* (starred columns are borrowed from) 1 0 1 1 1 1 1 – 1 0 1 0 1 1 ---------------- = 0 1 1 0 1 0 0 888: 799: 516: 2970:(6.25 in decimal) ------------------- 1 . 0 1 1 0 1 ← Corresponds to a 'one' in 6903:

An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities

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is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in

2844:* * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1 2031: 1402: 1256: 1146: 1084:

with binary tones are used to encode messages across Africa and Asia. Sets of binary combinations similar to the

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is given by the binary representation of the second number. This method can be seen in use, for instance, in the

823: 612: 388: 5518:

Non-integers can be represented by using negative powers, which are set off from the other digits by means of a

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Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:

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To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:

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000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)

6906:(Macmillan, Dover Publications, reprinted with corrections ed.). New York: Cambridge University Press. 1009: 1318: 692: 476: 80: 6691: 1880:

090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)

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B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50

5876: 5281: 3086: 1238: 872: 383: 299: 6006: 5768:{\displaystyle {\frac {1_{10}}{3_{10}}}={\frac {1_{2}}{11_{2}}}=0.01010101{\overline {01}}\ldots \,_{2}} 5635: 3235: 3189: 2635:

carry the 1 until it is one digit past the "string" below 1 1 1 0 1 1 1 1 1 0

1611:. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or 967: 955: 494: 6461:. Contemporary ethnography (1st ed.). Philadelphia: University of Pennsylvania Press. p. 25. 6046: 5937: 3674: 2631:

Traditional Carry Method Long Carry Method vs.

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Arbeitsbuch Informatik – eine praxisorientierte Einführung in die Datenverarbeitung mit Projektaufgabe

5317:= (12 × 16) + (0 × 16) + (14 × 16) + (7 × 16) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,383 3462:. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the 1855:. When the available symbols for this position are exhausted, the least significant digit is reset to 6664: 6607: 5971: 2596:

zeros. That concept follows, logically, just as in the decimal system, where adding 1 to a string of

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binary numeral—the binary representation has a finite number of terms after the radix point. Other

3185: 3181: 3034: 3030: 2867: 2812: 2712: 2708: 2479: 2465: 2450: 1643: 1137: 1017: 783: 671: 536: 487: 306: 238: 93: 54: 3201:

left of a binary number is the equivalent of multiplication by a (positive, integral) power of 2.

6714: 6587: 5943: 5900: 3173: 3066: 1321:. His logical calculus was to become instrumental in the design of digital electronic circuitry. 936: 921: 584: 344: 181: 176: 123: 3727: 3644: 3564: 2034:. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 6847: 6641: 3214: 2738: 1652:

6b100101 (a prefix indicating number of bits in binary format, common in programming languages)

939:, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the 7219: 7170: 7166:

Informatik für Ingenieure und Naturwissenschaftler: PC- und Mikrocomputertechnik, Rechnernetze

7140: 6907: 6880: 6874: 6853: 6823: 6754: 6579: 6548: 6462: 6428: 6383: 6359: 6334: 6311: 6278: 6272: 6226: 6196: 6168: 6138: 6132: 6111: 6085: 6079: 6016: 5961: 5887: 3177: 3167: 2007: 1589: 1367: 1261: 1126: 856: 827: 666: 656: 644: 624: 579: 574: 510: 349: 321: 228: 161: 151: 138: 103: 98: 6220: 6162: 6105: 3192:

may be performed on corresponding bits in two binary numerals provided as input. The logical

3138:. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. 7132: 7063: 6932: 6746: 6706: 6672: 6538: 6528: 5956: 5922: 5880: 5671: 3641:, etc. So if there is a 1 in the first place after the decimal, then the number is at least 3198: 3040: 2770: 2572: 1623: 1375: 1073: 994: 569: 463: 223: 211: 156: 146: 113: 88: 6494: 4439:

Another way of converting from binary to decimal, often quicker for a person familiar with

2718: 2628:), using the traditional carry method on the left, and the long carry method on the right: 2456:. Addition, subtraction, multiplication, and division can be performed on binary numerals. 1180:(published in 1703). The full title of Leibniz's article is translated into English as the 775:

that has a finite representation in the binary numeral system, that is, the quotient of an

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is the number of decimal digits in the second, least-significant piece before conversion.

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1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 ----- 0 0 1

2855: 2475: 1600: 1359: 1339: 772: 681: 651: 594: 564: 549: 316: 284: 256: 233: 216: 75: 7097: 2552:

again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 11

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The mathematics of harmony: from Euclid to contemporary mathematics and computer science

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00, 0101, 0110, 0111, (rightmost three bits start over, and the next bit is incremented)

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The modern binary number system was studied in Europe in the 16th and 17th centuries by

6543: 6516: 6418: 6075: 6071: 4506: 4486: 4466: 4446: 2885: 2859: 2453: 1859:, and the next digit of higher significance (one position to the left) is incremented ( 1582: 1383: 1346: 1325: 1248: 1206: 1141: 819: 756: 743: 676: 619: 599: 554: 166: 133: 118: 44: 6398: 6222:

Groundbreaking Scientific Experiments, Inventions, and Discoveries of the 18th Century

2544:). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10 844: 7252: 7042: 6999:"Pioneers – The people and ideas that made a difference – George Stibitz (1904–1995)" 6793: 6718: 6637: 5996: 5976: 5523: 4532: 3463: 3094: 3071: 2592:

is any integer length), adding 1 will result in the number 1 followed by a string of

1398: 1390: 1230: 1121: 1049: 482: 311: 251: 186: 128: 108: 7046: 7232: 7228: 7223: 6750: 5340: 2956: 2944:) --------- 0 0 0 0 ← Corresponds to the rightmost 'zero' in 2568: 1585: 1557: 1379: 1306: 1234: 947: 940: 928: 860: 849: 634: 559: 6805: 6333:. Blackwell ancient religions (1. publ ed.). Malden, Mass.: Wiley-Blackwell. 1655:#b100101 (a prefix indicating binary format, common in Lisp programming languages) 1297: 899: 7164: 7126: 6783: 6422: 5243:

Binary may be converted to and from hexadecimal more easily. This is because the

5966: 5519: 5427: 5344: 5211: 5169: 5127: 5085: 5039: 4548: 4440: 3134:, as shown on the top line, while the remainder, shown on the bottom line, is 10 2817: 2011: 1569: 1394: 1371: 1218: 1140:

for doing binary calculations using a non-positional representation by letters.

1133: 1113: 1093: 1021: 875:, approximately 2400 BC, and its fully developed hieroglyphic form dates to the 604: 469: 428: 418: 1899:

This counter shows how to count in binary from numbers zero through thirty-one.

7136: 6954: 5933: 2446: 2015: 1593: 803: 413: 171: 6998: 6583: 6315: 3176:. When a string of binary symbols is manipulated in this way, it is called a 2908:

that was used. The sum of all these partial products gives the final result.

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0 cross out the "string", + 1 0 1 0 1 1 0 0 1 1 + 1 0

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0b100101 (a prefix indicating binary format, common in programming languages)

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The scribes of ancient Egypt used two different systems for their fractions,

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Converting from octal to binary proceeds in the same fashion as it does for

2003: 1895: 1350: 1081: 1069: 951: 423: 6979:. Math & Computer Science Department, Denison University. 30 April 2004 6710: 6552: 5897:

1.0110101000001001111001100110011111110... is the binary representation of

1552: 1229:, a popular idea that would be followed closely by his successors such as 1225:

The relation was a central idea to his universal concept of a language or

7169:(in German). Vol. 2 (5 ed.). Vieweg, reprint: Springer-Verlag. 6901: 5986: 3193: 3119: 2933:

For example, the binary numbers 1011 and 1010 are multiplied as follows:

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0, 0011, (rightmost two bits start over, and the next bit is incremented)

1097: 971: 807: 6936: 6591: 6567: 6299: 5678:, with a finite sequence of digits repeating indefinitely. For instance 17: 6026: 5872: 3078: 2470: 2059: 1840: 1829: 1578: 1310: 1243: 1088:

have also been used in traditional African divination systems, such as

1077: 990: 909: 776: 408: 393: 7067: 6976: 6676: 3254:

organized in a multi-column table. For example, to convert 10010101101

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2 for any two bits x and y allows for very fast calculation, as well.

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to express binary values. In this clock, each column of LEDs shows a

1053: 978: 932: 752: 734: 398: 3067:

Division algorithm § Integer division (unsigned) with remainder

1089: 1033: 798:, or binary digit. Because of its straightforward implementation in 1048:

which has 64. The Ifá originated in 15th century West Africa among

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dates from the 9th century BC in China. The binary notation in the

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Arithmetic values thought to have been represented by parts of the

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The binary addition table is similar to, but not the same as, the

2469: 2018:. As a result, 1/10 does not have a finite binary representation ( 1902: 1894: 1612: 1551: 1314: 1296: 1183: 1156: 868: 864: 843: 787: 739: 403: 365: 326: 6248:"Yijing hexagram sequences: The Shao Yong square (Fuxi sequence)" 5506:= (1 × 8) + (2 × 8) + (7 × 8) = (1 × 64) + (2 × 8) + (7 × 1) = 87 3093:, or 27 in decimal. The procedure is the same as that of decimal 1353:, completed a relay-based computer he dubbed the "Model K" (for " 760: 6743:

Handbook of the History and Philosophy of Mathematical Practice

2952:+ 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0 1599:

In keeping with the customary representation of numerals using

1938:, (rightmost bit starts over, and the next bit is incremented) 1414: 1329: 795: 764: 7023:. Computer History Association of California. 6 February 1995 6849:

Leibniz: What Kind of Rationalist?: What Kind of Rationalist?

6655:

Shirley, John W. (1951). "Binary numeration before Leibniz".

6517:"Mangarevan invention of binary steps for easier calculation" 5674:

have binary representation, but instead of terminating, they

2888:

in binary is similar to its decimal counterpart. Two numbers

2528:

In this example, two numerals are being added together: 01101

1251:, who visited China in 1685 as a missionary. Leibniz saw the 993:(c. 2nd century BC) developed a binary system for describing 6931:(Thesis). Cambridge: Massachusetts Institute of Technology. 5526:

in the decimal system). For example, the binary number 11.01

2482:, which adds two bits together, producing sum and carry bits 2030:). This causes 10 × 1/10 not to precisely equal 1 in binary 1058:

Masterpieces of the Oral and Intangible Heritage of Humanity

5886:

Binary numerals that neither terminate nor recur represent

4997: 2974:+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in 6745:, Cham: Springer International Publishing, pp. 1–31, 6605: 6568:"Diversity in the Numeral Systems of Australian Languages" 5339:

numeral system, since octal uses a radix of 8, which is a

4552: 1642:%100101 (a prefix indicating binary format; also known as 1622:

100101b (a suffix indicating binary format; also known as

7098:"Introducing binary – Revision 1 – GCSE Computer Science" 7047:"Konrad Zuse's Legacy: The Architecture of the Z1 and Z3" 6300:"Mapping the Entrails: The Practice of Greek Hepatoscopy" 4955: 4913: 4867: 4825: 4783: 4741: 4695: 4653: 4611: 4567: 3982:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1} 3863:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1} 2896:

can be multiplied by partial products: for each digit in

2525:

0 1 1 0 1 + 1 0 1 1 1 ------------- = 1 0 0 1 0 0 = 36

791: 7158: 7156: 7120: 7118: 2955:

Binary numbers can also be multiplied with bits after a

1603:, binary numbers are commonly written using the symbols 3921:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1} 3802:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1} 3074:

in binary is again similar to its decimal counterpart.

