Binary number - Knowledge (XXG)

6804:: "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." 1154: 1900: 5934: 4431: 841: 896: 4170: 1294: 1549: 2467: 2515:. 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: 1169:, 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.. 3234:. 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) 4426:{\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}}} 3249:

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

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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

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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

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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

882:

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

1112:

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

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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

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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:

2837:. 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. 5863: 2607:

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

4157:... . 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 1353:

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

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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

4531:. 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 1016:. 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 2603:

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

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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.

5773: 1413:(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: 1217:

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

2659:). 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. 3664:, 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. 3983: 3864: 3922: 3803: 1053: 4175: 3635: 5679: 1121:

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

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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:

2046:+ ... = 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. 5339:(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 2647:

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

806:, 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. 5660: 3705: 2866:

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

1663:, 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 863:, 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 1859:), 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: 1330: 1178:"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 5915: 3745: 3662: 3582: 950:(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 2761: 1255:

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

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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

2974:+ 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) 2729: 1276: 4517: 4497: 4477: 4457: 3211: 1655:

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

712: 2544:. 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 1370:. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were 3550:

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:

1667:(a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct 859:

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

7186: 7274: 7140: 6907: 6880: 6853: 6754: 6462: 6428: 6359: 6334: 6226: 6196: 6168: 1923:

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.

4161:. 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. 2833:

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.

2978: 7269: 7170: 993:. He described meters in the form of short and long syllables (the latter equal in length to two short syllables). They were known as 265: 1184:. Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 6823: 6383: 6138: 6111: 6085: 3147: 1328:

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

826:. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. 5871:. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 1008:(8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to 6397: 1153: 56: 7067: 6007: 4527:

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

1847:. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the 1097: 879: 5858:{\displaystyle {\frac {12_{10}}{17_{10}}}={\frac {1100_{2}}{10001_{2}}}=0.1011010010110100{\overline {10110100}}\ldots \,_{2}} 3933: 3814: 7264: 6278: 5987: 5890:

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

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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.

1222: 872: 452: 3587: 2552:. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100 1592:", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use. 3146:

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

7254: 3223: 2843:* (starred columns are borrowed from) 1 0 1 1 1 1 1 – 1 0 1 0 1 1 ---------------- = 0 1 1 0 1 0 0 884: 795: 512: 2966:(6.25 in decimal) ------------------- 1 . 0 1 1 0 1 ← Corresponds to a 'one' in 6899:

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

2840:* * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1 2027: 1398: 1252: 1142: 1080:

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

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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)

6902:(Macmillan, Dover Publications, reprinted with corrections ed.). New York: Cambridge University Press. 1005: 1314: 688: 472: 69: 6687: 1876:

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

5872: 5277: 3082: 1234: 868: 372: 288: 6002: 5764:{\displaystyle {\frac {1_{10}}{3_{10}}}={\frac {1_{2}}{11_{2}}}=0.01010101{\overline {01}}\ldots \,_{2}} 5631: 3231: 3185: 2631:

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

1607:. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or 963: 951: 490: 6457:. Contemporary ethnography (1st ed.). Philadelphia: University of Pennsylvania Press. p. 25. 6042: 5933: 3670: 2627:

Traditional Carry Method Long Carry Method vs.

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Arbeitsbuch Informatik – eine praxisorientierte Einführung in die Datenverarbeitung mit Projektaufgabe

5313:= (12 × 16) + (0 × 16) + (14 × 16) + (7 × 16) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,383 3458:. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the 1851:. When the available symbols for this position are exhausted, the least significant digit is reset to 6660: 6603: 5967: 2592:

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

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

6710: 6583: 5939: 5896: 3169: 3062: 1317:. His logical calculus was to become instrumental in the design of digital electronic circuitry. 932: 917: 580: 333: 170: 165: 112: 3723: 3640: 3560: 2030:. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 6843: 6637: 3210: 2734: 1648:

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

935:, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the 7215: 7166: 7162:

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

7136: 6903: 6876: 6870: 6849: 6819: 6750: 6575: 6544: 6458: 6424: 6379: 6355: 6330: 6307: 6274: 6268: 6222: 6192: 6164: 6134: 6128: 6107: 6081: 6075: 6012: 5957: 5883: 3173: 3163: 2003: 1585: 1363: 1257: 1122: 852: 823: 662: 652: 640: 620: 575: 570: 506: 338: 310: 217: 150: 140: 127: 92: 87: 6216: 6158: 6101: 3188:

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

3134:. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. 7128: 7059: 6928: 6742: 6702: 6668: 6534: 6524: 5952: 5918: 5876: 5667: 3637:, etc. So if there is a 1 in the first place after the decimal, then the number is at least 3194: 3036: 2766: 2568: 1619: 1371: 1069: 990: 565: 459: 212: 200: 145: 135: 102: 77: 6490: 4435:

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

2714: 2624:), using the traditional carry method on the left, and the long carry method on the right: 2452:. Addition, subtraction, multiplication, and division can be performed on binary numerals. 1176:(published in 1703). The full title of Leibniz's article is translated into English as the 771:

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

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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

6539: 6512: 6414: 6071: 6067: 4502: 4482: 4462: 4442: 2881: 2855: 2449: 1855:, and the next digit of higher significance (one position to the left) is incremented ( 1578: 1379: 1342: 1321: 1244: 1202: 1137: 815: 752: 739: 672: 615: 595: 550: 423: 155: 122: 107: 33: 6394: 6218:

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

2540:). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10 840: 7248: 7038: 6995:"Pioneers – The people and ideas that made a difference – George Stibitz (1904–1995)" 6789: 6714: 6633: 5992: 5972: 5519: 4528: 3459: 3090: 3067: 2588:

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

1394: 1386: 1226: 1117: 1045: 478: 300: 240: 175: 117: 97: 7042: 7228: 7224: 7219: 6746: 5336: 2952: 2940:) --------- 0 0 0 0 ← Corresponds to the rightmost 'zero' in 2564: 1581: 1553: 1375: 1302: 1230: 943: 936: 924: 856: 845: 630: 555: 6801: 6329:. Blackwell ancient religions (1. publ ed.). Malden, Mass.: Wiley-Blackwell. 1651:#b100101 (a prefix indicating binary format, common in Lisp programming languages) 1293: 895: 7160: 7122: 6779: 6418: 5239:

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

5962: 5515: 5423: 5340: 5207: 5165: 5123: 5081: 5035: 4544: 4436: 3130:, as shown on the top line, while the remainder, shown on the bottom line, is 10 2813: 2007: 1565: 1390: 1367: 1214: 1136:

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

1129: 1109: 1089: 1017: 871:, approximately 2400 BC, and its fully developed hieroglyphic form dates to the 600: 465: 417: 407: 1895:

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

7132: 6950: 5929: 2442: 2011: 1589: 799: 402: 160: 6994: 6579: 6311: 3172:. When a string of binary symbols is manipulated in this way, it is called a 2904:

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

1999: 1891: 1346: 1077: 1065: 947: 412: 6975:. Math & Computer Science Department, Denison University. 30 April 2004 6706: 6548: 5893:

1.0110101000001001111001100110011111110... is the binary representation of

1548: 1225:, a popular idea that would be followed closely by his successors such as 1221:

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

7165:(in German). Vol. 2 (5 ed.). Vieweg, reprint: Springer-Verlag. 6897: 5982: 3189: 3115: 2929:

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)

1093: 967: 803: 6932: 6587: 6563: 6295: 5674:, with a finite sequence of digits repeating indefinitely. For instance 6022: 5868: 3074: 2466: 2055: 1836: 1825: 1574: 1306: 1239: 1084:

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

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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

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Division algorithm § Integer division (unsigned) with remainder

1085: 1029: 794:, or binary digit. Because of its straightforward implementation in 1044:

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

2465: 2014:. As a result, 1/10 does not have a finite binary representation ( 1898: 1890: 1608: 1547: 1310: 1292: 1179: 1152: 864: 860: 839: 783: 735: 392: 354: 315: 6244:"Yijing hexagram sequences: The Shao Yong square (Fuxi sequence)" 5502:= (1 × 8) + (2 × 8) + (7 × 8) = (1 × 64) + (2 × 8) + (7 × 1) = 87 3089:, or 27 in decimal. The procedure is the same as that of decimal 1349:, completed a relay-based computer he dubbed the "Model K" (for " 756: 6739:

Handbook of the History and Philosophy of Mathematical Practice

2948:+ 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0 1595:

In keeping with the customary representation of numerals using

1934:, (rightmost bit starts over, and the next bit is incremented) 1410: 1325: 791: 760: 7019:. Computer History Association of California. 6 February 1995 6845:

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

6651:

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

6513:"Mangarevan invention of binary steps for easier calculation" 5670:

have binary representation, but instead of terminating, they

2884:

in binary is similar to its decimal counterpart. Two numbers

2524:

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

1247:, who visited China in 1685 as a missionary. Leibniz saw the 989:(c. 2nd century BC) developed a binary system for describing 6927:(Thesis). Cambridge: Massachusetts Institute of Technology. 5522:

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

2478:, which adds two bits together, producing sum and carry bits 2026:). This causes 10 × 1/10 not to precisely equal 1 in binary 1054:

Masterpieces of the Oral and Intangible Heritage of Humanity

5882:

Binary numerals that neither terminate nor recur represent

4993: 2970:+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in 6741:, Cham: Springer International Publishing, pp. 1–31, 6601: 6564:"Diversity in the Numeral Systems of Australian Languages" 5335:

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

4548: 1638:%100101 (a prefix indicating binary format; also known as 1618:

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

7094:"Introducing binary – Revision 1 – GCSE Computer Science" 7043:"Konrad Zuse's Legacy: The Architecture of the Z1 and Z3" 6296:"Mapping the Entrails: The Practice of Greek Hepatoscopy" 4951: 4909: 4863: 4821: 4779: 4737: 4691: 4649: 4607: 4563: 3978:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1} 3859:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1} 2892:

can be multiplied by partial products: for each digit in

2521:

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

787: 7154: 7152: 7116: 7114: 2951:

Binary numbers can also be multiplied with bits after a

1599:, binary numbers are commonly written using the symbols 3917:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1} 3798:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1} 3070:

in binary is again similar to its decimal counterpart.