2600:

9s will result in the number 1 followed by a string of

7131:(in German). Vieweg-Verlag, reprint: Springer-Verlag. 6078:, eds. (2009), "Myth No. 2: the Horus eye fractions", 3940: 3882: 3821: 3763: 3730: 3677: 3647: 3594: 3567: 2567:

When computers must add two numbers, the rule that: x

2511:

7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 10) )

2508:

5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 10) )

1675:), but this does not make its binary nature explicit. 6951:"National Inventors Hall of Fame – George R. Stibitz" 6515:

Bender, Andrea; Beller, Sieghard (16 December 2013).

6277:. Stuttgart: Franz Steiner Verlag. pp. 165–170. 5903: 5780: 5686: 5638: 4509: 4489: 4469: 4449: 4177: 4001:... is equivalent to the repeating binary fraction 0. 3043: 2948:+ 1 0 1 1 ← Corresponds to the next 'one' in 2773: 2741: 2721: 2633:

1 1 1 1 1 1 1 1 (carried digits) 1 ← 1 ←

1269:

is not easy to impart to the pagans, is the creation

6161:

Edward Hacker; Steve Moore; Lorraine Patsco (2002).

5925:, another irrational. It has no discernible pattern. 2966:(5.625 in decimal) × 1 1 0 . 0 1 2499:

1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 2) )

7163:Küveler, Gerd; Schwoch, Dietrich (4 October 2007). 6929:

A symbolic analysis of relay and switching circuits

6737:Strickland, Lloyd (2020), Sriraman, Bharath (ed.), 6699:

Mitteilungen der deutschen Mathematiker-Vereinigung

3557:

In a fractional binary number such as 0.11010110101

3029:The binary multiplication table is the same as the 1335:

A Symbolic Analysis of Relay and Switching Circuits

5913: 5861: 5767: 5658: 4515: 4495: 4475: 4455: 4429: 3981: 3920: 3862: 3801: 3743: 3703: 3660: 3633: 3580: 3049: 2791: 2759: 2727: 6271:Zhonglian, Shi; Wenzhao, Li; Poser, Hans (2000). 6156: 6154: 6081:The Oxford Handbook of the History of Mathematics 3230:to its base-2 (binary) equivalent, the number is 1958:000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ... 1639:(a subscript indicating base-2 (binary) notation) 1338:, Shannon's thesis essentially founded practical 806:, the binary system is used by almost all modern 6794:"Bouvet and Leibniz: A Scholarly Correspondence" 6134:How Mathematics Happened: The First 50,000 Years 3634:{\textstyle ({\frac {1}{2}})^{2}={\frac {1}{4}}} 1176:His most well known work appears in his article 6692:"Leibniz, Caramuel, Harriot und das Dualsystem" 6521:Proceedings of the National Academy of Sciences 6047:"3.3. Binary and Its Advantages — CS160 Reader" 2850:Subtracting a positive number is equivalent to 1413:Any number can be represented by a sequence of 1267: 1209:in 1700, who had made himself an expert on the 6459:Vodún: secrecy and the search for divine power 6358:. Boca Raton, Florida: CRC Press. p. 37. 6356:Microcontroller programming: the microchip PIC 6110:, Cambridge University Press, pp. 42–43, 1632:bin 100101 (a prefix indicating binary format) 859:(not related to the binary number system) and 6186: 6184: 2866:to handle negative numbers—most commonly the 1044:, but has up to 256 binary signs, unlike the 710: 27:Number expressed in the base-2 numeral system 8: 7237:Sir Francis Bacon's BiLiteral Cypher system 6632: 6630: 6214: 6212: 5496:= (6 × 8) + (5 × 8) = (6 × 8) + (5 × 1) = 53 4157:This is also a repeating binary fraction 0.0 3205:Conversion to and from other numeral systems 1619:100101 binary (explicit statement of format) 4443:, is to do so indirectly—first converting ( 2449:in binary is much like arithmetic in other 1629:100101B (a suffix indicating binary format) 1247:through his contact with the French Jesuit 1205:While corresponding with the Jesuit priest 7125:Küveler, Gerd; Schwoch, Dietrich (2013) . 1615:. The following notations are equivalent: 717: 703: 60: 31: 6542: 6532: 6354:Sanchez, Julio; Canton, Maria P. (2007). 6107:Numerical Notation: A Comparative History 5904: 5902: 5853: 5852: 5838: 5824: 5814: 5808: 5797: 5787: 5781: 5779: 5759: 5758: 5744: 5730: 5720: 5714: 5703: 5693: 5687: 5685: 5648: 5639: 5637: 4508: 4488: 4468: 4448: 4417: 4405: 4375: 4329: 4299: 4254: 4229: 4202: 4178: 4176: 3963: 3941: 3939: 3902: 3883: 3881: 3844: 3822: 3820: 3783: 3764: 3762: 3731: 3729: 3695: 3681: 3676: 3648: 3646: 3621: 3612: 3598: 3593: 3568: 3566: 3222:to binary notation results in (101100101) 3042: 2772: 2740: 2720: 1832:counting system as a frame of reference. 6841: 6839: 6822:. Taylor & Francis. pp. 245–8. 6195:. Oxford University Press. p. 227. 6084:, Oxford University Press, p. 790, 5349: 4014: 3713: 3213: 2992: 2670: 2052: 1682: 1309:published a landmark paper detailing an 898: 7055:IEEE Annals of the History of Computing 6191:Redmond, Geoffrey; Hon, Tze-Ki (2014). 6038: 5335:Binary is also easily converted to the 43: 6778: 6776: 6481: 6137:, Prometheus Books, pp. 135–136, 3997:Thus the repeating decimal fraction 0. 2975: 2971: 2967: 2963: 2949: 2945: 2941: 2937: 2927: 2926:is 1, the partial product is equal to 2923: 2916: 2905: 2901: 2897: 2893: 2889: 2601: 2597: 2593: 2589: 2585: 1241:. Leibniz was first introduced to the 6732: 6730: 6728: 6566:Bowern, Claire; Zentz, Jason (2012). 5292:= 0101 0010 grouped with padding = 52 4483:in hexadecimal) and then converting ( 1386:, who wrote about it in his memoirs. 1178:Explication de l'Arithmétique Binaire 1117:science and artificial intelligence. 7: 6225:. Greenwood Publishing. p. 29. 3101:goes into the first three digits 110 1919:), except that only the two symbols 1915:(the rightmost one, also called the 808:computers and computer-based devices 2919:is 0, the partial product is also 0 1255:hexagrams as an affirmation of the 927:It is based on taoistic duality of 755:that uses only two symbols for the 7021:"George Robert Stibitz – Obituary" 6164:I Ching: An Annotated Bibliography 5659:{\displaystyle {\frac {p}{2^{a}}}} 3704:{\textstyle ({\frac {1}{3}})_{10}} 1393:, which was designed and built by 25: 7001:. Kerry Redshaw. 20 February 2006 6304:The American Journal of Philology 891:, which dates to around 1650 BC. 794:. Each digit is referred to as a 7245:, predates binary number system. 7077:from the original on 3 July 2022 6953:. 20 August 2008. Archived from 5936: 2983:Booth's multiplication algorithm 1328:produced his master's thesis at 1201:1 0 0 0 numerical value 2 1198:0 1 0 0 numerical value 2 1195:0 0 1 0 numerical value 2 1192:0 0 0 1 numerical value 2 1169:His first known work on binary, 6927:Shannon, Claude Elwood (1940). 6876:Leibniz, Mysticism and Religion 6421:; Olsen, Scott Anthony (2009). 6012:Redundant binary representation 2900:, the product of that digit in 1305:In 1854, British mathematician 1108:Western predecessors to Leibniz 1102:Indigenous Australian languages 1068:The residents of the island of 1036:is an African divination system 1008:Pingala's Hindu classic titled 937:64 hexagrams ("sixty-four" gua) 884:ancient Egyptian multiplication 782:The base-2 numeral system is a 6879:. Springer. pp. 149–150. 6751:10.1007/978-3-030-19071-2_90-1 5992:Linear-feedback shift register 4414: 4399: 4341: 4322: 3692: 3678: 3609: 3595: 1843:counting uses the ten symbols 1056:added Ifá to its list of the " 1: 6644:. London. pp. Chapter 1. 6642:"The Advancement of Learning" 6329:Johnston, Sarah Iles (2008). 6104:Chrisomalis, Stephen (2010), 6051:computerscience.chemeketa.edu 5626:For a total of 3.25 decimal. 3085:, or 5 in decimal, while the 2864:signed number representations 2809:signed number representations 2523:1 1 1 1 1 (carried digits) 1364:American Mathematical Society 1136:described a system he called 7279:Power-of-two numeral systems 6378:W. S. Anglin and J. Lambek, 6131:Rudman, Peter Strom (2007), 5883:2 + 2 + 2 + ... which is 1. 5843: 5749: 4304: 4259: 4207: 2820:works in much the same way: 2584:digits composed entirely of 2385:1/16 + 1/128 + 1/1024 . . . 1592:may be used. A "positive", " 1397:between 1935 and 1938, used 1362:. In a demonstration to the 1265:or creation out of nothing. 1092:among others, as well as in 977:Divination at Ancient Greek 840:Ancient Egyptian mathematics 800:digital electronic circuitry 751:, a method for representing 6977:"George Stibitz : Bio" 6739:"Leibniz on Number Systems" 6657:American Journal of Physics 6572:Anthropological Linguistics 6457:Landry, Timothy R. (2019). 6399:Math for Poets and Drummers 5914:{\displaystyle {\sqrt {2}}} 5488:And from octal to decimal: 3744:{\textstyle {\frac {1}{3}}} 3661:{\textstyle {\frac {1}{2}}} 3581:{\textstyle {\frac {1}{2}}} 3437: 3421: 3405: 3389: 3373: 3357: 3341: 3325: 3309: 3293: 3277: 3242:is expressed as (101100101) 2368:1/16 + 1/128 + 1/256 . . . 1568:numeral of the traditional 1317:that would become known as 1227:characteristica universalis 1171:“On the Binary Progression" 1076:were using a hybrid binary- 877:Nineteenth Dynasty of Egypt 7295: 6219:Jonathan Shectman (2003). 5455:And from binary to octal: 5328: 5069: 4897: 4725: 4583: 4546: 3226:To convert from a base-10 3165: 3077:In the example below, the 3064: 2806: 2463: 2435:1/32 + 1/64 + 1/128 . . . 2351:1/16 + 1/64 + 1/256 . . . 2334:1/16 + 1/64 + 1/128 . . . 2317:1/16 + 1/32 + 1/256 . . . 1166:unrelated to mathematics. 970:divided the outer edge of 889:Rhind Mathematical Papyrus 837: 437:Non-standard radices/bases 7274:Gottfried Wilhelm Leibniz 7241:23 September 2016 at the 7137:10.1007/978-3-322-92907-5 6852:. Springer. p. 415. 6167:. Routledge. p. 13. 5514:Representing real numbers 5481:grouped with padding = 23 5280:bits at the left (called 4163:floating point arithmetic 3174:Boolean logical operators 2760:{\displaystyle 1\lor 1=1} 2735:. The difference is that 2620:) and 1 0 1 0 1 1 0 0 1 1 2292:1/16 + 1/32 + 1/64 . . . 2275:1/16 + 1/32 + 1/64 . . . 2242:1/8 + 1/64 + 1/512 . . . 2225:1/8 + 1/32 + 1/128 . . . 2208:1/8 + 1/16 + 1/128 . . . 2067:Fractional approximation 2032:floating-point arithmetic 1690: 1685: 1147:Juan Caramuel y Lobkowitz 1112:In the late 13th century 917:is used to interpret its 879:, approximately 1200 BC. 824:Juan Caramuel y Lobkowitz 6331:Ancient Greek divination 5302:= 1101 1101 grouped = DD 3180:; the logical operators 2940:) × 1 0 1 0 ( 2183:1/8 + 1/16 + 1/32 . . . 2150:1/4 + 1/16 + 1/64 . . . 1279:Leibniz's letter to the 7229:Conversion of Fractions 6900:Boole, George (2009) . 6818:Aiton, Eric J. (1985). 