2596:

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

7127:(in German). Vieweg-Verlag, reprint: Springer-Verlag. 6074:, eds. (2009), "Myth No. 2: the Horus eye fractions", 3936: 3878: 3817: 3759: 3726: 3673: 3643: 3590: 3563: 2563:

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

2507:

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

2504:

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

1671:), but this does not make its binary nature explicit. 6947:"National Inventors Hall of Fame – George R. Stibitz" 6511:

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

6273:. Stuttgart: Franz Steiner Verlag. pp. 165–170. 5899: 5776: 5682: 5634: 4505: 4485: 4465: 4445: 4173: 3997:... is equivalent to the repeating binary fraction 0. 3039: 2944:+ 1 0 1 1 ← Corresponds to the next 'one' in 2769: 2737: 2717: 2629:

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

1265:

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

6157:

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

5921:, another irrational. It has no discernible pattern. 2962:(5.625 in decimal) × 1 1 0 . 0 1 2495:

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

7159:Küveler, Gerd; Schwoch, Dietrich (4 October 2007). 6925:

A symbolic analysis of relay and switching circuits

6733:Strickland, Lloyd (2020), Sriraman, Bharath (ed.), 6695:

Mitteilungen der deutschen Mathematiker-Vereinigung

3553:

In a fractional binary number such as 0.11010110101

3025:The binary multiplication table is the same as the 1331:

A Symbolic Analysis of Relay and Switching Circuits

5909: 5857: 5763: 5654: 4511: 4491: 4471: 4451: 4425: 3977: 3916: 3858: 3797: 3739: 3699: 3656: 3629: 3576: 3045: 2787: 2755: 2723: 6267:Zhonglian, Shi; Wenzhao, Li; Poser, Hans (2000). 6152: 6150: 6077:The Oxford Handbook of the History of Mathematics 3226:to its base-2 (binary) equivalent, the number is 1954:000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ... 1635:(a subscript indicating base-2 (binary) notation) 1334:, Shannon's thesis essentially founded practical 802:, the binary system is used by almost all modern 6790:"Bouvet and Leibniz: A Scholarly Correspondence" 6130:How Mathematics Happened: The First 50,000 Years 3630:{\textstyle ({\frac {1}{2}})^{2}={\frac {1}{4}}} 1172:His most well known work appears in his article 6688:"Leibniz, Caramuel, Harriot und das Dualsystem" 6517:Proceedings of the National Academy of Sciences 6043:"3.3. Binary and Its Advantages — CS160 Reader" 2846:Subtracting a positive number is equivalent to 1409:Any number can be represented by a sequence of 1263: 1205:in 1700, who had made himself an expert on the 6455:Vodún: secrecy and the search for divine power 6354:. Boca Raton, Florida: CRC Press. p. 37. 6352:Microcontroller programming: the microchip PIC 6106:, Cambridge University Press, pp. 42–43, 1628:bin 100101 (a prefix indicating binary format) 855:(not related to the binary number system) and 6182: 6180: 2862:to handle negative numbers—most commonly the 1040:, but has up to 256 binary signs, unlike the 706: 16:Number expressed in the base-2 numeral system 8: 7233:Sir Francis Bacon's BiLiteral Cypher system 6628: 6626: 6210: 6208: 5492:= (6 × 8) + (5 × 8) = (6 × 8) + (5 × 1) = 53 4153:This is also a repeating binary fraction 0.0 3201:Conversion to and from other numeral systems 1615:100101 binary (explicit statement of format) 4439:, is to do so indirectly—first converting ( 2445:in binary is much like arithmetic in other 1625:100101B (a suffix indicating binary format) 1243:through his contact with the French Jesuit 1201:While corresponding with the Jesuit priest 7121:Küveler, Gerd; Schwoch, Dietrich (2013) . 1611:. The following notations are equivalent: 713: 699: 49: 20: 6538: 6528: 6350:Sanchez, Julio; Canton, Maria P. (2007). 6103:Numerical Notation: A Comparative History 5900: 5898: 5849: 5848: 5834: 5820: 5810: 5804: 5793: 5783: 5777: 5775: 5755: 5754: 5740: 5726: 5716: 5710: 5699: 5689: 5683: 5681: 5644: 5635: 5633: 4504: 4484: 4464: 4444: 4413: 4401: 4371: 4325: 4295: 4250: 4225: 4198: 4174: 4172: 3959: 3937: 3935: 3898: 3879: 3877: 3840: 3818: 3816: 3779: 3760: 3758: 3727: 3725: 3691: 3677: 3672: 3644: 3642: 3617: 3608: 3594: 3589: 3564: 3562: 3218:to binary notation results in (101100101) 3038: 2768: 2736: 2716: 1828:counting system as a frame of reference. 6837: 6835: 6818:. Taylor & Francis. pp. 245–8. 6191:. Oxford University Press. p. 227. 6080:, Oxford University Press, p. 790, 5345: 4010: 3709: 3209: 2988: 2666: 2048: 1678: 1305:published a landmark paper detailing an 894: 7051:IEEE Annals of the History of Computing 6187:Redmond, Geoffrey; Hon, Tze-Ki (2014). 6034: 5331:Binary is also easily converted to the 32: 6774: 6772: 6477: 6133:, Prometheus Books, pp. 135–136, 3993:Thus the repeating decimal fraction 0. 2971: 2967: 2963: 2959: 2945: 2941: 2937: 2933: 2923: 2922:is 1, the partial product is equal to 2919: 2912: 2901: 2897: 2893: 2889: 2885: 2597: 2593: 2589: 2585: 2581: 1237:. Leibniz was first introduced to the 6728: 6726: 6724: 6562:Bowern, Claire; Zentz, Jason (2012). 5288:= 0101 0010 grouped with padding = 52 4479:in hexadecimal) and then converting ( 1382:, who wrote about it in his memoirs. 1174:Explication de l'Arithmétique Binaire 1113:science and artificial intelligence. 7: 6221:. Greenwood Publishing. p. 29. 3097:goes into the first three digits 110 1915:), except that only the two symbols 1911:(the rightmost one, also called the 804:computers and computer-based devices 2915:is 0, the partial product is also 0 1251:hexagrams as an affirmation of the 923:It is based on taoistic duality of 751:that uses only two symbols for the 7017:"George Robert Stibitz – Obituary" 6160:I Ching: An Annotated Bibliography 5655:{\displaystyle {\frac {p}{2^{a}}}} 3700:{\textstyle ({\frac {1}{3}})_{10}} 1389:, which was designed and built by 14: 6997:. Kerry Redshaw. 20 February 2006 6300:The American Journal of Philology 887:, which dates to around 1650 BC. 790:. Each digit is referred to as a 7241:, predates binary number system. 7073:from the original on 3 July 2022 6949:. 20 August 2008. Archived from 5932: 2979:Booth's multiplication algorithm 1324:produced his master's thesis at 1197:1 0 0 0 numerical value 2 1194:0 1 0 0 numerical value 2 1191:0 0 1 0 numerical value 2 1188:0 0 0 1 numerical value 2 1165:His first known work on binary, 6923:Shannon, Claude Elwood (1940). 6872:Leibniz, Mysticism and Religion 6417:; Olsen, Scott Anthony (2009). 6008:Redundant binary representation 2896:, the product of that digit in 1301:In 1854, British mathematician 1104:Western predecessors to Leibniz 1098:Indigenous Australian languages 1064:The residents of the island of 1032:is an African divination system 1004:Pingala's Hindu classic titled 933:64 hexagrams ("sixty-four" gua) 880:ancient Egyptian multiplication 778:The base-2 numeral system is a 6875:. Springer. pp. 149–150. 6747:10.1007/978-3-030-19071-2_90-1 5988:Linear-feedback shift register 4410: 4395: 4337: 4318: 3688: 3674: 3605: 3591: 1839:counting uses the ten symbols 1052:added Ifá to its list of the " 1: 6640:. London. pp. Chapter 1. 6638:"The Advancement of Learning" 6325:Johnston, Sarah Iles (2008). 6100:Chrisomalis, Stephen (2010), 6047:computerscience.chemeketa.edu 5622:For a total of 3.25 decimal. 3081:, or 5 in decimal, while the 2860:signed number representations 2805:signed number representations 2519:1 1 1 1 1 (carried digits) 1360:American Mathematical Society 1132:described a system he called 7275:Power-of-two numeral systems 6374:W. S. Anglin and J. Lambek, 6127:Rudman, Peter Strom (2007), 5879:2 + 2 + 2 + ... which is 1. 5839: 5745: 4300: 4255: 4203: 2816:works in much the same way: 2580:digits composed entirely of 2381:1/16 + 1/128 + 1/1024 . . . 1588:may be used. A "positive", " 1393:between 1935 and 1938, used 1358:. In a demonstration to the 1261:or creation out of nothing. 1088:among others, as well as in 973:Divination at Ancient Greek 836:Ancient Egyptian mathematics 796:digital electronic circuitry 747:, a method for representing 6973:"George Stibitz : Bio" 6735:"Leibniz on Number Systems" 6653:American Journal of Physics 6568:Anthropological Linguistics 6453:Landry, Timothy R. (2019). 6395:Math for Poets and Drummers 5910:{\displaystyle {\sqrt {2}}} 5484:And from octal to decimal: 3740:{\textstyle {\frac {1}{3}}} 3657:{\textstyle {\frac {1}{2}}} 3577:{\textstyle {\frac {1}{2}}} 3433: 3417: 3401: 3385: 3369: 3353: 3337: 3321: 3305: 3289: 3273: 3238:is expressed as (101100101) 2364:1/16 + 1/128 + 1/256 . . . 1564:numeral of the traditional 1313:that would become known as 1223:characteristica universalis 1167:“On the Binary Progression" 1072:were using a hybrid binary- 873:Nineteenth Dynasty of Egypt 7291: 6215:Jonathan Shectman (2003). 5451:And from binary to octal: 5324: 5065: 4893: 4721: 4579: 4542: 3222:To convert from a base-10 3161: 3073:In the example below, the 3060: 2802: 2459: 2431:1/32 + 1/64 + 1/128 . . . 2347:1/16 + 1/64 + 1/256 . . . 2330:1/16 + 1/64 + 1/128 . . . 2313:1/16 + 1/32 + 1/256 . . . 1162:unrelated to mathematics. 966:divided the outer edge of 885:Rhind Mathematical Papyrus 833: 433:Non-standard radices/bases 7270:Gottfried Wilhelm Leibniz 7237:23 September 2016 at the 7133:10.1007/978-3-322-92907-5 6848:. Springer. p. 415. 6163:. Routledge. p. 13. 5510:Representing real numbers 5477:grouped with padding = 23 5276:bits at the left (called 4159:floating point arithmetic 3170:Boolean logical operators 2756:{\displaystyle 1\lor 1=1} 2731:. The difference is that 2616:) and 1 0 1 0 1 1 0 0 1 1 2288:1/16 + 1/32 + 1/64 . . . 2271:1/16 + 1/32 + 1/64 . . . 2238:1/8 + 1/64 + 1/512 . . . 2221:1/8 + 1/32 + 1/128 . . . 2204:1/8 + 1/16 + 1/128 . . . 2063:Fractional approximation 2028:floating-point arithmetic 1686: 1681: 1143:Juan Caramuel y Lobkowitz 1108:In the late 13th century 913:is used to interpret its 875:, approximately 1200 BC. 820:Juan Caramuel y Lobkowitz 6327:Ancient Greek divination 5298:= 1101 1101 grouped = DD 3176:; the logical operators 2936:) × 1 0 1 0 ( 2179:1/8 + 1/16 + 1/32 . . . 2146:1/4 + 1/16 + 1/64 . . . 1275:Leibniz's letter to the 7225:Conversion of Fractions 6896:Boole, George (2009) . 6814:Aiton, Eric J. (1985). 6530:10.1073/pnas.1309160110 6491:"Ifa Divination System" 6294:Collins, Derek (2008). 