6534:10.1073/pnas.1309160110 6495:"Ifa Divination System" 6298:Collins, Derek (2008). 5631:dyadic rational numbers 3234:. The remainder is the 3097:; here, the divisor 101 2133:1/4 + 1/8 + 1/16 . . . 693:List of numeral systems 6873:Yuen-Ting Lai (1998). 6711:10.1515/dmvm-2008-0009 6380:The Heritage of Thales 5915: 5863: 5769: 5660: 4517: 4503:in hexadecimal) into ( 4497: 4477: 4457: 4431: 3983: 3922: 3864: 3803: 3745: 3705: 3662: 3635: 3582: 3223: 3051: 3050:{\displaystyle \land } 2793: 2792:{\displaystyle 1+1=10} 2761: 2729: 2483: 1908: 1900: 1573: 1403:floating-point numbers 1302: 1290: 1162: 933:Eight trigrams (Bagua) 904: 873:Fifth Dynasty of Egypt 852: 7269:Elementary arithmetic 6846:J.E.H. Smith (2008). 6690:Ineichen, R. (2008). 6614:, Fidora et al. 2011 6007:Reduction of summands 5916: 5864: 5770: 5661: 4518: 4498: 4478: 4458: 4432: 3984: 3923: 3865: 3804: 3746: 3706: 3663: 3636: 3583: 3561:, the first digit is 3236:least-significant bit 3217: 3052: 2875:A − B = A + not B + 1 2807:Further information: 2794: 2762: 2730: 2728:{\displaystyle \lor } 2473: 2006:in binary arithmetic 1906: 1898: 1877: ... 1555: 1300: 1239:modern symbolic logic 1160: 1104:use a base-2 system. 956:least significant bit 902: 847: 749:binary numeral system 68:Hindu–Arabic numerals 6820:Leibniz: A Biography 6620:8 April 2019 at the 6610:3 April 2014 at the 6427:. World Scientific. 6404:16 June 2012 at the 6193:Teaching the I Ching 5972:Binary-coded decimal 5901: 5778: 5684: 5636: 4507: 4487: 4467: 4447: 4175: 3938: 3880: 3819: 3761: 3728: 3675: 3645: 3592: 3565: 3041: 2989:Multiplication table 2771: 2739: 2719: 1566:binary-coded decimal 1080:system before 1450. 882:The method used for 771:may also refer to a 590:Prehistoric counting 373:Common radices/bases 55:Place-value notation 7264:Computer arithmetic 7045:(April–June 1997). 6669:1951AmJPh..19..452S 6076:Stedall, Jacqueline 4008:Or for example, 0.1 3218:Conversion of (357) 3035:logical conjunction 2827:0 − 1 → 1, borrow 1 2713:logical disjunction 2466:Adder (electronics) 2451:positional notation 2402:1/16 + 1/256 . . . 2376:0.0714285714285... 2100:1/2 + 1/4 + 1/8... 1644:Motorola convention 1590:magnetic polarities 1138:location arithmetic 1018:positional notation 1005:(heavy) syllables. 989:The Indian scholar 963:Classical antiquity 784:positional notation 779:by a power of two. 537:Sign-value notation 7197:on 23 October 2017 6382:, Springer, 1995, 5944:Mathematics portal 5911: 5888:irrational numbers 5879:is the sum of the 5859: 5836:0.1011010010110100 5765: 5656: 4513: 4493: 4473: 4453: 4427: 4425: 3979: 3918: 3860: 3799: 3741: 3701: 3658: 3631: 3578: 3458:The result is 1197 3224: 3162:Bitwise operations 3047: 2962:1 0 1 . 1 0 1 2789: 2757: 2725: 2484: 2359:0.076923076923... 2233:0.142857142857... 2022:has prime factors 1909: 1901: 1679:Counting in binary 1574: 1349:, then working at 1345:In November 1937, 1303: 1293:Later developments 1283:attached with the 1163: 1100:. The majority of 943:of ancient China. 905: 857:Egyptian fractions 853: 200:East Asian systems 7259:Binary arithmetic 7146:978-3-528-04952-2 7068:10.1109/85.586067 6913:978-1-108-00153-3 6886:978-0-7923-5223-5 6859:978-1-4020-8668-7 6760:978-3-030-19071-2 6677:10.1119/1.1933042 6604:(see Bonner 2007 6468:978-0-8122-5074-9 6434:978-981-277-582-5 6365:978-0-8493-7189-9 6340:978-1-4051-1573-5 6246:Marshall, Steve. 6232:978-0-313-32015-6 6202:978-0-19-976681-9 6174:978-0-415-93969-0 6017:Repeating decimal 5962:Bitwise operation 5909: 5846: 5830: 5803: 5752: 5736: 5709: 5654: 5624: 5623: 5424: 5423: 5241: 5240: 4516:{\displaystyle x} 4496:{\displaystyle x} 4476:{\displaystyle x} 4463:in binary) into ( 4456:{\displaystyle x} 4307: 4262: 4210: 4155: 4154: 4012:, in binary, is: 3995: 3994: 3971: 3949: 3910: 3891: 3852: 3830: 3791: 3772: 3739: 3711:, in binary, is: 3689: 3656: 3629: 3606: 3576: 3552: 3551: 3456: 3455: 3249:Binary to decimal 3210:Decimal to binary 3178:bitwise operation 3168:Bitwise operation 3154:. An example is: 3027: 3026: 2705: 2704: 2579:Long carry method 2515:This is known as 2442:Binary arithmetic 2439: 2438: 1980:= + + + + + 1971:= + + + + + 1963:form as follows: 1886:00, 101, 102, ... 1825: 1824: 1550: 1549: 1368:Dartmouth College 1281:Duke of Brunswick 1262:creatio ex nihilo 1161:Gottfried Leibniz 1014:science of meters 972:divination livers 828:Gottfried Leibniz 759:: typically "0" ( 737:expressed in the 727: 726: 526: 525: 16:(Redirected from 7286: 7207: 7206: 7204: 7202: 7193:. Archived from 7187: 7181: 7180: 7179:. 9783834891914. 7160: 7151: 7150: 7149:. 9783322929075. 7122: 7113: 7112: 7110: 7108: 7094: 7088: 7086: 7084: 7082: 7076: 7051: 7039: 7033: 7032: 7030: 7028: 7017: 7011: 7010: 7008: 7006: 6995: 6989: 6988: 6986: 6984: 6973: 6967: 6966: 6964: 6962: 6947: 6941: 6940: 6924: 6918: 6917: 6897: 6891: 6890: 6870: 6864: 6863: 6843: 6834: 6833: 6815: 6809: 6803: 6797: 6796:, Swiderski 1980 6791: 6785: 6780: 6771: 6770: 6769: 6767: 6734: 6723: 6722: 6696: 6687: 6681: 6680: 6652: 6646: 6645: 6634: 6625: 6602: 6596: 6595: 6563: 6557: 6556: 6546: 6536: 6527:(4): 1322–1327. 6512: 6506: 6505: 6503: 6501: 6491: 6485: 6479: 6473: 6472: 6454: 6448: 6445: 6439: 6438: 6415: 6409: 6396: 6390: 6376: 6370: 6369: 6351: 6345: 6344: 6326: 6320: 6319: 6295: 6289: 6288: 6268: 6262: 6261: 6256: 6254: 6243: 6237: 6236: 6216: 6207: 6206: 6188: 6179: 6178: 6158: 6149: 6147: 6128: 6122: 6120: 6101: 6095: 6094: 6068: 6062: 6061: 6059: 6057: 6043: 6022:Two's complement 5957:Balanced ternary 5946: 5941: 5940: 5923:square root of 2 5920: 5918: 5917: 5912: 5910: 5905: 5890:. For instance, 5881:geometric series 5868: 5866: 5865: 5860: 5858: 5857: 5847: 5839: 5831: 5829: 5828: 5819: 5818: 5809: 5804: 5802: 5801: 5792: 5791: 5782: 5774: 5772: 5771: 5766: 5764: 5763: 5753: 5745: 5737: 5735: 5734: 5725: 5724: 5715: 5710: 5708: 5707: 5698: 5697: 5688: 5672:rational numbers 5665: 5663: 5662: 5657: 5655: 5653: 5652: 5640: 5615: 5614: 5610: 5587: 5586: 5582: 5533: 5532: 5350: 5284:). For example: 4553: 4522: 4520: 4519: 4514: 4502: 4500: 4499: 4494: 4482: 4480: 4479: 4474: 4462: 4460: 4459: 4454: 4436: 4434: 4433: 4428: 4426: 4422: 4421: 4409: 4379: 4334: 4333: 4308: 4300: 4263: 4255: 4234: 4233: 4211: 4203: 4160: 4015: 4004: 4000: 3988: 3986: 3985: 3980: 3972: 3964: 3950: 3942: 3927: 3925: 3924: 3919: 3911: 3903: 3892: 3884: 3869: 3867: 3866: 3861: 3853: 3845: 3831: 3823: 3808: 3806: 3805: 3800: 3792: 3784: 3773: 3765: 3750: 3748: 3747: 3742: 3740: 3732: 3714: 3710: 3708: 3707: 3702: 3700: 3699: 3690: 3682: 3667: 3665: 3664: 3659: 3657: 3649: 3640: 3638: 3637: 3632: 3630: 3622: 3617: 3616: 3607: 3599: 3587: 3585: 3584: 3579: 3577: 3569: 3469: 3468: 3261: 3260: 3199:arithmetic shift 3056: 3054: 3053: 3048: 2993: 2922:If the digit in 2915:If the digit in 2876: 2868:two's complement 2862:. Computers use 2813:two's complement 2798: 2796: 2795: 2790: 2766: 2764: 2763: 2758: 2734: 2732: 2731: 2726: 2671: 2634: 2524: 2432: 2428: 2424: 2417: 2413: 2399: 2382: 2365: 2348: 2331: 2314: 2307: 2303: 2289: 2272: 2268: 2264: 2257: 2253: 2239: 2222: 2205: 2198: 2194: 2180: 2176: 2172: 2165: 2161: 2147: 2130: 2126: 2122: 2115: 2111: 2097: 2093: 2089: 2082: 2078: 2053: 1874:0, 011, 012, ... 1836:Decimal counting 1683: 1624:Intel convention 1420: 1419: 1376:John von Neumann 1288: 1074:French Polynesia 719: 712: 705: 508: 492: 474: 464:balanced ternary 461: 448: 61: 32: 21: 7294: 7293: 7289: 7288: 7287: 7285: 7284: 7283: 7249: 7248: 7243:Wayback Machine 7216: 7211: 7210: 7200: 7198: 7189: 7188: 7184: 7177: 7162: 7161: 7154: 7147: 7124: 7123: 7116: 7106: 7104: 7096: 7095: 7091: 7080: 7078: 7074: 7049: 7041: 7040: 7036: 7026: 7024: 7019: 7018: 7014: 7004: 7002: 6997: 6996: 6992: 6982: 6980: 6975: 6974: 6970: 6960: 6958: 6949: 6948: 6944: 6926: 6925: 6921: 6914: 6899: 6898: 6894: 6887: 6872: 6871: 6867: 6860: 6845: 6844: 6837: 6830: 6817: 6816: 6812: 6804: 6800: 6792: 6788: 6781: 6774: 6765: 6763: 6761: 6736: 6735: 6726: 6694: 6689: 6688: 6684: 6654: 6653: 6649: 6636: 6635: 6628: 6622:Wayback Machine 6612:Wayback Machine 6603: 6599: 6565: 6564: 6560: 6514: 6513: 6509: 6499: 6497: 6493: 6492: 6488: 6480: 6476: 6469: 6456: 6455: 6451: 6446: 6442: 6435: 6419:Stakhov, Alexey 6417: 6416: 6412: 6406:Wayback Machine 6397: 6393: 6377: 6373: 6366: 6353: 6352: 6348: 6341: 6328: 6327: 6323: 6297: 6296: 6292: 6285: 6270: 6269: 6265: 6252: 6250: 6245: 6244: 6240: 6233: 6218: 6217: 6210: 6203: 6190: 6189: 6182: 6175: 6160: 6159: 6152: 6145: 6130: 6129: 6125: 6118: 6103: 6102: 6098: 6092: 6072:Robson, Eleanor 6070: 6069: 6065: 6055: 6053: 6045: 6044: 6040: 6035: 5942: 5935: 5932: 5899: 5898: 5851: 5820: 5810: 5793: 5783: 5776: 5775: 5757: 5726: 5716: 5699: 5689: 5682: 5681: 5644: 5634: 5633: 5612: 5608: 5607: 5584: 