5627:dyadic rational numbers 3230:. The remainder is the 3093:; here, the divisor 101 2129:1/4 + 1/8 + 1/16 . . . 689:List of numeral systems 6869:Yuen-Ting Lai (1998). 6707:10.1515/dmvm-2008-0009 6376:The Heritage of Thales 5911: 5859: 5765: 5656: 4513: 4499:in hexadecimal) into ( 4493: 4473: 4453: 4427: 3979: 3918: 3860: 3799: 3741: 3701: 3658: 3631: 3578: 3219: 3047: 3046:{\displaystyle \land } 2789: 2788:{\displaystyle 1+1=10} 2757: 2725: 2479: 1904: 1896: 1569: 1399:floating-point numbers 1298: 1286: 1158: 929:Eight trigrams (Bagua) 900: 869:Fifth Dynasty of Egypt 848: 7265:Elementary arithmetic 6842:J.E.H. Smith (2008). 6686:Ineichen, R. (2008). 6610:, Fidora et al. 2011 6003:Reduction of summands 5912: 5860: 5766: 5657: 4514: 4494: 4474: 4454: 4428: 3980: 3919: 3861: 3800: 3742: 3702: 3659: 3632: 3579: 3557:, the first digit is 3232:least-significant bit 3213: 3048: 2871:A − B = A + not B + 1 2803:Further information: 2790: 2758: 2726: 2724:{\displaystyle \lor } 2469: 2002:in binary arithmetic 1902: 1894: 1873: ... 1551: 1296: 1235:modern symbolic logic 1156: 1100:use a base-2 system. 952:least significant bit 898: 843: 745:binary numeral system 57:Hindu–Arabic numerals 6816:Leibniz: A Biography 6616:8 April 2019 at the 6606:3 April 2014 at the 6423:. World Scientific. 6400:16 June 2012 at the 6189:Teaching the I Ching 5968:Binary-coded decimal 5897: 5774: 5680: 5632: 4503: 4483: 4463: 4443: 4171: 3934: 3876: 3815: 3757: 3724: 3671: 3641: 3588: 3561: 3037: 2985:Multiplication table 2767: 2735: 2715: 1562:binary-coded decimal 1076:system before 1450. 878:The method used for 767:may also refer to a 586:Prehistoric counting 362:Common radices/bases 44:Place-value notation 7260:Computer arithmetic 7041:(April–June 1997). 6665:1951AmJPh..19..452S 6072:Stedall, Jacqueline 4004:Or for example, 0.1 3214:Conversion of (357) 3031:logical conjunction 2823:0 − 1 → 1, borrow 1 2709:logical disjunction 2462:Adder (electronics) 2447:positional notation 2398:1/16 + 1/256 . . . 2372:0.0714285714285... 2096:1/2 + 1/4 + 1/8... 1640:Motorola convention 1586:magnetic polarities 1134:location arithmetic 1014:positional notation 1001:(heavy) syllables. 985:The Indian scholar 959:Classical antiquity 780:positional notation 775:by a power of two. 533:Sign-value notation 7193:on 23 October 2017 6378:, Springer, 1995, 5940:Mathematics portal 5907: 5884:irrational numbers 5875:is the sum of the 5855: 5832:0.1011010010110100 5761: 5652: 4509: 4489: 4469: 4449: 4423: 4421: 3975: 3914: 3856: 3795: 3737: 3697: 3654: 3627: 3574: 3454:The result is 1197 3220: 3158:Bitwise operations 3043: 2958:1 0 1 . 1 0 1 2785: 2753: 2721: 2480: 2355:0.076923076923... 2229:0.142857142857... 2018:has prime factors 1905: 1897: 1675:Counting in binary 1570: 1345:, then working at 1341:In November 1937, 1299: 1289:Later developments 1279:attached with the 1159: 1096:. The majority of 939:of ancient China. 901: 853:Egyptian fractions 849: 189:East Asian systems 7255:Binary arithmetic 7142:978-3-528-04952-2 7064:10.1109/85.586067 6909:978-1-108-00153-3 6882:978-0-7923-5223-5 6855:978-1-4020-8668-7 6756:978-3-030-19071-2 6673:10.1119/1.1933042 6600:(see Bonner 2007 6464:978-0-8122-5074-9 6430:978-981-277-582-5 6361:978-0-8493-7189-9 6336:978-1-4051-1573-5 6242:Marshall, Steve. 6228:978-0-313-32015-6 6198:978-0-19-976681-9 6170:978-0-415-93969-0 6013:Repeating decimal 5958:Bitwise operation 5905: 5842: 5826: 5799: 5748: 5732: 5705: 5650: 5620: 5619: 5420: 5419: 5237: 5236: 4512:{\displaystyle x} 4492:{\displaystyle x} 4472:{\displaystyle x} 4459:in binary) into ( 4452:{\displaystyle x} 4303: 4258: 4206: 4151: 4150: 4008:, in binary, is: 3991: 3990: 3967: 3945: 3906: 3887: 3848: 3826: 3787: 3768: 3735: 3707:, in binary, is: 3685: 3652: 3625: 3602: 3572: 3548: 3547: 3452: 3451: 3245:Binary to decimal 3206:Decimal to binary 3174:bitwise operation 3164:Bitwise operation 3150:. An example is: 3023: 3022: 2701: 2700: 2575:Long carry method 2511:This is known as 2438:Binary arithmetic 2435: 2434: 1976:= + + + + + 1967:= + + + + + 1959:form as follows: 1882:00, 101, 102, ... 1821: 1820: 1546: 1545: 1364:Dartmouth College 1277:Duke of Brunswick 1258:creatio ex nihilo 1157:Gottfried Leibniz 1010:science of meters 968:divination livers 824:Gottfried Leibniz 755:: typically "0" ( 733:expressed in the 723: 722: 522: 521: 7282: 7203: 7202: 7200: 7198: 7189:. Archived from 7183: 7177: 7176: 7175:. 9783834891914. 7156: 7147: 7146: 7145:. 9783322929075. 7118: 7109: 7108: 7106: 7104: 7090: 7084: 7082: 7080: 7078: 7072: 7047: 7035: 7029: 7028: 7026: 7024: 7013: 7007: 7006: 7004: 7002: 6991: 6985: 6984: 6982: 6980: 6969: 6963: 6962: 6960: 6958: 6943: 6937: 6936: 6920: 6914: 6913: 6893: 6887: 6886: 6866: 6860: 6859: 6839: 6830: 6829: 6811: 6805: 6799: 6793: 6792:, Swiderski 1980 6787: 6781: 6776: 6767: 6766: 6765: 6763: 6730: 6719: 6718: 6692: 6683: 6677: 6676: 6648: 6642: 6641: 6630: 6621: 6598: 6592: 6591: 6559: 6553: 6552: 6542: 6532: 6523:(4): 1322–1327. 6508: 6502: 6501: 6499: 6497: 6487: 6481: 6475: 6469: 6468: 6450: 6444: 6441: 6435: 6434: 6411: 6405: 6392: 6386: 6372: 6366: 6365: 6347: 6341: 6340: 6322: 6316: 6315: 6291: 6285: 6284: 6264: 6258: 6257: 6252: 6250: 6239: 6233: 6232: 6212: 6203: 6202: 6184: 6175: 6174: 6154: 6145: 6143: 6124: 6118: 6116: 6097: 6091: 6090: 6064: 6058: 6057: 6055: 6053: 6039: 6018:Two's complement 5953:Balanced ternary 5942: 5937: 5936: 5919:square root of 2 5916: 5914: 5913: 5908: 5906: 5901: 5886:. For instance, 5877:geometric series 5864: 5862: 5861: 5856: 5854: 5853: 5843: 5835: 5827: 5825: 5824: 5815: 5814: 5805: 5800: 5798: 5797: 5788: 5787: 5778: 5770: 5768: 5767: 5762: 5760: 5759: 5749: 5741: 5733: 5731: 5730: 5721: 5720: 5711: 5706: 5704: 5703: 5694: 5693: 5684: 5668:rational numbers 5661: 5659: 5658: 5653: 5651: 5649: 5648: 5636: 5611: 5610: 5606: 5583: 5582: 5578: 5529: 5528: 5346: 5280:). For example: 4549: 4518: 4516: 4515: 4510: 4498: 4496: 4495: 4490: 4478: 4476: 4475: 4470: 4458: 4456: 4455: 4450: 4432: 4430: 4429: 4424: 4422: 4418: 4417: 4405: 4375: 4330: 4329: 4304: 4296: 4259: 4251: 4230: 4229: 4207: 4199: 4156: 4011: 4000: 3996: 3984: 3982: 3981: 3976: 3968: 3960: 3946: 3938: 3923: 3921: 3920: 3915: 3907: 3899: 3888: 3880: 3865: 3863: 3862: 3857: 3849: 3841: 3827: 3819: 3804: 3802: 3801: 3796: 3788: 3780: 3769: 3761: 3746: 3744: 3743: 3738: 3736: 3728: 3710: 3706: 3704: 3703: 3698: 3696: 3695: 3686: 3678: 3663: 3661: 3660: 3655: 3653: 3645: 3636: 3634: 3633: 3628: 3626: 3618: 3613: 3612: 3603: 3595: 3583: 3581: 3580: 3575: 3573: 3565: 3465: 3464: 3257: 3256: 3195:arithmetic shift 3052: 3050: 3049: 3044: 2989: 2918:If the digit in 2911:If the digit in 2872: 2864:two's complement 2858:. Computers use 2809:two's complement 2794: 2792: 2791: 2786: 2762: 2760: 2759: 2754: 2730: 2728: 2727: 2722: 2667: 2630: 2520: 2428: 2424: 2420: 2413: 2409: 2395: 2378: 2361: 2344: 2327: 2310: 2303: 2299: 2285: 2268: 2264: 2260: 2253: 2249: 2235: 2218: 2201: 2194: 2190: 2176: 2172: 2168: 2161: 2157: 2143: 2126: 2122: 2118: 2111: 2107: 2093: 2089: 2085: 2078: 2074: 2049: 1870:0, 011, 012, ... 1832:Decimal counting 1679: 1620:Intel convention 1416: 1415: 1372:John von Neumann 1284: 1070:French Polynesia 715: 708: 701: 504: 488: 470: 460:balanced ternary 457: 444: 50: 21: 7290: 7289: 7285: 7284: 7283: 7281: 7280: 7279: 7245: 7244: 7239:Wayback Machine 7212: 7207: 7206: 7196: 7194: 7185: 7184: 7180: 7173: 7158: 7157: 7150: 7143: 7120: 7119: 7112: 7102: 7100: 7092: 7091: 7087: 7076: 7074: 7070: 7045: 7037: 7036: 7032: 7022: 7020: 7015: 7014: 7010: 7000: 6998: 6993: 6992: 6988: 6978: 6976: 6971: 6970: 6966: 6956: 6954: 6945: 6944: 6940: 6922: 6921: 6917: 6910: 6895: 6894: 6890: 6883: 6868: 6867: 6863: 6856: 6841: 6840: 6833: 6826: 6813: 6812: 6808: 6800: 6796: 6788: 6784: 6777: 6770: 6761: 6759: 6757: 6732: 6731: 6722: 6690: 6685: 6684: 6680: 6650: 6649: 6645: 6632: 6631: 6624: 6618:Wayback Machine 6608:Wayback Machine 6599: 6595: 6561: 6560: 6556: 6510: 6509: 6505: 6495: 6493: 6489: 6488: 6484: 6476: 6472: 6465: 6452: 6451: 6447: 6442: 6438: 6431: 6415:Stakhov, Alexey 6413: 6412: 6408: 6402:Wayback Machine 6393: 6389: 6373: 6369: 6362: 6349: 6348: 6344: 6337: 6324: 6323: 6319: 6293: 6292: 6288: 6281: 6266: 6265: 6261: 6248: 6246: 6241: 6240: 6236: 6229: 6214: 6213: 6206: 6199: 6186: 6185: 6178: 6171: 6156: 6155: 6148: 6141: 6126: 6125: 6121: 6114: 6099: 6098: 6094: 6088: 6068:Robson, Eleanor 6066: 6065: 6061: 6051: 6049: 6041: 6040: 6036: 6031: 5938: 5931: 5928: 5895: 5894: 5847: 5816: 5806: 5789: 5779: 5772: 5771: 5753: 5722: 5712: 5695: 5685: 5678: 5677: 5640: 5630: 5629: 5608: 5604: 5603: 5580: 5576: 5575: 5525: 5512: 5505: 5501: 5495: 5491: 5480: 5476: 5472: 5466: 5462: 5458: 5447: 5443: 5437: 5433: 5329: 5323: 5316: 5312: 5301: 5297: 5291: 5287: 5268: 5264: 5258: 5254: 5221: 5211: 5201: 5179: 5169: 5159: 5137: 5127: 5117: 5095: 5085: 5075: 5049: 5039: 5029: 5007: 4997: 4987: 4965: 4955: 4945: 4923: 4913: 4903: 4877: 4867: 4857: 4835: 4825: 4815: 4793: 4783: 4773: 4751: 4741: 4731: 4705: 4695: 4685: 4663: 4653: 4643: 4621: 4611: 4601: 4577: 4567: 4557: 4547: 4541: 4501: 4500: 4481: 4480: 4461: 4460: 4441: 4440: 4420: 4419: 4409: 4393: 4386: 4380: 4379: 4366: 4359: 4353: 4352: 4347: 4340: 4321: 4309: 4308: 4288: 4283: 4276: 4264: 4263: 4243: 4238: 4231: 4221: 4212: 4211: 4193: 4188: 4181: 4169: 4168: 4154: 4007: 3998: 3994: 3932: 3931: 3874: 3873: 3813: 3812: 3755: 3754: 3722: 3721: 3687: 3669: 3668: 3639: 3638: 3604: 3586: 3585: 3559: 3558: 3556: 3457: 3253: 3247: 3241: 3237: 3217: 3208: 3203: 3166: 3160: 3155: 3144: 3133: 3129: 3125: 3121: 3112: 3106: 3100: 3096: 3088: 3080: 3065: 3059: 3035: 3034: 2987: 2975: 2973: 2969: 2965: 2961: 2949: 2947: 2943: 2939: 2935: 2925: 2921: 2914: 2903: 2899: 2895: 2891: 2887: 2879: 2870: 2852:negative number 2844: 2841: 2811: 2801: 2765: 2764: 2733: 2732: 2713: 2712: 2665: 2658: 2654: 2648: 2628: 2623: 2619: 2615: 2611: 2605: 2599: 2595: 2591: 2587: 2583: 2577: 2559: 2555: 2551: 2547: 2543: 2539: 2535: 2531: 2527: 2522: 2518: 2472:circuit diagram 2464: 2458: 2450:numeral systems 2440: 2426: 2422: 2418: 2411: 2407: 2393: 2376: 2359: 2342: 2325: 2308: 2301: 2297: 2283: 2266: 2262: 2258: 2251: 2247: 2233: 2216: 2199: 2192: 2188: 2174: 2170: 2166: 2159: 2155: 2141: 2124: 2120: 2116: 2109: 2105: 2091: 2087: 2083: 2076: 2072: 1997: 1989: 1985: 1975: 1966: 1889: 1887:Binary counting 1834: 1688: 1683: 1677: 1634: 1597:Arabic numerals 1407: 1356:complex numbers 1336:digital circuit 1315:Boolean algebra 1291: 1285: 1274: 1151: 1106: 1062: 1036:Similar to the 1026: 983: 961: 893: 838: 832: 812: 769:rational number 753:natural numbers 719: 683: 682: 605: 591:Proto-cuneiform 536: 535: 524: 523: 518: 517: 502: 486: 468: 455: 442: 429: 358: 357: 345: 344: 325: 285: 270: 261: 260: 251: 250: 232: 191: 190: 181: 180: 132: 74: 60: 59: 47: 46: 34:Numeral systems 17: 12: 11: 5: 7288: 7286: 7278: 7277: 7272: 7267: 7262: 7257: 7247: 7246: 7243: 7242: 7231: 7222: 7211: 7210:External links 7208: 7205: 7204: 7178: 7172:978-3834891914 7171: 7148: 7141: 7110: 7085: 7030: 7008: 6986: 6964: 6953:on 9 July 2010 6938: 6915: 6908: 6888: 6881: 6861: 6854: 6831: 6824: 6806: 6794: 6782: 6768: 6755: 6720: 6678: 6659:(8): 452–454. 6643: 6634:Bacon, Francis 6622: 6593: 6574:(2): 133–160. 6554: 6503: 6482: 6480:, p. 154. 6470: 6463: 6445: 6436: 6429: 6406: 6387: 6367: 6360: 6342: 6335: 6317: 6306:(3): 319–345. 6286: 6279: 6259: 6234: 6227: 6204: 6197: 6176: 6169: 6146: 6139: 6119: 6112: 6092: 6086: 6059: 6033: 6032: 6030: 6027: 6026: 6025: 6020: 6015: 6010: 6005: 6000: 5995: 5990: 5985: 5980: 5975: 5970: 5965: 5960: 5955: 5950: 5944: 5943: 5927: 5924: 5923: 5922: 5904: 5891: 5852: 5846: 5841: 5838: 5833: 5830: 5823: 5819: 5813: 5809: 5803: 5796: 5792: 5786: 5782: 5758: 5752: 5747: 5744: 5739: 5736: 5729: 5725: 5719: 5715: 5709: 5702: 5698: 5692: 5688: 5647: 5643: 5639: 5618: 5617: 5600: 5593: 5592: 5589: 5572: 5565: 5564: 5561: 5554: 5547: 5546: 5543: 5536: 5523: 5511: 5508: 5507: 5506: 5503: 5499: 5496: 5493: 5489: 5482: 5481: 5478: 5474: 5470: 5467: 5464: 5460: 5456: 5449: 5448: 5445: 5441: 5438: 5435: 5431: 5418: 5417: 5414: 5410: 5409: 5406: 5402: 5401: 5398: 5394: 5393: 5390: 5386: 5385: 5382: 5378: 5377: 5374: 5370: 5369: 5366: 5362: 5361: 5358: 5354: 5353: 5350: 5325:Main article: 5322: 5319: 5318: 5317: 5314: 5310: 5303: 5302: 5299: 5295: 5292: 5289: 5285: 5270: 5269: 5266: 5262: 5259: 5256: 5252: 5235: 5234: 5231: 5228: 5225: 5222: 5219: 5216: 5213: 5209: 5205: 5202: 5199: 5193: 5192: 5189: 5186: 5183: 5180: 5177: 5174: 5171: 5167: 5163: 5160: 5157: 5151: 5150: 5147: 5144: 5141: 5138: 5135: 5132: 5129: 5125: 5121: 5118: 5115: 5109: 5108: 5105: 5102: 5099: 5096: 5093: 5090: 5087: 5083: 5079: 5076: 5073: 5067: 5066: 5063: 5062: 5059: 5056: 5053: 5050: 5047: 5044: 5041: 5037: 5033: 5030: 5027: 5021: 5020: 5017: 5014: 5011: 5008: 5005: 5002: 4999: 4995: 4991: 4988: 4985: 4979: 4978: 4975: 4972: 4969: 4966: 4963: 4960: 4957: 4953: 4949: 4946: 4943: 4937: 4936: 4933: 4930: 4927: 4924: 4921: 4918: 4915: 4911: 4907: 4904: 4901: 4895: 4894: 4891: 4890: 4887: 4884: 4881: 4878: 4875: 4872: 4869: 4865: 4861: 4858: 4855: 4849: 4848: 4845: 4842: 4839: 4836: 4833: 4830: 4827: 4823: 4819: 4816: 4813: 4807: 4806: 4803: 4800: 4797: 4794: 4791: 4788: 4785: 4781: 4777: 4774: 4771: 4765: 4764: 4761: 4758: 4755: 4752: 4749: 4746: 4743: 4739: 4735: 4732: 4729: 4723: 4722: 4719: 4718: 4715: 4712: 4709: 4706: 4703: 4700: 4697: 4693: 4689: 4686: 4683: 4677: 4676: 4673: 4670: 4667: 4664: 4661: 4658: 4655: 4651: 4647: 4644: 4641: 4635: 4634: 4631: 4628: 4625: 4622: 4619: 4616: 4613: 4609: 4605: 4602: 4599: 4593: 4592: 4589: 4586: 4583: 4580: 4578: 4575: 4572: 4569: 4565: 4561: 4558: 4555: 4543:Main article: 4540: 4537: 4508: 4488: 4468: 4448: 4416: 4412: 4408: 4404: 4400: 4397: 4394: 4392: 4389: 4387: 4385: 4382: 4381: 4378: 4374: 4370: 4367: 4365: 4362: 4360: 4358: 4355: 4354: 4351: 4348: 4346: 4343: 4341: 4339: 4336: 4333: 4328: 4324: 4320: 4317: 4314: 4311: 4310: 4307: 4302: 4299: 4294: 4291: 4289: 4287: 4284: 4282: 4279: 4277: 4275: 4272: 4269: 4266: 4265: 4262: 4257: 4254: 4249: 4246: 4244: 4242: 4239: 4237: 4234: 4232: 4228: 4224: 4220: 4217: 4214: 4213: 4210: 4205: 4202: 4197: 4194: 4192: 4189: 4187: 4184: 4182: 4180: 4177: 4176: 4149: 4148: 4145: 4137: 4136: 4133: 4125: 4124: 4121: 4113: 4112: 4109: 4101: 4100: 4097: 4089: 4088: 4085: 4077: 4076: 4073: 4065: 4064: 4061: 4053: 4052: 4049: 4041: 4040: 4037: 4029: 4028: 4025: 4019: 4018: 4015: 4005: 3989: 3988: 3985: 3974: 3971: 3966: 3963: 3958: 3955: 3952: 3949: 3944: 3941: 3928: 3927: 3924: 3913: 3910: 3905: 3902: 3897: 3894: 3891: 3886: 3883: 3870: 3869: 3866: 3855: 3852: 3847: 3844: 3839: 3836: 3833: 3830: 3825: 3822: 3809: 3808: 3805: 3794: 3791: 3786: 3783: 3778: 3775: 3772: 3767: 3764: 3751: 3750: 3747: 3734: 3731: 3718: 3717: 3714: 3694: 3690: 3684: 3681: 3676: 3651: 3648: 3624: 3621: 3616: 3611: 3607: 3601: 3598: 3593: 3571: 3568: 3554: 3546: 3545: 3542: 3539: 3536: 3533: 3530: 3527: 3524: 3521: 3518: 3515: 3512: 3509: 3508:Decimal 3505: 3504: 3502: 3499: 3496: 3493: 3490: 3487: 3484: 3481: 3478: 3475: 3472: 3469: 3455: 3450: 3449: 3443: 3438: 3435: 3431: 3430: 3427: 3422: 3419: 3415: 3414: 3411: 3406: 3403: 3399: 3398: 3395: 3390: 3387: 3383: 3382: 3379: 3374: 3371: 3367: 3366: 3363: 3358: 3355: 3351: 3350: 3347: 3342: 3339: 3335: 3334: 3331: 3326: 3323: 3319: 3318: 3315: 3310: 3307: 3303: 3302: 3299: 3294: 3291: 3287: 3286: 3283: 3278: 3275: 3271: 3270: 3267: 3264: 3261: 3251: 3246: 3243: 3239: 3235: 3228:divided by two 3215: 3207: 3204: 3202: 3199: 3162:Main article: 3159: 3156: 3152: 3143: 3140: 3131: 3127: 3123: 3122:divided by 101 3119: 3110: 3104: 3098: 3094: 3086: 3078: 3058: 3055: 3042: 3021: 3020: 3017: 3014: 3010: 3009: 3006: 3003: 2999: 2998: 2995: 2992: 2986: 2983: 2957: 2931: 2927: 2926: 2916: 2882:Multiplication 2878: 2877:Multiplication 2875: 2874: 2873: 2856:absolute value 2842: 2839: 2831: 2830: 2827: 2824: 2821: 2800: 2797: 2784: 2781: 2778: 2775: 2772: 2752: 2749: 2746: 2743: 2740: 2720: 2699: 2698: 2695: 2692: 2688: 2687: 2684: 2681: 2677: 2676: 2673: 2670: 2664: 2663:Addition table 2661: 2656: 2652: 2626: 2621: 2617: 2613: 2609: 2602: 2576: 2573: 2557: 2553: 2549: 2545: 2541: 2537: 2533: 2529: 2525: 2517: 2509: 2508: 2505: 2497: 2496: 2493: 2490: 2487: 2460:Main article: 2457: 2454: 2439: 2436: 2433: 2432: 2429: 2415: 2404: 2400: 2399: 2396: 2390: 2387: 2383: 2382: 2379: 2373: 2370: 2366: 2365: 2362: 2356: 2353: 2349: 2348: 2345: 2339: 2336: 2332: 2331: 2328: 2322: 2319: 2315: 2314: 2311: 2305: 2294: 2290: 2289: 2286: 2280: 2277: 2273: 2272: 2269: 2255: 2244: 2240: 2239: 2236: 2230: 2227: 2223: 2222: 2219: 2213: 2210: 2206: 2205: 2202: 2196: 2185: 2181: 2180: 2177: 2163: 2152: 2148: 2147: 2144: 2138: 2135: 2131: 2130: 2127: 2113: 2102: 2098: 2097: 2094: 2080: 2069: 2065: 2064: 2061: 2058: 2053: 1996: 1993: 1992: 1991: 1987: 1983: 1978: 1977: 1973: 1969: 1968: 1964: 1956: 1955: 1949: 1942: 1935: 1928: 1888: 1885: 1884: 1883: 1877: 1874: 1871: 1864: 1833: 1830: 1819: 1818: 1815: 1811: 1810: 1807: 1803: 1802: 1799: 1795: 1794: 1791: 1787: 1786: 1783: 1779: 1778: 1775: 1771: 1770: 1767: 1763: 1762: 1759: 1755: 1754: 1751: 1747: 1746: 1743: 1739: 1738: 1735: 1731: 1730: 1727: 1723: 1722: 1719: 1715: 1714: 1711: 1707: 1706: 1703: 1699: 1698: 1695: 1691: 1690: 1685: 1676: 1673: 1659:, rather than 1653: 1652: 1649: 1646: 1643: 1636: 1632: 1629: 1626: 1623: 1616: 1544: 1543: 1540: 1537: 1534: 1531: 1528: 1525: 1522: 1519: 1516: 1512: 1511: 1508: 1505: 1502: 