5580: 5579: 5529: 5516: 5509: 5505: 5499: 5495: 5484: 5480: 5476: 5470: 5466: 5462: 5451: 5447: 5441: 5437: 5333: 5327: 5320: 5316: 5305: 5301: 5295: 5291: 5272: 5268: 5262: 5258: 5225: 5215: 5205: 5183: 5173: 5163: 5141: 5131: 5121: 5099: 5089: 5079: 5053: 5043: 5033: 5011: 5001: 4991: 4969: 4959: 4949: 4927: 4917: 4907: 4881: 4871: 4861: 4839: 4829: 4819: 4797: 4787: 4777: 4755: 4745: 4735: 4709: 4699: 4689: 4667: 4657: 4647: 4625: 4615: 4605: 4581: 4571: 4561: 4551: 4545: 4505: 4504: 4485: 4484: 4465: 4464: 4445: 4444: 4424: 4423: 4413: 4397: 4390: 4384: 4383: 4370: 4363: 4357: 4356: 4351: 4344: 4325: 4313: 4312: 4292: 4287: 4280: 4268: 4267: 4247: 4242: 4235: 4225: 4216: 4215: 4197: 4192: 4185: 4173: 4172: 4158: 4011: 4002: 3998: 3936: 3935: 3878: 3877: 3817: 3816: 3759: 3758: 3726: 3725: 3691: 3673: 3672: 3643: 3642: 3608: 3590: 3589: 3563: 3562: 3560: 3461: 3257: 3251: 3245: 3241: 3221: 3212: 3207: 3170: 3164: 3159: 3148: 3137: 3133: 3129: 3125: 3116: 3110: 3104: 3100: 3092: 3084: 3069: 3063: 3039: 3038: 2991: 2979: 2977: 2973: 2969: 2965: 2953: 2951: 2947: 2943: 2939: 2929: 2925: 2918: 2907: 2903: 2899: 2895: 2891: 2883: 2874: 2856:negative number 2848: 2845: 2815: 2805: 2769: 2768: 2737: 2736: 2717: 2716: 2669: 2662: 2658: 2652: 2632: 2627: 2623: 2619: 2615: 2609: 2603: 2599: 2595: 2591: 2587: 2581: 2563: 2559: 2555: 2551: 2547: 2543: 2539: 2535: 2531: 2526: 2522: 2476:circuit diagram 2468: 2462: 2454:numeral systems 2444: 2430: 2426: 2422: 2415: 2411: 2397: 2380: 2363: 2346: 2329: 2312: 2305: 2301: 2287: 2270: 2266: 2262: 2255: 2251: 2237: 2220: 2203: 2196: 2192: 2178: 2174: 2170: 2163: 2159: 2145: 2128: 2124: 2120: 2113: 2109: 2095: 2091: 2087: 2080: 2076: 2001: 1993: 1989: 1979: 1970: 1893: 1891:Binary counting 1838: 1692: 1687: 1681: 1638: 1601:Arabic numerals 1411: 1360:complex numbers 1340:digital circuit 1319:Boolean algebra 1295: 1289: 1278: 1155: 1110: 1066: 1040:Similar to the 1030: 987: 965: 897: 842: 836: 816: 773:rational number 757:natural numbers 723: 687: 686: 609: 595:Proto-cuneiform 540: 539: 528: 527: 522: 521: 506: 490: 472: 459: 446: 433: 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: 7292: 7290: 7282: 7281: 7276: 7271: 7266: 7261: 7251: 7250: 7247: 7246: 7235: 7226: 7215: 7214:External links 7212: 7209: 7208: 7182: 7176:978-3834891914 7175: 7152: 7145: 7114: 7089: 7034: 7012: 6990: 6968: 6957:on 9 July 2010 6942: 6919: 6912: 6892: 6885: 6865: 6858: 6835: 6828: 6810: 6798: 6786: 6772: 6759: 6724: 6682: 6663:(8): 452–454. 6647: 6638:Bacon, Francis 6626: 6597: 6578:(2): 133–160. 6558: 6507: 6486: 6484:, p. 154. 6474: 6467: 6449: 6440: 6433: 6410: 6391: 6371: 6364: 6346: 6339: 6321: 6310:(3): 319–345. 6290: 6283: 6263: 6238: 6231: 6208: 6201: 6180: 6173: 6150: 6143: 6123: 6116: 6096: 6090: 6063: 6037: 6036: 6034: 6031: 6030: 6029: 6024: 6019: 6014: 6009: 6004: 5999: 5994: 5989: 5984: 5979: 5974: 5969: 5964: 5959: 5954: 5948: 5947: 5931: 5928: 5927: 5926: 5908: 5895: 5856: 5850: 5845: 5842: 5837: 5834: 5827: 5823: 5817: 5813: 5807: 5800: 5796: 5790: 5786: 5762: 5756: 5751: 5748: 5743: 5740: 5733: 5729: 5723: 5719: 5713: 5706: 5702: 5696: 5692: 5651: 5647: 5643: 5622: 5621: 5604: 5597: 5596: 5593: 5576: 5569: 5568: 5565: 5558: 5551: 5550: 5547: 5540: 5527: 5515: 5512: 5511: 5510: 5507: 5503: 5500: 5497: 5493: 5486: 5485: 5482: 5478: 5474: 5471: 5468: 5464: 5460: 5453: 5452: 5449: 5445: 5442: 5439: 5435: 5422: 5421: 5418: 5414: 5413: 5410: 5406: 5405: 5402: 5398: 5397: 5394: 5390: 5389: 5386: 5382: 5381: 5378: 5374: 5373: 5370: 5366: 5365: 5362: 5358: 5357: 5354: 5329:Main article: 5326: 5323: 5322: 5321: 5318: 5314: 5307: 5306: 5303: 5299: 5296: 5293: 5289: 5274: 5273: 5270: 5266: 5263: 5260: 5256: 5239: 5238: 5235: 5232: 5229: 5226: 5223: 5220: 5217: 5213: 5209: 5206: 5203: 5197: 5196: 5193: 5190: 5187: 5184: 5181: 5178: 5175: 5171: 5167: 5164: 5161: 5155: 5154: 5151: 5148: 5145: 5142: 5139: 5136: 5133: 5129: 5125: 5122: 5119: 5113: 5112: 5109: 5106: 5103: 5100: 5097: 5094: 5091: 5087: 5083: 5080: 5077: 5071: 5070: 5067: 5066: 5063: 5060: 5057: 5054: 5051: 5048: 5045: 5041: 5037: 5034: 5031: 5025: 5024: 5021: 5018: 5015: 5012: 5009: 5006: 5003: 4999: 4995: 4992: 4989: 4983: 4982: 4979: 4976: 4973: 4970: 4967: 4964: 4961: 4957: 4953: 4950: 4947: 4941: 4940: 4937: 4934: 4931: 4928: 4925: 4922: 4919: 4915: 4911: 4908: 4905: 4899: 4898: 4895: 4894: 4891: 4888: 4885: 4882: 4879: 4876: 4873: 4869: 4865: 4862: 4859: 4853: 4852: 4849: 4846: 4843: 4840: 4837: 4834: 4831: 4827: 4823: 4820: 4817: 4811: 4810: 4807: 4804: 4801: 4798: 4795: 4792: 4789: 4785: 4781: 4778: 4775: 4769: 4768: 4765: 4762: 4759: 4756: 4753: 4750: 4747: 4743: 4739: 4736: 4733: 4727: 4726: 4723: 4722: 4719: 4716: 4713: 4710: 4707: 4704: 4701: 4697: 4693: 4690: 4687: 4681: 4680: 4677: 4674: 4671: 4668: 4665: 4662: 4659: 4655: 4651: 4648: 4645: 4639: 4638: 4635: 4632: 4629: 4626: 4623: 4620: 4617: 4613: 4609: 4606: 4603: 4597: 4596: 4593: 4590: 4587: 4584: 4582: 4579: 4576: 4573: 4569: 4565: 4562: 4559: 4547:Main article: 4544: 4541: 4512: 4492: 4472: 4452: 4420: 4416: 4412: 4408: 4404: 4401: 4398: 4396: 4393: 4391: 4389: 4386: 4385: 4382: 4378: 4374: 4371: 4369: 4366: 4364: 4362: 4359: 4358: 4355: 4352: 4350: 4347: 4345: 4343: 4340: 4337: 4332: 4328: 4324: 4321: 4318: 4315: 4314: 4311: 4306: 4303: 4298: 4295: 4293: 4291: 4288: 4286: 4283: 4281: 4279: 4276: 4273: 4270: 4269: 4266: 4261: 4258: 4253: 4250: 4248: 4246: 4243: 4241: 4238: 4236: 4232: 4228: 4224: 4221: 4218: 4217: 4214: 4209: 4206: 4201: 4198: 4196: 4193: 4191: 4188: 4186: 4184: 4181: 4180: 4153: 4152: 4149: 4141: 4140: 4137: 4129: 4128: 4125: 4117: 4116: 4113: 4105: 4104: 4101: 4093: 4092: 4089: 4081: 4080: 4077: 4069: 4068: 4065: 4057: 4056: 4053: 4045: 4044: 4041: 4033: 4032: 4029: 4023: 4022: 4019: 4009: 3993: 3992: 3989: 3978: 3975: 3970: 3967: 3962: 3959: 3956: 3953: 3948: 3945: 3932: 3931: 3928: 3917: 3914: 3909: 3906: 3901: 3898: 3895: 3890: 3887: 3874: 3873: 3870: 3859: 3856: 3851: 3848: 3843: 3840: 3837: 3834: 3829: 3826: 3813: 3812: 3809: 3798: 3795: 3790: 3787: 3782: 3779: 3776: 3771: 3768: 3755: 3754: 3751: 3738: 3735: 3722: 3721: 3718: 3698: 3694: 3688: 3685: 3680: 3655: 3652: 3628: 3625: 3620: 3615: 3611: 3605: 3602: 3597: 3575: 3572: 3558: 3550: 3549: 3546: 3543: 3540: 3537: 3534: 3531: 3528: 3525: 3522: 3519: 3516: 3513: 3512:Decimal 3509: 3508: 3506: 3503: 3500: 3497: 3494: 3491: 3488: 3485: 3482: 3479: 3476: 3473: 3459: 3454: 3453: 3447: 3442: 3439: 3435: 3434: 3431: 3426: 3423: 3419: 3418: 3415: 3410: 3407: 3403: 3402: 3399: 3394: 3391: 3387: 3386: 3383: 3378: 3375: 3371: 3370: 3367: 3362: 3359: 3355: 3354: 3351: 3346: 3343: 3339: 3338: 3335: 3330: 3327: 3323: 3322: 3319: 3314: 3311: 3307: 3306: 3303: 3298: 3295: 3291: 3290: 3287: 3282: 3279: 3275: 3274: 3271: 3268: 3265: 3255: 3250: 3247: 3243: 3239: 3232:divided by two 3219: 3211: 3208: 3206: 3203: 3166:Main article: 3163: 3160: 3156: 3147: 3144: 3135: 3131: 3127: 3126:divided by 101 3123: 3114: 3108: 3102: 3098: 3090: 3082: 3062: 3059: 3046: 3025: 3024: 3021: 3018: 3014: 3013: 3010: 3007: 3003: 3002: 2999: 2996: 2990: 2987: 2961: 2935: 2931: 2930: 2920: 2886:Multiplication 2882: 2881:Multiplication 2879: 2878: 2877: 2860:absolute value 2846: 2843: 2835: 2834: 2831: 2828: 2825: 2804: 2801: 2788: 2785: 2782: 2779: 2776: 2756: 2753: 2750: 2747: 2744: 2724: 2703: 2702: 2699: 2696: 2692: 2691: 2688: 2685: 2681: 2680: 2677: 2674: 2668: 2667:Addition table 2665: 2660: 2656: 2630: 2625: 2621: 2617: 2613: 2606: 2580: 2577: 2561: 2557: 2553: 2549: 2545: 2541: 2537: 2533: 2529: 2521: 2513: 2512: 2509: 2501: 2500: 2497: 2494: 2491: 2464:Main article: 2461: 2458: 2443: 2440: 2437: 2436: 2433: 2419: 2408: 2404: 2403: 2400: 2394: 2391: 2387: 2386: 2383: 2377: 2374: 2370: 2369: 2366: 2360: 2357: 2353: 2352: 2349: 2343: 2340: 2336: 2335: 2332: 2326: 2323: 2319: 2318: 2315: 2309: 2298: 2294: 2293: 2290: 2284: 2281: 2277: 2276: 2273: 2259: 2248: 2244: 2243: 2240: 2234: 2231: 2227: 2226: 2223: 2217: 2214: 2210: 2209: 2206: 2200: 2189: 2185: 2184: 2181: 2167: 2156: 2152: 2151: 2148: 2142: 2139: 2135: 2134: 2131: 2117: 2106: 2102: 2101: 2098: 2084: 2073: 2069: 2068: 2065: 2062: 2057: 2000: 1997: 1996: 1995: 1991: 1987: 1982: 1981: 1977: 1973: 1972: 1968: 1960: 1959: 1953: 1946: 1939: 1932: 1892: 1889: 1888: 1887: 1881: 1878: 1875: 1868: 1837: 1834: 1823: 1822: 1819: 1815: 1814: 1811: 1807: 1806: 1803: 1799: 1798: 1795: 1791: 1790: 1787: 1783: 1782: 1779: 1775: 1774: 1771: 1767: 1766: 1763: 1759: 1758: 1755: 1751: 1750: 1747: 1743: 1742: 1739: 1735: 1734: 1731: 1727: 1726: 1723: 1719: 1718: 1715: 1711: 1710: 1707: 1703: 1702: 1699: 1695: 1694: 1689: 1680: 1677: 1663:, rather than 1657: 1656: 1653: 1650: 1647: 1640: 1636: 1633: 1630: 1627: 1620: 