1499: 1496: 1493: 1490: 1487: 1484: 1480: 1479: 1476: 1473: 1470: 1467: 1464: 1461: 1458: 1455: 1452: 1448: 1447: 1444: 1441: 1438: 1435: 1432: 1429: 1426: 1423: 1420: 1406: 1405:Representation 1403: 1380:Norbert Wiener 1362:conference at 1343:George Stibitz 1322:Claude Shannon 1290: 1287: 1272: 1245:Joachim Bouvet 1203:Joachim Bouvet 1199: 1198: 1195: 1192: 1189: 1150: 1147: 1138:Thomas Harriot 1123:Bacon's cipher 1105: 1102: 1061: 1060:Other cultures 1058: 1025: 1022: 982: 979: 960: 957: 892: 889: 831: 828: 816:Thomas Harriot 811: 808: 740:numeral system 721: 720: 718: 717: 710: 703: 695: 692: 691: 685: 684: 681: 680: 675: 670: 665: 660: 655: 650: 645: 644: 643: 638: 633: 623: 618: 612: 611: 604: 603: 598: 593: 588: 583: 578: 573: 568: 563: 558: 553: 548: 542: 541: 540:Non-alphabetic 537: 531: 530: 529: 526: 525: 520: 519: 516: 515: 510: 497: 481: 476: 463: 450: 436: 435: 428: 427: 420: 415: 410: 405: 400: 395: 390: 385: 380: 375: 370: 364: 363: 359: 352: 351: 350: 347: 346: 343: 342: 336: 330: 329: 324: 323: 318: 313: 308: 303: 298: 292: 291: 289:Post-classical 284: 283: 277: 276: 269: 268: 262: 258: 257: 256: 253: 252: 249: 248: 243: 237: 236: 231: 230: 225: 220: 215: 210: 209: 208: 197: 196: 192: 188: 187: 186: 183: 182: 179: 178: 173: 168: 163: 158: 153: 148: 143: 138: 131: 130: 125: 120: 115: 110: 105: 100: 95: 90: 85: 80: 73: 72: 70:Eastern Arabic 67: 65:Western Arabic 61: 55: 54: 53: 48: 42: 41: 40: 37: 36: 30: 29: 15: 13: 10: 9: 6: 4: 3: 2: 7287: 7276: 7273: 7271: 7268: 7266: 7263: 7261: 7258: 7256: 7253: 7252: 7250: 7240: 7236: 7232: 7230: 7226: 7223: 7221: 7217: 7216:Binary System 7214: 7213: 7209: 7192: 7188: 7187:"Base System" 7182: 7179: 7174: 7168: 7164: 7163: 7155: 7153: 7149: 7144: 7138: 7134: 7130: 7126: 7125: 7117: 7115: 7111: 7099: 7095: 7089: 7086: 7069: 7065: 7061: 7057: 7053: 7052: 7044: 7040: 7034: 7031: 7018: 7012: 7009: 6996: 6990: 6987: 6974: 6968: 6965: 6952: 6948: 6942: 6939: 6934: 6930: 6926: 6919: 6916: 6911: 6905: 6901: 6900: 6892: 6889: 6884: 6878: 6874: 6873: 6865: 6862: 6857: 6851: 6847: 6846: 6838: 6836: 6832: 6827: 6825:0-85274-470-6 6821: 6817: 6810: 6807: 6803: 6798: 6795: 6791: 6786: 6783: 6780: 6775: 6773: 6769: 6758: 6752: 6748: 6744: 6740: 6736: 6729: 6727: 6725: 6721: 6716: 6712: 6708: 6704: 6700: 6697:(in German). 6696: 6689: 6682: 6679: 6674: 6670: 6666: 6662: 6658: 6654: 6647: 6644: 6639: 6635: 6629: 6627: 6623: 6619: 6615: 6612: 6609: 6605: 6602: 6597: 6594: 6589: 6585: 6581: 6577: 6573: 6569: 6565: 6558: 6555: 6550: 6546: 6541: 6536: 6531: 6526: 6522: 6518: 6514: 6507: 6504: 6492: 6486: 6483: 6479: 6474: 6471: 6466: 6460: 6456: 6449: 6446: 6440: 6437: 6432: 6426: 6422: 6421: 6416: 6410: 6407: 6403: 6399: 6396: 6391: 6388: 6385: 6384:0-387-94544-X 6381: 6377: 6371: 6368: 6363: 6357: 6353: 6346: 6343: 6338: 6332: 6328: 6321: 6318: 6313: 6309: 6305: 6301: 6297: 6290: 6287: 6282: 6276: 6272: 6271: 6263: 6260: 6256: 6245: 6238: 6235: 6230: 6224: 6220: 6219: 6211: 6209: 6205: 6200: 6194: 6190: 6183: 6181: 6177: 6172: 6166: 6162: 6161: 6153: 6151: 6147: 6142: 6140:9781615921768 6136: 6132: 6131: 6123: 6120: 6115: 6113:9780521878180 6109: 6105: 6104: 6096: 6093: 6089: 6087:9780199213122 6083: 6079: 6078: 6073: 6069: 6063: 6060: 6048: 6044: 6038: 6035: 6028: 6024: 6021: 6019: 6016: 6014: 6011: 6009: 6006: 6004: 6001: 5999: 5996: 5994: 5993:Offset binary 5991: 5989: 5986: 5984: 5981: 5979: 5976: 5974: 5973:Finger binary 5971: 5969: 5966: 5964: 5961: 5959: 5956: 5954: 5951: 5949: 5946: 5945: 5941: 5935: 5930: 5925: 5920: 5902: 5892: 5889: 5888: 5887: 5885: 5880: 5878: 5874: 5870: 5865: 5850: 5844: 5836: 5831: 5828: 5821: 5817: 5811: 5807: 5801: 5794: 5790: 5784: 5780: 5756: 5750: 5742: 5737: 5734: 5727: 5723: 5717: 5713: 5707: 5700: 5696: 5690: 5686: 5675: 5673: 5669: 5665: 5645: 5641: 5637: 5628: 5623: 5615: 5601: 5598: 5595: 5594: 5590: 5587: 5573: 5570: 5567: 5566: 5562: 5559: 5555: 5552: 5549: 5548: 5544: 5541: 5537: 5534: 5531: 5530: 5527: 5521: 5520:decimal point 5517: 5509: 5497: 5487: 5486: 5485: 5468: 5454: 5453: 5452: 5439: 5429: 5428: 5427: 5425: 5415: 5412: 5411: 5407: 5404: 5403: 5399: 5396: 5395: 5391: 5388: 5387: 5383: 5380: 5379: 5375: 5372: 5371: 5367: 5364: 5363: 5359: 5356: 5355: 5351: 5348: 5347: 5344: 5342: 5338: 5334: 5328: 5320: 5308: 5307: 5306: 5293: 5283: 5282: 5281: 5279: 5275: 5260: 5250: 5249: 5248: 5245: 5242: 5232: 5229: 5226: 5223: 5217: 5214: 5212: 5206: 5203: 5198: 5195: 5194: 5190: 5187: 5184: 5181: 5175: 5172: 5170: 5164: 5161: 5156: 5153: 5152: 5148: 5145: 5142: 5139: 5133: 5130: 5128: 5122: 5119: 5114: 5111: 5110: 5106: 5103: 5100: 5097: 5091: 5088: 5086: 5080: 5077: 5072: 5069: 5068: 5064: 5060: 5057: 5054: 5051: 5045: 5042: 5040: 5034: 5031: 5026: 5023: 5022: 5018: 5015: 5012: 5009: 5003: 5000: 4998: 4992: 4989: 4984: 4981: 4980: 4976: 4973: 4970: 4967: 4961: 4958: 4956: 4950: 4947: 4942: 4939: 4938: 4934: 4931: 4928: 4925: 4919: 4916: 4914: 4908: 4905: 4900: 4897: 4896: 4892: 4888: 4885: 4882: 4879: 4873: 4870: 4868: 4862: 4859: 4854: 4851: 4850: 4846: 4843: 4840: 4837: 4831: 4828: 4826: 4820: 4817: 4812: 4809: 4808: 4804: 4801: 4798: 4795: 4789: 4786: 4784: 4778: 4775: 4770: 4767: 4766: 4762: 4759: 4756: 4753: 4747: 4744: 4742: 4736: 4733: 4728: 4725: 4724: 4720: 4716: 4713: 4710: 4707: 4701: 4698: 4696: 4690: 4687: 4682: 4679: 4678: 4674: 4671: 4668: 4665: 4659: 4656: 4654: 4648: 4645: 4640: 4637: 4636: 4632: 4629: 4626: 4623: 4617: 4614: 4612: 4606: 4603: 4598: 4595: 4594: 4590: 4587: 4584: 4581: 4573: 4570: 4568: 4562: 4559: 4554: 4551: 4550: 4546: 4538: 4536: 4534: 4530: 4526: 4520: 4519:in decimal). 4506: 4486: 4466: 4446: 4438: 4433: 4414: 4406: 4402: 4398: 4390: 4388: 4383: 4376: 4372: 4368: 4363: 4361: 4356: 4349: 4344: 4342: 4334: 4331: 4326: 4322: 4315: 4312: 4305: 4297: 4292: 4290: 4285: 4280: 4278: 4273: 4270: 4267: 4260: 4252: 4247: 4245: 4240: 4235: 4233: 4226: 4222: 4218: 4215: 4208: 4200: 4195: 4190: 4185: 4183: 4178: 4166: 4162: 4160: 4147:0.0001100110 4146: 4143: 4139: 4138: 4134: 4131: 4127: 4126: 4122: 4119: 4115: 4114: 4110: 4107: 4103: 4102: 4098: 4095: 4091: 4090: 4086: 4083: 4079: 4078: 4074: 4071: 4067: 4066: 4062: 4059: 4055: 4054: 4050: 4047: 4043: 4042: 4038: 4035: 4031: 4030: 4026: 4024: 4021: 4020: 4016: 4013: 4012: 4009: 4002: 3986: 3972: 3969: 3964: 3961: 3956: 3953: 3950: 3947: 3942: 3939: 3930: 3929: 3925: 3911: 3908: 3903: 3900: 3895: 3892: 3889: 3884: 3881: 3872: 3871: 3867: 3853: 3850: 3845: 3842: 3837: 3834: 3831: 3828: 3823: 3820: 3811: 3810: 3806: 3792: 3789: 3784: 3781: 3776: 3773: 3770: 3765: 3762: 3753: 3752: 3748: 3732: 3729: 3720: 3719: 3715: 3712: 3711: 3708: 3692: 3682: 3679: 3667:For example, 3665: 3649: 3646: 3622: 3619: 3614: 3609: 3599: 3596: 3584:, the second 3569: 3566: 3551: 3543: 3540: 3537: 3534: 3531: 3528: 3525: 3522: 3519: 3516: 3513: 3510: 3507: 3506: 3503: 3500: 3497: 3494: 3491: 3488: 3485: 3482: 3479: 3476: 3473: 3470: 3468:Binary 3467: 3466: 3463: 3461: 3460:Horner scheme 3448: 3444: 3442: 3439: 3436: 3432: 3428: 3426: 3423: 3420: 3416: 3412: 3410: 3407: 3404: 3400: 3396: 3394: 3391: 3388: 3384: 3380: 3378: 3375: 3372: 3368: 3364: 3362: 3359: 3356: 3352: 3348: 3346: 3343: 3340: 3336: 3332: 3330: 3327: 3324: 3320: 3316: 3314: 3311: 3308: 3304: 3300: 3298: 3295: 3292: 3288: 3284: 3282: 3279: 3276: 3272: 3269:= Next value 3268: 3265: 3262: 3259: 3258: 3255: 3244: 3242: 3233: 3229: 3225: 3212: 3205: 3200: 3198: 3196: 3191: 3187: 3183: 3179: 3175: 3171: 3165: 3157: 3151: 3149: 3141: 3139: 3135: 3117: 3109: 3103: 3092: 3091:long division 3084: 3076: 3071: 3069: 3068:Long division 3064: 3056: 3054: 3040: 3032: 3028: 3018: 3015: 3012: 3011: 3007: 3004: 3001: 3000: 2996: 2993: 2991: 2990: 2984: 2982: 2980: 2956: 2954: 2930: 2917: 2910: 2909: 2908: 2905: 2883: 2876: 2869: 2868: 2867: 2865: 2861: 2857: 2853: 2849: 2838: 2836: 2828: 2825: 2822: 2819: 2818: 2817: 2815: 2810: 2806: 2798: 2796: 2782: 2779: 2776: 2773: 2770: 2750: 2747: 2744: 2741: 2738: 2718: 2710: 2706: 2696: 2693: 2690: 2689: 2685: 2682: 2679: 2678: 2674: 2671: 2669: 2668: 2662: 2660: 2646: 2642: 2638: 2634: 2625: 2601: 2574: 2572: 2570: 2566: 2561: 2516: 2514: 2506: 2503: 2502: 2501: 2494: 2491: 2488: 2485: 2484: 2483: 2477: 2474:for a binary 2473: 2468: 2463: 2455: 2453: 2451: 2448: 2444: 2437: 2430: 2416: 2414:0.0624999... 2405: 2402: 2401: 2397: 2391: 2388: 2385: 2384: 2380: 2374: 2371: 2368: 2367: 2363: 2357: 2354: 2351: 2350: 2346: 2340: 2337: 2334: 2333: 2329: 2323: 2320: 2317: 2316: 2312: 2306: 2295: 2292: 2291: 2287: 2281: 2278: 2275: 2274: 2270: 2256: 2245: 2242: 2241: 2237: 2231: 2228: 2225: 2224: 2220: 2214: 2211: 2208: 2207: 2203: 2197: 2186: 2183: 2182: 2178: 2164: 2153: 2150: 2149: 2145: 2139: 2136: 2133: 2132: 2128: 2114: 2103: 2100: 2099: 2095: 2081: 2070: 2067: 2066: 2062: 2059: 2057: 2054: 2051: 2050: 2047: 2045: 2041: 2037: 2033: 2029: 2025: 2021: 2017: 2013: 2009: 2005: 2001: 1994: 1990: 1980: 1979: 1971: 1970: 1962: 1961: 1960: 1953: 1950: 1947: 1943: 1940: 1936: 1933: 1929: 1926: 1925: 1924: 1922: 1918: 1914: 1910: 1901: 1893: 1886: 1881: 1878: 1875: 1872: 1869: 1865: 1862: 1861: 1860: 1858: 1854: 1850: 1846: 1842: 1838: 1831: 1829: 1827: 1816: 1813: 1812: 1808: 1805: 1804: 1800: 1797: 1796: 1792: 1789: 1788: 1784: 1781: 1780: 1776: 1773: 1772: 1768: 1765: 1764: 1760: 1757: 1756: 1752: 1749: 1748: 1744: 1741: 1740: 1736: 1733: 1732: 1728: 1725: 1724: 1720: 1717: 1716: 1712: 1709: 1708: 1704: 1701: 1700: 1696: 1693: 1692: 1680: 1674: 1672: 1670: 1666: 1662: 1658: 1657:one zero zero 1650: 1647: 1644: 1641: 1637: 1630: 1627: 1624: 1621: 1617: 1614: 1613: 1612: 1610: 1606: 1602: 1598: 1593: 1591: 1587: 1583: 1580: 1576: 1567: 1563: 1559: 1555: 1550: 1541: 1538: 1535: 1532: 1529: 1526: 1523: 1520: 1517: 1514: 1513: 1509: 1506: 1503: 1500: 1497: 1494: 1491: 1488: 1485: 1482: 1481: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1450: 1449: 1445: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1417: 1414: 1412: 1404: 1402: 1400: 1396: 1395:Boolean logic 1392: 1388: 1383: 1381: 1377: 1373: 1369: 1365: 1361: 1357: 1352: 1348: 1344: 1339: 1337: 1333: 1332: 1327: 1323: 1318: 1316: 1312: 1308: 1304: 1295: 1288: 1282: 1278: 1271: 1268: 1262: 1260: 1259: 1254: 1250: 1246: 1242: 1241: 1236: 1232: 1228: 1227:Gottlob Frege 1224: 1219: 1216: 1212: 1208: 1204: 1196: 1193: 1190: 1187: 1186: 1185: 1183: 1181: 1175: 1170: 1168: 1163: 1155: 1148: 1146: 1144: 1139: 1135: 1131: 1126: 1124: 1119: 1118:Francis Bacon 1114: 1111: 1103: 1101: 1099: 1095: 1091: 1087: 1083: 1079: 1075: 1071: 1067: 1059: 1057: 1055: 1051: 1047: 1046:Yoruba people 1043: 1039: 1035: 1031: 1023: 1021: 1019: 1015: 1011: 1007: 1006:Chandaḥśāstra 1002: 1000: 996: 992: 988: 980: 978: 976: 971: 969: 965: 958: 956: 953: 949: 945: 940: 938: 934: 931:and a set of 930: 926: 921: 919: 916: 912: 908: 907: 897: 890: 888: 886: 881: 876: 874: 870: 866: 862: 858: 854: 847: 842: 837: 829: 827: 825: 821: 817: 809: 807: 805: 801: 797: 793: 789: 785: 781: 776: 774: 770: 766: 765:binary number 762: 758: 754: 750: 746: 742: 741: 737: 732: 728: 727:binary number 716: 711: 709: 704: 702: 697: 696: 694: 693: 690: 687: 686: 679: 676: 674: 671: 669: 666: 664: 661: 659: 656: 654: 651: 649: 646: 642: 639: 637: 634: 632: 629: 628: 627: 626:Alphasyllabic 624: 622: 619: 617: 614: 613: 610: 607: 606: 602: 599: 597: 594: 592: 589: 587: 584: 582: 579: 577: 574: 572: 569: 567: 564: 562: 559: 557: 554: 552: 549: 547: 544: 543: 539: 538: 534: 528: 527: 514: 511: 508: 501: 498: 495: 494: 485: 482: 480: 477: 474: 467: 464: 461: 454: 451: 448: 441: 438: 437: 434: 431: 430: 425: 421: 419: 416: 414: 411: 409: 406: 404: 401: 399: 396: 394: 391: 389: 386: 384: 381: 379: 376: 374: 371: 369: 366: 365: 361: 360: 356: 349: 348: 340: 337: 335: 332: 331: 327: 326: 322: 319: 317: 314: 312: 309: 307: 304: 302: 299: 297: 294: 293: 290: 287: 286: 282: 279: 278: 275: 272: 271: 267: 264: 263: 259:Other systems 255: 254: 247: 244: 242: 241:Counting rods 239: 238: 234: 233: 229: 226: 224: 221: 219: 216: 214: 211: 207: 204: 203: 202: 199: 198: 194: 193: 185: 184: 177: 174: 172: 169: 167: 164: 162: 159: 157: 154: 152: 149: 147: 144: 142: 139: 137: 134: 133: 129: 126: 124: 121: 119: 116: 114: 111: 109: 106: 104: 101: 99: 96: 94: 91: 89: 86: 84: 81: 79: 76: 75: 71: 68: 66: 63: 62: 58: 52: 51: 45: 39: 38: 35: 31: 27: 23: 22: 19: 7229:cut-the-knot 7220:cut-the-knot 7195:. Retrieved 7191:the original 7181: 7161: 7123: 7101:. Retrieved 7097: 7088: 7075:. Retrieved 7055: 7049: 7033: 7021:. Retrieved 7011: 6999:. Retrieved 6989: 6977:. Retrieved 6967: 6955:. Retrieved 6951:the original 6941: 6933:1721.1/11173 6924: 6918: 6898: 6891: 6871: 6864: 6844: 6815: 6809: 6797: 6785: 6760:, retrieved 6738: 6701:(1): 12–15. 6698: 6694: 6681: 6656: 6652: 6646: 6596: 6571: 6567: 6557: 6520: 6516: 6506: 6494:. Retrieved 6485: 6473: 6454: 6448: 6439: 6419: 6409: 6404:(pdf, 145KB) 6390: 6375: 6370: 6351: 6345: 6326: 6320: 6303: 6299: 6289: 6269: 6262: 6254: 6249:15 September 6247:. Retrieved 6237: 6217: 6188: 6159: 6129: 6122: 6102: 6095: 6076: 6062: 6050:. Retrieved 6046: 6037: 5881: 5866: 5676: 5671: 5663: 5624: 5621: 5613: 5596: 5585: 5568: 5557: 5550: 5539: 5532: 5513: 5483: 5463:grouped = 54 5450: 5421: 5337:power of two 5330: 5304: 5273: 5271: 5246: 5238: 5196: 5154: 5112: 5070: 5024: 4982: 4940: 4898: 4852: 4810: 4768: 4726: 4680: 4638: 4596: 4552: 4532: 4529:concatenated 4524: 4521: 4434: 4167: 4163: 4152: 4141: 4135:0.000110011 4129: 4117: 4105: 4093: 4081: 4069: 4057: 4045: 4033: 4022: 4003: 3992: 3666: 3552: 3549: 3453: 3446: 3440: 3424: 3408: 3392: 3376: 3360: 3344: 3328: 3312: 3296: 3280: 3260:Prior value 3254:to decimal: 3248: 3221: 3167: 3145: 3136: 3113: 3107: 3072: 3066: 3024: 2976: 2953:binary point 2950: 2928: 2906: 2880: 2847: 2845: 2834: 2832: 2812: 2702: 2649: 2644: 2640: 2636: 2632: 2606: 2584:ones (where 2578: 2567:y = (x + y) 2562: 2523: 2512: 2510: 2498: 2481: 2441: 2360:000100111011 2321:0.090909... 2254:0.124999... 2043: 2039: 2035: 2031: 2023: 2019: 2015: 2006:only if the 1998: 1981: 1957: 1951: 1945: 1938: 1931: 1920: 1916: 1912: 1908: 1906: 1879: 1867: 1856: 1852: 1848: 1844: 1840: 1835: 1822: 1668: 1664: 1660: 1656: 1654: 1604: 1600: 1594: 1571: 1554:binary clock 1408: 1384: 1376:John Mauchly 1350: 1340: 1329: 1319: 1303:George Boole 1300: 1297:George Boole 1280: 1266: 1264: 1256: 1253:universality 1248: 1238: 1231:George Boole 1220: 1210: 1206: 1200: 1177: 1173: 1171: 1166: 1164: 1160: 1127: 1115: 1107: 1081: 1063: 1041: 1037: 1033: 1027: 1018:place values 1009: 1003: 998: 997:(light) and 994: 984: 972: 962: 944:Song dynasty 941: 937:Zhou dynasty 925:yin and yang 922: 910: 904: 902: 899:Daoist Bagua 877: 850: 846:Eye of Horus 813: 777: 764: 744: 734: 726: 724: 492: 453:Signed-digit 367: 328:Contemporary 195:Contemporary 18: 7058:(2): 5–16. 7039:Rojas, Raúl 6478:Landry 2019 5963:Binary code 5873:0.111111... 5664:terminating 5516:radix point 5424:hexadecimal 5341:hexadecimal 5265:= 1110 0111 5255:= 0011 1010 4545:Hexadecimal 4539:Hexadecimal 4437:hexadecimal 4123:0.00011001 3142:Square root 3027:truth table 2932:1 0 1 1 ( 2814:Subtraction 2799:Subtraction 2705:truth table 2532:) and 10111 2338:0.08333... 2162:0.24999... 2008:denominator 1849:first digit 1665:one hundred 1661:one hundred 1566:sexagesimal 1397:and binary 1391:Konrad Zuse 1387:Z1 computer 1233:in forming 1215:mathematics 1145:, in 1700. 1130:John Napier 1110:Ramon Llull 1048:. In 2008, 920:technique. 800:logic gates 759:) and "1" ( 631:Akṣarapallī 601:Tally marks 500:Non-integer 7249:Categories 7083:(12 pages) 6280:3515074481 6029:References 5738:0.01010101 5518:(called a 4369:1100010101 4350:1100010101 4241:1100101110 4140:0.2 × 2 = 4128:0.6 × 2 = 4116:0.8 × 2 = 4111:0.0001100 4104:0.4 × 2 = 4092:0.2 × 2 = 4080:0.6 × 2 = 4068:0.8 × 2 = 4056:0.4 × 2 = 4044:0.2 × 2 = 4032:0.1 × 2 = 4014:Converting 3713:Converting 3114:Thus, the 3061:See also: 3033:operation 2711:operation 2643:0 1 1 0 0 2476:half adder 2443:Arithmetic 2389:0.0666... 2326:0001011101 2304:0.0999... 2212:0.1666... 2195:0.1999... 2112:0.4999... 2012:power of 2 1556:might use 1309:system of 1078:Slit drums 918:divination 915:quaternary 834:See also: 668:Glagolitic 641:Kaṭapayādi 609:Alphabetic 513:Asymmetric 355:radix/base 296:Cistercian 281:Babylonian 228:Vietnamese 83:Devanagari 7197:31 August 6762:20 August 6715:179000299 6580:0003-5483 6312:0002-9475 5998:Quibinary 5978:Gray code 5845:… 5840:¯ 5751:… 5746:¯ 5556:(1 × 1 = 5538:(1 × 2 = 5473:= 010 011 5459:= 101 100 5444:= 001 111 5434:= 110 101 4332:− 4316:× 4306:… 4301:¯ 4271:× 4261:… 4256:¯ 4219:× 4209:… 4204:¯ 4099:0.000110 3970:≥ 3948:× 3890:× 3851:≥ 3829:× 3771:× 3266:Next bit 3041:∧ 2977:See also 2854:of equal 2835:borrowing 2829:1 − 1 → 0 2826:1 − 0 → 1 2820:0 − 0 → 0 2742:∨ 2719:∨ 2637:1 1 1 1 1 2492:1 + 0 → 1 2489:0 + 1 → 1 2486:0 + 0 → 0 2279:0.111... 2137:0.333... 2079:0.999... 2052:Fraction 2004:terminate 2000:Fractions 1995:Fractions 1913:first bit 1347:Bell Labs 1320:In 1937, 1307:algebraic 1283:hexagrams 1267:ex nihilo 1128:In 1617, 1116:In 1605, 1066:Mangareva 964:Etruscans 948:Shao Yong 857:Horus-Eye 636:Āryabhaṭa 581:Kharosthi 473:factorial 440:Bijective 341:(Iñupiaq) 171:Sundanese 166:Mongolian 113:Malayalam 7235:Archived 7068:Archived 6636:(1605). 6614:Archived 6604:Archived 6588:23621076 6549:24344278 6398:Archived 5983:IEEE 754 5926:See also 5837:10110100 5294:11011101 4087:0.00011 3118:of 11011 3116:quotient 3085:is 11011 3083:dividend 3057:Division 2763:, while 2513:carrying 2456:Addition 1857:overflow 1843:through 1579:magnetic 1575:voltages 1368:teletype 1338:design. 