1548: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1526: 1523: 1520: 1516: 1515: 1512: 1509: 1506: 1503: 1500: 1497: 1494: 1491: 1488: 1484: 1483: 1480: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1452: 1451: 1448: 1445: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1410: 1409:Representation 1407: 1384:Norbert Wiener 1366:conference at 1347:George Stibitz 1326:Claude Shannon 1294: 1291: 1276: 1249:Joachim Bouvet 1207:Joachim Bouvet 1203: 1202: 1199: 1196: 1193: 1154: 1151: 1142:Thomas Harriot 1127:Bacon's cipher 1109: 1106: 1065: 1064:Other cultures 1062: 1029: 1026: 986: 983: 964: 961: 896: 893: 835: 832: 820:Thomas Harriot 815: 812: 744:numeral system 725: 724: 722: 721: 714: 707: 699: 696: 695: 689: 688: 685: 684: 679: 674: 669: 664: 659: 654: 649: 648: 647: 642: 637: 627: 622: 616: 615: 608: 607: 602: 597: 592: 587: 582: 577: 572: 567: 562: 557: 552: 546: 545: 544:Non-alphabetic 541: 535: 534: 533: 530: 529: 524: 523: 520: 519: 514: 501: 485: 480: 467: 454: 440: 439: 432: 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: 14: 13: 10: 9: 6: 4: 3: 2: 7291: 7280: 7277: 7275: 7272: 7270: 7267: 7265: 7262: 7260: 7257: 7256: 7254: 7244: 7240: 7236: 7234: 7230: 7227: 7225: 7221: 7220:Binary System 7218: 7217: 7213: 7196: 7192: 7191:"Base System" 7186: 7183: 7178: 7172: 7168: 7167: 7159: 7157: 7153: 7148: 7142: 7138: 7134: 7130: 7129: 7121: 7119: 7115: 7103: 7099: 7093: 7090: 7073: 7069: 7065: 7061: 7057: 7056: 7048: 7044: 7038: 7035: 7022: 7016: 7013: 7000: 6994: 6991: 6978: 6972: 6969: 6956: 6952: 6946: 6943: 6938: 6934: 6930: 6923: 6920: 6915: 6909: 6905: 6904: 6896: 6893: 6888: 6882: 6878: 6877: 6869: 6866: 6861: 6855: 6851: 6850: 6842: 6840: 6836: 6831: 6829:0-85274-470-6 6825: 6821: 6814: 6811: 6807: 6802: 6799: 6795: 6790: 6787: 6784: 6779: 6777: 6773: 6762: 6756: 6752: 6748: 6744: 6740: 6733: 6731: 6729: 6725: 6720: 6716: 6712: 6708: 6704: 6701:(in German). 6700: 6693: 6686: 6683: 6678: 6674: 6670: 6666: 6662: 6658: 6651: 6648: 6643: 6639: 6633: 6631: 6627: 6623: 6619: 6616: 6613: 6609: 6606: 6601: 6598: 6593: 6589: 6585: 6581: 6577: 6573: 6569: 6562: 6559: 6554: 6550: 6545: 6540: 6535: 6530: 6526: 6522: 6518: 6511: 6508: 6496: 6490: 6487: 6483: 6478: 6475: 6470: 6464: 6460: 6453: 6450: 6444: 6441: 6436: 6430: 6426: 6425: 6420: 6414: 6411: 6407: 6403: 6400: 6395: 6392: 6389: 6388:0-387-94544-X 6385: 6381: 6375: 6372: 6367: 6361: 6357: 6350: 6347: 6342: 6336: 6332: 6325: 6322: 6317: 6313: 6309: 6305: 6301: 6294: 6291: 6286: 6280: 6276: 6275: 6267: 6264: 6260: 6249: 6242: 6239: 6234: 6228: 6224: 6223: 6215: 6213: 6209: 6204: 6198: 6194: 6187: 6185: 6181: 6176: 6170: 6166: 6165: 6157: 6155: 6151: 6146: 6144:9781615921768 6140: 6136: 6135: 6127: 6124: 6119: 6117:9780521878180 6113: 6109: 6108: 6100: 6097: 6093: 6091:9780199213122 6087: 6083: 6082: 6077: 6073: 6067: 6064: 6052: 6048: 6042: 6039: 6032: 6028: 6025: 6023: 6020: 6018: 6015: 6013: 6010: 6008: 6005: 6003: 6000: 5998: 5997:Offset binary 5995: 5993: 5990: 5988: 5985: 5983: 5980: 5978: 5977:Finger binary 5975: 5973: 5970: 5968: 5965: 5963: 5960: 5958: 5955: 5953: 5950: 5949: 5945: 5939: 5934: 5929: 5924: 5906: 5896: 5893: 5892: 5891: 5889: 5884: 5882: 5878: 5874: 5869: 5854: 5848: 5840: 5835: 5832: 5825: 5821: 5815: 5811: 5805: 5798: 5794: 5788: 5784: 5760: 5754: 5746: 5741: 5738: 5731: 5727: 5721: 5717: 5711: 5704: 5700: 5694: 5690: 5679: 5677: 5673: 5669: 5649: 5645: 5641: 5632: 5627: 5619: 5605: 5602: 5599: 5598: 5594: 5591: 5577: 5574: 5571: 5570: 5566: 5563: 5559: 5556: 5553: 5552: 5548: 5545: 5541: 5538: 5535: 5534: 5531: 5525: 5524:decimal point 5521: 5513: 5501: 5491: 5490: 5489: 5472: 5458: 5457: 5456: 5443: 5433: 5432: 5431: 5429: 5419: 5416: 5415: 5411: 5408: 5407: 5403: 5400: 5399: 5395: 5392: 5391: 5387: 5384: 5383: 5379: 5376: 5375: 5371: 5368: 5367: 5363: 5360: 5359: 5355: 5352: 5351: 5348: 5346: 5342: 5338: 5332: 5324: 5312: 5311: 5310: 5297: 5287: 5286: 5285: 5283: 5279: 5264: 5254: 5253: 5252: 5249: 5246: 5236: 5233: 5230: 5227: 5221: 5218: 5216: 5210: 5207: 5202: 5199: 5198: 5194: 5191: 5188: 5185: 5179: 5176: 5174: 5168: 5165: 5160: 5157: 5156: 5152: 5149: 5146: 5143: 5137: 5134: 5132: 5126: 5123: 5118: 5115: 5114: 5110: 5107: 5104: 5101: 5095: 5092: 5090: 5084: 5081: 5076: 5073: 5072: 5068: 5064: 5061: 5058: 5055: 5049: 5046: 5044: 5038: 5035: 5030: 5027: 5026: 5022: 5019: 5016: 5013: 5007: 5004: 5002: 4996: 4993: 4988: 4985: 4984: 4980: 4977: 4974: 4971: 4965: 4962: 4960: 4954: 4951: 4946: 4943: 4942: 4938: 4935: 4932: 4929: 4923: 4920: 4918: 4912: 4909: 4904: 4901: 4900: 4896: 4892: 4889: 4886: 4883: 4877: 4874: 4872: 4866: 4863: 4858: 4855: 4854: 4850: 4847: 4844: 4841: 4835: 4832: 4830: 4824: 4821: 4816: 4813: 4812: 4808: 4805: 4802: 4799: 4793: 4790: 4788: 4782: 4779: 4774: 4771: 4770: 4766: 4763: 4760: 4757: 4751: 4748: 4746: 4740: 4737: 4732: 4729: 4728: 4724: 4720: 4717: 4714: 4711: 4705: 4702: 4700: 4694: 4691: 4686: 4683: 4682: 4678: 4675: 4672: 4669: 4663: 4660: 4658: 4652: 4649: 4644: 4641: 4640: 4636: 4633: 4630: 4627: 4621: 4618: 4616: 4610: 4607: 4602: 4599: 4598: 4594: 4591: 4588: 4585: 4577: 4574: 4572: 4566: 4563: 4558: 4555: 4554: 4550: 4542: 4540: 4538: 4534: 4530: 4524: 4523:in decimal). 4510: 4490: 4470: 4450: 4442: 4437: 4418: 4410: 4406: 4402: 4394: 4392: 4387: 4380: 4376: 4372: 4367: 4365: 4360: 4353: 4348: 4346: 4338: 4335: 4330: 4326: 4319: 4316: 4309: 4301: 4296: 4294: 4289: 4284: 4282: 4277: 4274: 4271: 4264: 4256: 4251: 4249: 4244: 4239: 4237: 4230: 4226: 4222: 4219: 4212: 4204: 4199: 4194: 4189: 4187: 4182: 4170: 4166: 4164: 4151:0.0001100110 4150: 4147: 4143: 4142: 4138: 4135: 4131: 4130: 4126: 4123: 4119: 4118: 4114: 4111: 4107: 4106: 4102: 4099: 4095: 4094: 4090: 4087: 4083: 4082: 4078: 4075: 4071: 4070: 4066: 4063: 4059: 4058: 4054: 4051: 4047: 4046: 4042: 4039: 4035: 4034: 4030: 4028: 4025: 4024: 4020: 4017: 4016: 4013: 4006: 3990: 3976: 3973: 3968: 3965: 3960: 3957: 3954: 3951: 3946: 3943: 3934: 3933: 3929: 3915: 3912: 3907: 3904: 3899: 3896: 3893: 3888: 3885: 3876: 3875: 3871: 3857: 3854: 3849: 3846: 3841: 3838: 3835: 3832: 3827: 3824: 3815: 3814: 3810: 3796: 3793: 3788: 3785: 3780: 3777: 3774: 3769: 3766: 3757: 3756: 3752: 3736: 3733: 3724: 3723: 3719: 3716: 3715: 3712: 3696: 3686: 3683: 3671:For example, 3669: 3653: 3650: 3626: 3623: 3618: 3613: 3603: 3600: 3588:, the second 3573: 3570: 3555: 3547: 3544: 3541: 3538: 3535: 3532: 3529: 3526: 3523: 3520: 3517: 3514: 3511: 3510: 3507: 3504: 3501: 3498: 3495: 3492: 3489: 3486: 3483: 3480: 3477: 3474: 3472:Binary 3471: 3470: 3467: 3465: 3464:Horner scheme 3452: 3448: 3446: 3443: 3440: 3436: 3432: 3430: 3427: 3424: 3420: 3416: 3414: 3411: 3408: 3404: 3400: 3398: 3395: 3392: 3388: 3384: 3382: 3379: 3376: 3372: 3368: 3366: 3363: 3360: 3356: 3352: 3350: 3347: 3344: 3340: 3336: 3334: 3331: 3328: 3324: 3320: 3318: 3315: 3312: 3308: 3304: 3302: 3299: 3296: 3292: 3288: 3286: 3283: 3280: 3276: 3273:= Next value 3272: 3269: 3266: 3263: 3262: 3259: 3248: 3246: 3237: 3233: 3229: 3216: 3209: 3204: 3202: 3200: 3195: 3191: 3187: 3183: 3179: 3175: 3169: 3161: 3155: 3153: 3145: 3143: 3139: 3121: 3113: 3107: 3096: 3095:long division 3088: 3080: 3075: 3073: 3072:Long division 3068: 3060: 3058: 3044: 3036: 3032: 3022: 3019: 3016: 3015: 3011: 3008: 3005: 3004: 3000: 2997: 2995: 2994: 2988: 2986: 2984: 2960: 2958: 2934: 2921: 2914: 2913: 2912: 2909: 2887: 2880: 2873: 2872: 2871: 2869: 2865: 2861: 2857: 2853: 2842: 2840: 2832: 2829: 2826: 2823: 2822: 2821: 2819: 2814: 2810: 2802: 2800: 2786: 2783: 2780: 2777: 2774: 2754: 2751: 2748: 2745: 2742: 2722: 2714: 2710: 2700: 2697: 2694: 2693: 2689: 2686: 2683: 2682: 2678: 2675: 2673: 2672: 2666: 2664: 2650: 2646: 2642: 2638: 2629: 2605: 2578: 2576: 2574: 2570: 2565: 2520: 2518: 2510: 2507: 2506: 2505: 2498: 2495: 2492: 2489: 2488: 2487: 2481: 2478:for a binary 2477: 2472: 2467: 2459: 2457: 2455: 2452: 2448: 2441: 2434: 2420: 2418:0.0624999... 2409: 2406: 2405: 2401: 2395: 2392: 2389: 2388: 2384: 2378: 2375: 2372: 2371: 2367: 2361: 2358: 2355: 2354: 2350: 2344: 2341: 2338: 2337: 2333: 2327: 2324: 2321: 2320: 2316: 2310: 2299: 2296: 2295: 2291: 2285: 2282: 2279: 2278: 2274: 2260: 2249: 2246: 2245: 2241: 2235: 2232: 2229: 2228: 2224: 2218: 2215: 2212: 2211: 2207: 2201: 2190: 2187: 2186: 2182: 2168: 2157: 2154: 2153: 2149: 2143: 2140: 2137: 2136: 2132: 2118: 2107: 2104: 2103: 2099: 2085: 2074: 2071: 2070: 2066: 2063: 2061: 2058: 2055: 2054: 2051: 2049: 2045: 2041: 2037: 2033: 2029: 2025: 2021: 2017: 2013: 2009: 2005: 1998: 1994: 1984: 1983: 1975: 1974: 1966: 1965: 1964: 1957: 1954: 1951: 1947: 1944: 1940: 1937: 1933: 1930: 1929: 1928: 1926: 1922: 1918: 1914: 1905: 1897: 1890: 1885: 1882: 1879: 1876: 1873: 1869: 1866: 1865: 1864: 1862: 1858: 1854: 1850: 1846: 1842: 1835: 1833: 1831: 1820: 1817: 1816: 1812: 1809: 1808: 1804: 1801: 1800: 1796: 1793: 1792: 1788: 1785: 1784: 1780: 1777: 1776: 1772: 1769: 1768: 1764: 1761: 1760: 1756: 1753: 1752: 1748: 1745: 1744: 1740: 1737: 1736: 1732: 1729: 1728: 1724: 1721: 1720: 1716: 1713: 1712: 1708: 1705: 1704: 1700: 1697: 1696: 1684: 1678: 1676: 1674: 1670: 1666: 1662: 1661:one zero zero 1654: 1651: 1648: 1645: 1641: 1634: 1631: 1628: 1625: 1621: 1618: 1617: 1616: 1614: 1610: 1606: 1602: 1597: 1595: 1591: 1587: 1584: 1580: 1571: 1567: 1563: 1559: 1554: 1545: 1542: 1539: 1536: 1533: 1530: 1527: 1524: 1521: 1518: 1517: 1513: 1510: 1507: 1504: 1501: 1498: 1495: 1492: 1489: 1486: 1485: 1481: 1478: 1475: 1472: 1469: 1466: 1463: 1460: 1457: 1454: 1453: 1449: 1446: 1443: 1440: 1437: 1434: 1431: 1428: 1425: 1422: 1421: 1418: 1416: 1408: 1406: 1404: 1400: 1399:Boolean logic 1396: 1392: 1387: 1385: 1381: 1377: 1373: 1369: 1365: 1361: 1356: 1352: 1348: 1343: 1341: 1337: 1336: 1331: 1327: 1322: 1320: 1316: 1312: 1308: 1299: 1292: 1286: 1282: 1275: 1272: 1266: 1264: 1263: 1258: 1254: 1250: 1246: 1245: 1240: 1236: 1232: 1231:Gottlob Frege 1228: 1223: 1220: 1216: 1212: 1208: 1200: 1197: 1194: 1191: 1190: 1189: 1187: 1185: 1179: 1174: 1172: 1167: 1159: 1152: 1150: 1148: 1143: 1139: 1135: 1130: 1128: 1123: 1122:Francis Bacon 1118: 1115: 1107: 1105: 1103: 1099: 1095: 1091: 1087: 1083: 1079: 1075: 1071: 1063: 1061: 1059: 1055: 1051: 1050:Yoruba people 1047: 1043: 1039: 1035: 1027: 1025: 1023: 1019: 1015: 1011: 1010:Chandaḥśāstra 1006: 1004: 1000: 996: 992: 984: 982: 980: 975: 973: 969: 962: 960: 957: 953: 949: 944: 942: 938: 935:and a set of 934: 930: 925: 923: 920: 916: 912: 911: 901: 894: 892: 890: 885: 880: 878: 874: 870: 866: 862: 858: 851: 846: 841: 833: 831: 829: 825: 821: 813: 811: 809: 805: 801: 797: 793: 789: 785: 780: 778: 774: 770: 769:binary number 766: 762: 758: 754: 750: 746: 745: 741: 736: 732: 731:binary number 720: 715: 713: 708: 706: 701: 700: 698: 697: 694: 691: 690: 683: 680: 678: 675: 673: 670: 668: 665: 663: 660: 658: 655: 653: 650: 646: 643: 641: 638: 636: 633: 632: 631: 630:Alphasyllabic 628: 626: 623: 621: 618: 617: 614: 611: 610: 606: 603: 601: 598: 596: 593: 591: 588: 586: 583: 581: 578: 576: 573: 571: 568: 566: 563: 561: 558: 556: 553: 551: 548: 547: 543: 542: 538: 532: 531: 518: 515: 512: 505: 502: 499: 498: 489: 486: 484: 481: 478: 471: 468: 465: 458: 455: 452: 445: 442: 441: 438: 435: 434: 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: 7233:cut-the-knot 7224:cut-the-knot 7199:. Retrieved 7195:the original 7185: 7165: 7127: 7105:. Retrieved 7101: 7092: 7079:. Retrieved 7059: 7053: 7037: 7025:. Retrieved 7015: 7003:. Retrieved 6993: 6981:. Retrieved 6971: 6959:. Retrieved 6955:the original 6945: 6937:1721.1/11173 6928: 6922: 6902: 6895: 6875: 6868: 6848: 6819: 6813: 6801: 6789: 6764:, retrieved 6742: 6705:(1): 12–15. 6702: 6698: 6685: 6660: 6656: 6650: 6600: 6575: 6571: 6561: 6524: 6520: 6510: 6498:. Retrieved 6489: 6477: 6458: 6452: 6443: 6423: 6413: 6408:(pdf, 145KB) 6394: 6379: 6374: 6355: 6349: 6330: 6324: 6307: 6303: 6293: 6273: 6266: 6258: 6253:15 September 6251:. Retrieved 6241: 6221: 6192: 6163: 6133: 6126: 6106: 6099: 6080: 6066: 6054:. Retrieved 6050: 6041: 5885: 5870: 5680: 5675: 5667: 5628: 5625: 5617: 5600: 5589: 5572: 5561: 5554: 5543: 5536: 5517: 5487: 5467:grouped = 54 5454: 5425: 5341:power of two 5334: 5308: 5277: 5275: 5250: 5242: 5200: 5158: 5116: 5074: 5028: 4986: 4944: 4902: 4856: 4814: 4772: 4730: 4684: 4642: 4600: 4556: 4536: 4533:concatenated 4528: 4525: 4438: 4171: 4167: 4156: 4145: 4139:0.000110011 4133: 4121: 4109: 4097: 4085: 4073: 4061: 4049: 4037: 4026: 4007: 3996: 3670: 3556: 3553: 3457: 3450: 3444: 3428: 3412: 3396: 3380: 3364: 3348: 3332: 3316: 3300: 3284: 3264:Prior value 3258:to decimal: 3252: 3225: 3171: 3149: 3140: 3117: 3111: 3076: 3070: 3028: 2980: 2957:binary point 2954: 2932: 2910: 2884: 2851: 2849: 2838: 2836: 2816: 2706: 2653: 2648: 2644: 2640: 2636: 2610: 2588:ones (where 2582: 2571:y = (x + y) 2566: 2527: 2516: 2514: 2502: 2485: 2445: 2364:000100111011 2325:0.090909... 2258:0.124999... 2047: 2043: 2039: 2035: 2027: 2023: 2019: 2010:only if the 2002: 1985: 1961: 1955: 1949: 1942: 1935: 1924: 1920: 1916: 1912: 1910: 1883: 1871: 1860: 1856: 1852: 1848: 1844: 1839: 1826: 1672: 1668: 1664: 1660: 1658: 1608: 1604: 1598: 1575: 1558:binary clock 1412: 1388: 1380:John Mauchly 1354: 1344: 1333: 1323: 1307:George Boole 1304: 1301:George Boole 1284: 1270: 1268: 1260: 1257:universality 1252: 1242: 1235:George Boole 1224: 1214: 1210: 1204: 1181: 1177: 1175: 1170: 1168: 1164: 1131: 1119: 1111: 1085: 1067: 1045: 1041: 1037: 1031: 1022:place values 1013: 1007: 1002: 1001:(light) and 998: 988: 976: 966: 948:Song dynasty 945: 941:Zhou dynasty 929:yin and yang 926: 914: 908: 906: 903:Daoist Bagua 881: 854: 850:Eye of Horus 817: 781: 768: 748: 738: 730: 728: 496: 457:Signed-digit 378: 339:Contemporary 206:Contemporary 29: 7062:(2): 5–16. 7043:Rojas, Raúl 6482:Landry 2019 5967:Binary code 5877:0.111111... 5668:terminating 5520:radix point 5428:hexadecimal 5345:hexadecimal 5269:= 1110 0111 5259:= 0011 1010 4549:Hexadecimal 4543:Hexadecimal 4441:hexadecimal 4127:0.00011001 3146:Square root 3031:truth table 2936:1 0 1 1 ( 2818:Subtraction 2803:Subtraction 2709:truth table 2536:) and 10111 2342:0.08333... 2166:0.24999... 2012:denominator 1853:first digit 1669:one hundred 1665:one hundred 1570:sexagesimal 1401:and binary 1395:Konrad Zuse 1391:Z1 computer 1237:in forming 1219:mathematics 1149:, in 1700. 1134:John Napier 1114:Ramon Llull 1052:. In 2008, 924:technique. 804:logic gates 763:) and "1" ( 635:Akṣarapallī 605:Tally marks 504:Non-integer 7253:Categories 7087:(12 pages) 6284:3515074481 6033:References 5742:0.01010101 5522:(called a 4373:1100010101 4354:1100010101 4245:1100101110 4144:0.2 × 2 = 4132:0.6 × 2 = 4120:0.8 × 2 = 4115:0.0001100 4108:0.4 × 2 = 4096:0.2 × 2 = 4084:0.6 × 2 = 4072:0.8 × 2 = 4060:0.4 × 2 = 4048:0.2 × 2 = 4036:0.1 × 2 = 4018:Converting 3717:Converting 3118:Thus, the 3065:See also: 3037:operation 2715:operation 2647:0 1 1 0 0 2480:half adder 2447:Arithmetic 2393:0.0666... 2330:0001011101 2308:0.0999... 2216:0.1666... 2199:0.1999... 2116:0.4999... 2016:power of 2 1560:might use 1313:system of 1082:Slit drums 922:divination 919:quaternary 838:See also: 672:Glagolitic 645:Kaṭapayādi 613:Alphabetic 517:Asymmetric 366:radix/base 307:Cistercian 292:Babylonian 239:Vietnamese 94:Devanagari 7201:31 August 6766:20 August 6719:179000299 6584:0003-5483 6316:0002-9475 6002:Quibinary 5982:Gray code 5849:… 5844:¯ 5755:… 5750:¯ 5560:(1 × 1 = 5542:(1 × 2 = 5477:= 010 011 5463:= 101 100 5448:= 001 111 5438:= 110 101 4336:− 4320:× 4310:… 4305:¯ 4275:× 4265:… 4260:¯ 4223:× 4213:… 4208:¯ 4103:0.000110 3974:≥ 3952:× 3894:× 3855:≥ 3833:× 3775:× 3270:Next bit 3045:∧ 2981:See also 2858:of equal 2839:borrowing 2833:1 − 1 → 0 2830:1 − 0 → 1 2824:0 − 0 → 0 2746:∨ 2723:∨ 2641:1 1 1 1 1 2496:1 + 0 → 1 2493:0 + 1 → 1 2490:0 + 0 → 0 2283:0.111... 2141:0.333... 2083:0.999... 2056:Fraction 2008:terminate 2004:Fractions 1999:Fractions 1917:first bit 1351:Bell Labs 1324:In 1937, 1311:algebraic 1287:hexagrams 1271:ex nihilo 1132:In 1617, 1120:In 1605, 1070:Mangareva 968:Etruscans 952:Shao Yong 861:Horus-Eye 640:Āryabhaṭa 585:Kharosthi 477:factorial 444:Bijective 352:(Iñupiaq) 182:Sundanese 177:Mongolian 124:Malayalam 7239:Archived 7072:Archived 6640:(1605). 6618:Archived 6608:Archived 6592:23621076 6553:24344278 6402:Archived 5987:IEEE 754 5930:See also 5841:10110100 5298:11011101 4091:0.00011 3122:of 11011 3120:quotient 3089:is 11011 3087:dividend 3061:Division 2767:, while 2517:carrying 2460:Addition 1861:overflow 1847:through 1583:magnetic 1579:voltages 1372:teletype 1342:design. 1277:— 1274:Nothing. 1098:geomancy 1096:Western 1094:medieval 950:scholar 667:Georgian 657:Cyrillic 625:Armenian 580:Etruscan 575:Egyptian 483: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 18:Base two 7107:26 June 6806:Leibniz 6665:Bibcode 6544:3910603 6027:Unicode 5873:decimal 5666:have a 5611:⁄ 5583:⁄ 5530:means: 5356:Binary 5288:1010010 5282:padding 4079:0.0001 4021:Result 3991:0.0101 3720:Result 3228:integer 3079:divisor 3033:of the 2711:of the 2064:Binary 2060:Decimal 1841:Decimal 1830:decimal 1686:Decimal 1581:; on a 1285:I Ching 1253:I Ching 1244:I Ching 1215:I Ching 1211:I Ching 1153:Leibniz 1086:I Ching 1078:decimal 1046:I Ching 1042:I Ching 995:prosody 991:Pingala 915:I Ching 910:I Ching 814:History 786:with a 777:integer 753:numbers 570:Chuvash 488:Complex 285:Ancient 277:History 224:Hokkien 212:Chinese 157:Burmese 147:Tibetan 134:Kannada 114:Sinhala 89:Bengali 7173: 7143: 7081:3 July 7027:5 July 7005:5 July 6983:5 July 6961:5 July 6910: 6883: 6856: 6826: 6757: 6717: 6590: 6582: 6551: 6541: 6500:5 July 6465: 6431: 6386: 6362: 6337: 6314: 6281: 6229: 6199: 6171: 6141: 6114: 6088: 6056:22 May 5921:, the 5459:101100 4381:111110 4148:< 1 4112:< 1 4100:< 1 4067:0.000 4064:< 1 4052:< 1 4040:< 1 4005:... . 3930:0.010 3433:= 598 3417:= 299 3401:= 149 3267:× 2 + 3188:, and 3130:is 101 3081:is 101 2852:adding 2429:0.0000 2427: 2423: 2421:0.0001 2416: 2412: 2410:0.0625 2345:0.0001 2306: 2302: 2288:000111 2267: 2263: 2256: 2252: 2197: 2193: 2175: 2171: 2164: 2160: 2125: 2121: 2114: 2110: 2092: 2088: 2081: 2077: 2046:+ 1 × 2042:+ 0 × 2038:+ 1 × 1986:100101 1976:100101 1967:100101 1693:number 1691:Binary 1688:number 1635:100101 1054:UNESCO 1028:Africa 979:Dodona 826:, and 802:using 735:number 682:Hebrew 652:Coptic 565:Brahmi 550:Aegean 507: 491: 473: 460: 447: 317:Muisca 257:Tangut 234:Korean 217:Suzhou 129:Telugu 7075:(PDF) 7050:(PDF) 6715:S2CID 6695:(PDF) 6588:JSTOR 5952:ASCII 5822:10001 5676:recur 5606:(1 × 5595:plus 5578:(0 × 5567:plus 5549:plus 5473:10011 5353:Octal 5337:octal 5331:Octal 5325:Octal 5245:radix 4302:01110 4290:11001 4257:01110 4205:01110 4055:0.00 3872:0.01 3548:1197 3545:1×2 = 3542:0×2 + 3539:1×2 + 3536:1×2 + 3533:0×2 + 3530:1×2 + 3527:0×2 + 3524:1×2 + 3521:0×2 + 3518:0×2 + 3515:1×2 + 3441:× 2 + 3425:× 2 + 3409:× 2 + 3393:× 2 + 3385:= 74 3377:× 2 + 3369:= 37 3361:× 2 + 3353:= 18 3345:× 2 + 3329:× 2 + 3313:× 2 + 3297:× 2 + 3281:× 2 + 2659:(1649 2637:1 1 1 2407:1/16 2390:1/15 2373:1/14 2356:1/13 2339:1/12 2322:1/11 2297:1/10 2269:0.000 2261:0.001 2250:0.125 2219:0.001 2014:is a 1931:0000, 1673:value 1613:radix 1572:time. 1315:logic 1184:Fu Xi 999:laghu 985:India 895:China 869:hekat 865:Horus 834:Egypt 788:radix 767:). A 733:is a 677:Greek 662:Geʽez 620:Abjad 600:Roman 560:Aztec 555:Attic 470:Mixed 327:Quipu 312:Mayan 167:Khmer 119:Tamil 7203:2016 7171:ISBN 7141:ISBN 7109:2019 7083:2022 7029:2010 7007:2010 6985:2010 6963:2010 6908:ISBN 6881:ISBN 6854:ISBN 6824:ISBN 6768:2024 6755:ISBN 6580:ISSN 6549:PMID 6502:2017 6463:ISBN 6429:ISBN 6384:ISBN 6360:ISBN 6335:ISBN 6312:ISSN 6279:ISBN 6255:2022 6227:ISBN 6197:ISBN 6169:ISBN 6139:ISBN 6112:ISBN 6086:ISBN 6058:2024 5812:1100 5629:All 5618:0.25 5420:111 5412:110 5404:101 5396:100 5388:011 5380:010 5372:001 5364:000 5313:C0E7 4195:1100 4159:0011 4043:0.0 3913:< 3811:0.0 3794:< 3451:1197 3337:= 9 3321:= 4 3305:= 2 3289:= 1 3152:here 2892:and 2811:and 2624:(691 2616:(958 2604:0s: 2474:The 2398:0001 2313:0011 2280:1/9 2247:1/8 2230:1/7 2213:1/6 2204:0011 2188:1/5 2177:0.00 2169:0.01 2158:0.25 2155:1/4 2138:1/3 2105:1/2 2072:1/1 2026:and 1990:= 37 1923:and 1821:1111 1813:1110 1805:1101 1797:1100 1789:1011 1781:1010 1773:1001 1765:1000 1607:and 1586:disk 1562:LEDs 1415:bits 1389:The 1382:and 1233:and 1032:The 1003:guru 946:The 907:The 761:zero 740:base 332:Rumi 187:Thai 109:Odia 7231:at 7222:at 7133:doi 7102:BBC 7064:doi 6933:hdl 6747:doi 6707:doi 6673:doi 6539:PMC 6529:doi 6525:111 6308:129 5603:× 2 5575:× 2 5557:× 2 5539:× 2 5502:127 5224:oct 5214:dec 5204:hex 5182:oct 5172:dec 5162:hex 5140:oct 5130:dec 5120:hex 5098:oct 5088:dec 5078:hex 5052:oct 5042:dec 5032:hex 5010:oct 5000:dec 4990:hex 4968:oct 4958:dec 4948:hex 4926:oct 4916:dec 4906:hex 4880:oct 4870:dec 4860:hex 4838:oct 4828:dec 4818:hex 4796:oct 4786:dec 4776:hex 4754:oct 4744:dec 4734:hex 4708:oct 4698:dec 4688:hex 4666:oct 4656:dec 4646:hex 4624:oct 4614:dec 4604:hex 4580:oct 4570:dec 4560:hex 4403:789 4146:0.4 4136:≥ 1 4134:1.2 4124:≥ 1 4122:1.6 4110:0.8 4098:0.4 4088:≥ 1 4086:1.2 4076:≥ 1 4074:1.6 4062:0.8 4050:0.4 4038:0.2 4031:0. 4027:0.1 3753:0. 3438:598 3422:299 3406:149 3194:NOT 3190:XOR 3182:AND 2701:10 2573:mod 2569:xor 2564:). 2560:(36 2540:(23 2532:(13 2381:001 2379:0.0 2311:0.0 2300:0.1 2238:001 2191:0.2 2127:0.0 2119:0.1 2108:0.5 1934:000 1913:bit 1757:111 1749:110 1741:101 1733:100 1594:yes 1330:MIT 1129:.) 1090:Ifá 1072:in 1060:". 1034:Ifá 796:bit 790:of 765:one 747:or 742:-2 364:By 172:Lao 7255:: 7155:^ 7139:. 7117:^ 7100:. 7070:. 7060:19 7058:. 7052:. 6838:^ 6775:^ 6753:, 6741:, 6727:^ 6713:. 6703:16 6697:. 6671:. 6661:19 6659:. 6629:^ 6586:. 6576:54 6574:. 6570:. 6547:. 6537:. 6523:. 6519:. 6306:. 6302:. 6257:. 6211:^ 6183:^ 6153:^ 6074:; 6049:. 5799:10 5795:17 5789:10 5785:12 5747:01 5728:11 5705:10 5695:10 5620:) 5616:= 5588:= 5508:10 5498:10 5492:65 5444:17 5434:65 5430:: 5319:10 5315:16 5304:16 5294:16 5267:16 5265:E7 5257:16 5255:3A 5237:1 5222:17 5212:15 5195:0 5180:16 5170:14 5153:1 5138:15 5128:13 5111:0 5096:14 5086:12 5065:1 5050:13 5040:11 5023:0 5008:12 4998:10 4981:1 4966:11 4939:0 4924:10 4893:1 4851:0 4809:1 4767:0 4721:1 4679:0 4637:1 4595:0 4419:10 4411:62 4200:.1 4010:10 4003:01 3697:10 3466:. 3460:10 3449:= 3390:74 3374:37 3358:18 3244:2. 3240:10 3220:10 3186:OR 3184:, 3057:. 3023:1 3020:0 3017:1 3012:0 3009:0 3006:0 3001:1 2998:0 2985:. 2959:: 2854:a 2799:. 2787:10 2698:1 2695:1 2690:1 2687:0 2684:0 2679:1 2676:0 2661:10 2639:0 2626:10 2618:10 2562:10 2542:10 2534:10 2425:or 2414:or 2396:0. 2362:0. 2347:01 2328:0. 2304:or 2286:0. 2265:or 2254:or 2236:0. 2221:01 2202:0. 2195:or 2173:or 2162:or 2146:01 2144:0. 2123:or 2112:or 2094:0. 2090:or 2079:or 2020:10 1992:10 1941:00 1818:15 1810:14 1802:13 1794:12 1786:11 1778:10 1725:11 1717:10 1588:, 1556:A 1546:y 1514:☒ 1482:| 1450:1 1405:. 1378:, 1024:. 931:. 822:, 729:A 429:60 424:20 419:16 414:12 409:10 39:on 7205:. 7135:: 7111:. 7085:. 7066:: 7031:. 7009:. 6987:. 6965:. 6939:. 6935:: 6916:. 6889:. 6862:. 6832:. 6749:: 6721:. 6709:: 6679:. 6675:: 6667:: 6624:) 6594:. 6555:. 6531:: 6504:. 6471:. 6437:. 6368:. 6343:. 6318:. 6287:. 6235:. 6205:. 6177:. 6148:. 6121:. 6060:. 5907:2 5855:2 5833:= 5826:2 5816:2 5806:= 5761:2 5739:= 5732:2 5722:2 5718:1 5712:= 5701:3 5691:1 5650:a 5646:2 5642:p 5613:4 5609:1 5601:1 5592:) 5590:0 5585:2 5581:1 5573:0 5564:) 5562:1 5555:1 5546:) 5544:2 5537:1 5528:2 5504:8 5494:8 5483:8 5479:2 5475:2 5469:8 5465:2 5461:2 5450:2 5446:8 5440:2 5436:8 5417:7 5409:6 5401:5 5393:4 5385:3 5377:2 5369:1 5361:0 5300:2 5290:2 5278:0 5271:2 5261:2 5234:1 5231:1 5228:1 5219:= 5208:= 5201:F 5192:1 5189:1 5186:1 5177:= 5166:= 5159:E 5150:0 5147:1 5144:1 5135:= 5124:= 5117:D 5108:0 5105:1 5102:1 5093:= 5082:= 5075:C 5062:1 5059:0 5056:1 5047:= 5036:= 5029:B 5020:1 5017:0 5014:1 5005:= 4994:= 4987:A 4978:0 4975:0 4972:1 4963:= 4956:9 4952:= 4945:9 4936:0 4933:0 4930:1 4921:= 4914:8 4910:= 4903:8 4890:1 4887:1 4884:0 4878:7 4875:= 4868:7 4864:= 4857:7 4848:1 4845:1 4842:0 4836:6 4833:= 4826:6 4822:= 4815:6 4806:0 4803:1 4800:0 4794:5 4791:= 4784:5 4780:= 4773:5 4764:0 4761:1 4758:0 4752:4 4749:= 4742:4 4738:= 4731:4 4718:1 4715:0 4712:0 4706:3 4703:= 4696:3 4692:= 4685:3 4676:1 4673:0 4670:0 4664:2 4661:= 4654:2 4650:= 4643:2 4634:0 4631:0 4628:0 4622:1 4619:= 4612:1 4608:= 4601:1 4592:0 4589:0 4586:0 4578:0 4575:= 4568:0 4564:= 4557:0 4537:k 4529:k 4511:x 4491:x 4471:x 4451:x 4415:) 4407:/ 4400:( 4395:= 4388:x 4377:/ 4368:= 4361:x 4349:= 4342:) 4339:2 4331:6 4327:2 4323:( 4317:x 4297:. 4285:= 4278:2 4272:x 4252:. 4240:= 4231:6 4227:2 4220:x 4190:= 4183:x 3999:3 3977:1 3969:3 3966:1 3961:1 3958:= 3955:2 3947:3 3944:2 3916:1 3908:3 3905:2 3900:= 3897:2 3889:3 3886:1 3858:1 3850:3 3847:1 3842:1 3839:= 3836:2 3828:3 3825:2 3797:1 3789:3 3786:2 3781:= 3778:2 3770:3 3767:1 3737:3 3734:1 3693:) 3687:3 3684:1 3679:( 3654:2 3651:1 3627:4 3624:1 3619:= 3614:2 3610:) 3604:2 3601:1 3596:( 3574:2 3571:1 3559:2 3505:1 3502:0 3499:1 3496:1 3493:0 3490:1 3487:0 3484:1 3481:0 3478:0 3475:1 3445:1 3429:0 3413:1 3397:1 3381:0 3365:1 3349:0 3342:9 3333:1 3326:4 3317:0 3310:2 3301:0 3294:1 3285:1 3278:0 3256:2 3136:2 3132:2 3128:2 3124:2 3103:2 3099:2 3091:2 3083:2 2976:B 2972:B 2968:B 2964:A 2950:B 2946:B 2942:B 2938:A 2928:A 2924:B 2917:B 2906:B 2902:A 2898:B 2894:B 2890:A 2784:= 2781:1 2778:+ 2775:1 2755:1 2752:= 2749:1 2743:1 2657:2 2649:1 2645:1 2622:2 2614:2 2612:0 2602:n 2598:n 2594:n 2590:n 2586:n 2558:2 2554:2 2550:2 2546:2 2538:2 2530:2 2431:1 2271:1 2179:1 2129:1 2096:1 2086:1 2075:1 2048:2 2044:2 2040:2 2036:2 2028:5 2024:2 1988:2 1978:2 1969:2 1956:1 1950:1 1948:0 1943:1 1936:1 1925:1 1921:0 1884:1 1872:1 1870:0 1857:0 1849:9 1845:0 1770:9 1762:8 1754:7 1746:6 1738:5 1730:4 1722:3 1714:2 1709:1 1706:1 1701:0 1698:0 1646:) 1637:2 1626:) 1609:1 1605:0 1543:y 1540:n 1537:y 1534:y 1531:n 1528:n 1525:y 1522:n 1519:y 1511:☒ 1508:☐ 1505:☒ 1502:☒ 1499:☐ 1496:☐ 1493:☒ 1490:☐ 1487:☒ 1479:| 1476:― 1473:| 1470:| 1467:― 1464:― 1461:| 1458:― 1455:| 1447:1 1444:0 1441:1 1438:1 1435:0 1432:0 1429:1 1426:0 1423:1 1355:K 1186:" 1038:. 792:2 718:e 711:t 704:v 513:) 511:φ 509:( 500:) 497:i 495:2 493:( 479:) 475:( 466:) 462:( 453:) 451:1 449:( 404:8 399:6 394:5 389:4 384:3 379:2 20:)

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Index