1273:— 1270:Nothing. 1094:geomancy 1092:Western 1090:medieval 946:scholar 663:Georgian 653:Cyrillic 621:Armenian 576:Etruscan 571:Egyptian 479:Negative 339:Kaktovik 334:Cherokee 311:Pentadic 235:Historic 218:Japanese 151:Javanese 141:Balinese 128:Dzongkha 93:Gurmukhi 88:Gujarati 26:a series 24:Part of 7103:26 June 6802:Leibniz 6661:Bibcode 6540:3910603 6023:Unicode 5869:decimal 5662:have a 5607:⁄ 5579:⁄ 5526:means: 5352:Binary 5284:1010010 5278:padding 4075:0.0001 4017:Result 3987:0.0101 3716:Result 3224:integer 3075:divisor 3029:of the 2707:of the 2060:Binary 2056:Decimal 1837:Decimal 1826:decimal 1682:Decimal 1577:; on a 1281:I Ching 1249:I Ching 1240:I Ching 1211:I Ching 1207:I Ching 1149:Leibniz 1082:I Ching 1074:decimal 1042:I Ching 1038:I Ching 991:prosody 987:Pingala 911:I Ching 906:I Ching 810:History 782:with a 773:integer 749:numbers 566:Chuvash 484:Complex 274:Ancient 266:History 213:Hokkien 201:Chinese 146:Burmese 136:Tibetan 123:Kannada 103:Sinhala 78:Bengali 7169: 7139: 7077:3 July 7023:5 July 7001:5 July 6979:5 July 6957:5 July 6906: 6879: 6852: 6822: 6753: 6713: 6586: 6578: 6547: 6537: 6496:5 July 6461: 6427: 6382: 6358: 6333: 6310: 6277: 6225: 6195: 6167: 6137: 6110: 6084: 6052:22 May 5917:, the 5455:101100 4377:111110 4144:< 1 4108:< 1 4096:< 1 4063:0.000 4060:< 1 4048:< 1 4036:< 1 4001:... . 3926:0.010 3429:= 598 3413:= 299 3397:= 149 3263:× 2 + 3184:, and 3126:is 101 3077:is 101 2848:adding 2425:0.0000 2423: 2419: 2417:0.0001 2412: 2408: 2406:0.0625 2341:0.0001 2302: 2298: 2284:000111 2263: 2259: 2252: 2248: 2193: 2189: 2171: 2167: 2160: 2156: 2121: 2117: 2110: 2106: 2088: 2084: 2077: 2073: 2042:+ 1 × 2038:+ 0 × 2034:+ 1 × 1982:100101 1972:100101 1963:100101 1689:number 1687:Binary 1684:number 1631:100101 1050:UNESCO 1024:Africa 975:Dodona 822:, and 798:using 731:number 678:Hebrew 648:Coptic 561:Brahmi 546:Aegean 503: 487: 469: 456: 443: 306:Muisca 246:Tangut 223:Korean 206:Suzhou 118:Telugu 7071:(PDF) 7046:(PDF) 6711:S2CID 6691:(PDF) 6584:JSTOR 5948:ASCII 5818:10001 5672:recur 5602:(1 × 5591:plus 5574:(0 × 5563:plus 5545:plus 5469:10011 5349:Octal 5333:octal 5327:Octal 5321:Octal 5241:radix 4298:01110 4286:11001 4253:01110 4201:01110 4051:0.00 3868:0.01 3544:1197 3541:1×2 = 3538:0×2 + 3535:1×2 + 3532:1×2 + 3529:0×2 + 3526:1×2 + 3523:0×2 + 3520:1×2 + 3517:0×2 + 3514:0×2 + 3511:1×2 + 3437:× 2 + 3421:× 2 + 3405:× 2 + 3389:× 2 + 3381:= 74 3373:× 2 + 3365:= 37 3357:× 2 + 3349:= 18 3341:× 2 + 3325:× 2 + 3309:× 2 + 3293:× 2 + 3277:× 2 + 2655:(1649 2633:1 1 1 2403:1/16 2386:1/15 2369:1/14 2352:1/13 2335:1/12 2318:1/11 2293:1/10 2265:0.000 2257:0.001 2246:0.125 2215:0.001 2010:is a 1927:0000, 1669:value 1609:radix 1568:time. 1311:logic 1180:Fu Xi 995:laghu 981:India 891:China 865:hekat 861:Horus 830:Egypt 784:radix 763:). A 729:is a 673:Greek 658:Geʽez 616:Abjad 596:Roman 556:Aztec 551:Attic 466:Mixed 424:table 316:Quipu 301:Mayan 156:Khmer 108:Tamil 7199:2016 7167:ISBN 7137:ISBN 7105:2019 7079:2022 7025:2010 7003:2010 6981:2010 6959:2010 6904:ISBN 6877:ISBN 6850:ISBN 6820:ISBN 6764:2024 6751:ISBN 6576:ISSN 6545:PMID 6498:2017 6459:ISBN 6425:ISBN 6380:ISBN 6356:ISBN 6331:ISBN 6308:ISSN 6275:ISBN 6251:2022 6223:ISBN 6193:ISBN 6165:ISBN 6135:ISBN 6108:ISBN 6082:ISBN 6054:2022 5808:1100 5625:All 5614:0.25 5416:111 5408:110 5400:101 5392:100 5384:011 5376:010 5368:001 5360:000 5309:C0E7 4191:1100 4155:0011 4039:0.0 3909:< 3807:0.0 3790:< 3447:1197 3333:= 9 3317:= 4 3301:= 2 3285:= 1 3148:here 2888:and 2807:and 2620:(691 2612:(958 2600:0s: 2470:The 2394:0001 2309:0011 2276:1/9 2243:1/8 2226:1/7 2209:1/6 2200:0011 2184:1/5 2173:0.00 2165:0.01 2154:0.25 2151:1/4 2134:1/3 2101:1/2 2068:1/1 2022:and 1986:= 37 1919:and 1817:1111 1809:1110 1801:1101 1793:1100 1785:1011 1777:1010 1769:1001 1761:1000 1603:and 1582:disk 1558:LEDs 1411:bits 1385:The 1378:and 1229:and 1028:The 999:guru 942:The 903:The 757:zero 736:base 321:Rumi 176:Thai 98:Odia 7227:at 7218:at 7129:doi 7098:BBC 7060:doi 6929:hdl 6743:doi 6703:doi 6669:doi 6535:PMC 6525:doi 6521:111 6304:129 5599:× 2 5571:× 2 5553:× 2 5535:× 2 5498:127 5220:oct 5210:dec 5200:hex 5178:oct 5168:dec 5158:hex 5136:oct 5126:dec 5116:hex 5094:oct 5084:dec 5074:hex 5048:oct 5038:dec 5028:hex 5006:oct 4996:dec 4986:hex 4964:oct 4954:dec 4944:hex 4922:oct 4912:dec 4902:hex 4876:oct 4866:dec 4856:hex 4834:oct 4824:dec 4814:hex 4792:oct 4782:dec 4772:hex 4750:oct 4740:dec 4730:hex 4704:oct 4694:dec 4684:hex 4662:oct 4652:dec 4642:hex 4620:oct 4610:dec 4600:hex 4576:oct 4566:dec 4556:hex 4399:789 4142:0.4 4132:≥ 1 4130:1.2 4120:≥ 1 4118:1.6 4106:0.8 4094:0.4 4084:≥ 1 4082:1.2 4072:≥ 1 4070:1.6 4058:0.8 4046:0.4 4034:0.2 4027:0. 4023:0.1 3749:0. 3434:598 3418:299 3402:149 3190:NOT 3186:XOR 3178:AND 2697:10 2569:mod 2565:xor 2560:). 2556:(36 2536:(23 2528:(13 2377:001 2375:0.0 2307:0.0 2296:0.1 2234:001 2187:0.2 2123:0.0 2115:0.1 2104:0.5 1930:000 1909:bit 1753:111 1745:110 1737:101 1729:100 1590:yes 1326:MIT 1125:.) 1086:Ifá 1068:in 1056:". 1030:Ifá 792:bit 786:of 761:one 743:or 738:-2 353:By 161:Lao 7251:: 7151:^ 7135:. 7113:^ 7096:. 7066:. 7056:19 7054:. 7048:. 6834:^ 6771:^ 6749:, 6737:, 6723:^ 6709:. 6699:16 6693:. 6667:. 6657:19 6655:. 6625:^ 6582:. 6572:54 6570:. 6566:. 6543:. 6533:. 6519:. 6515:. 6302:. 6298:. 6253:. 6207:^ 6179:^ 6149:^ 6070:; 6045:. 5795:10 5791:17 5785:10 5781:12 5743:01 5724:11 5701:10 5691:10 5616:) 5612:= 5584:= 5504:10 5494:10 5488:65 5440:17 5430:65 5426:: 5315:10 5311:16 5300:16 5290:16 5263:16 5261:E7 5253:16 5251:3A 5233:1 5218:17 5208:15 5191:0 5176:16 5166:14 5149:1 5134:15 5124:13 5107:0 5092:14 5082:12 5061:1 5046:13 5036:11 5019:0 5004:12 4994:10 4977:1 4962:11 4935:0 4920:10 4889:1 4847:0 4805:1 4763:0 4717:1 4675:0 4633:1 4591:0 4415:10 4407:62 4196:.1 4006:10 3999:01 3693:10 3462:. 3456:10 3445:= 3386:74 3370:37 3354:18 3240:2. 3236:10 3216:10 3182:OR 3180:, 3053:. 3019:1 3016:0 3013:1 3008:0 3005:0 3002:0 2997:1 2994:0 2981:. 2955:: 2850:a 2795:. 2783:10 2694:1 2691:1 2686:1 2683:0 2680:0 2675:1 2672:0 2657:10 2635:0 2622:10 2614:10 2558:10 2538:10 2530:10 2421:or 2410:or 2392:0. 2358:0. 2343:01 2324:0. 2300:or 2282:0. 2261:or 2250:or 2232:0. 2217:01 2198:0. 2191:or 2169:or 2158:or 2142:01 2140:0. 2119:or 2108:or 2090:0. 2086:or 2075:or 2016:10 1988:10 1937:00 1814:15 1806:14 1798:13 1790:12 1782:11 1774:10 1721:11 1713:10 1584:, 1552:A 1542:y 1510:☒ 1478:| 1446:1 1401:. 1374:, 1020:. 927:. 818:, 725:A 418:60 413:20 408:16 403:12 398:10 28:on 7201:. 7131:: 7107:. 7081:. 7062:: 7027:. 7005:. 6983:. 6961:. 6935:. 6931:: 6912:. 6885:. 6858:. 6828:. 6745:: 6717:. 6705:: 6675:. 6671:: 6663:: 6620:) 6590:. 6551:. 6527:: 6500:. 6467:. 6433:. 6364:. 6339:. 6314:. 6283:. 6231:. 6201:. 6173:. 6144:. 6117:. 6056:. 5903:2 5851:2 5829:= 5822:2 5812:2 5802:= 5757:2 5735:= 5728:2 5718:2 5714:1 5708:= 5697:3 5687:1 5646:a 5642:2 5638:p 5609:4 5605:1 5597:1 5588:) 5586:0 5581:2 5577:1 5569:0 5560:) 5558:1 5551:1 5542:) 5540:2 5533:1 5524:2 5500:8 5490:8 5479:8 5475:2 5471:2 5465:8 5461:2 5457:2 5446:2 5442:8 5436:2 5432:8 5413:7 5405:6 5397:5 5389:4 5381:3 5373:2 5365:1 5357:0 5296:2 5286:2 5274:0 5267:2 5257:2 5230:1 5227:1 5224:1 5215:= 5204:= 5197:F 5188:1 5185:1 5182:1 5173:= 5162:= 5155:E 5146:0 5143:1 5140:1 5131:= 5120:= 5113:D 5104:0 5101:1 5098:1 5089:= 5078:= 5071:C 5058:1 5055:0 5052:1 5043:= 5032:= 5025:B 5016:1 5013:0 5010:1 5001:= 4990:= 4983:A 4974:0 4971:0 4968:1 4959:= 4952:9 4948:= 4941:9 4932:0 4929:0 4926:1 4917:= 4910:8 4906:= 4899:8 4886:1 4883:1 4880:0 4874:7 4871:= 4864:7 4860:= 4853:7 4844:1 4841:1 4838:0 4832:6 4829:= 4822:6 4818:= 4811:6 4802:0 4799:1 4796:0 4790:5 4787:= 4780:5 4776:= 4769:5 4760:0 4757:1 4754:0 4748:4 4745:= 4738:4 4734:= 4727:4 4714:1 4711:0 4708:0 4702:3 4699:= 4692:3 4688:= 4681:3 4672:1 4669:0 4666:0 4660:2 4657:= 4650:2 4646:= 4639:2 4630:0 4627:0 4624:0 4618:1 4615:= 4608:1 4604:= 4597:1 4588:0 4585:0 4582:0 4574:0 4571:= 4564:0 4560:= 4553:0 4533:k 4525:k 4507:x 4487:x 4467:x 4447:x 4411:) 4403:/ 4396:( 4391:= 4384:x 4373:/ 4364:= 4357:x 4345:= 4338:) 4335:2 4327:6 4323:2 4319:( 4313:x 4293:. 4281:= 4274:2 4268:x 4248:. 4236:= 4227:6 4223:2 4216:x 4186:= 4179:x 3995:3 3973:1 3965:3 3962:1 3957:1 3954:= 3951:2 3943:3 3940:2 3912:1 3904:3 3901:2 3896:= 3893:2 3885:3 3882:1 3854:1 3846:3 3843:1 3838:1 3835:= 3832:2 3824:3 3821:2 3793:1 3785:3 3782:2 3777:= 3774:2 3766:3 3763:1 3733:3 3730:1 3689:) 3683:3 3680:1 3675:( 3650:2 3647:1 3623:4 3620:1 3615:= 3610:2 3606:) 3600:2 3597:1 3592:( 3570:2 3567:1 3555:2 3501:1 3498:0 3495:1 3492:1 3489:0 3486:1 3483:0 3480:1 3477:0 3474:0 3471:1 3441:1 3425:0 3409:1 3393:1 3377:0 3361:1 3345:0 3338:9 3329:1 3322:4 3313:0 3306:2 3297:0 3290:1 3281:1 3274:0 3252:2 3132:2 3128:2 3124:2 3120:2 3099:2 3095:2 3087:2 3079:2 2972:B 2968:B 2964:B 2960:A 2946:B 2942:B 2938:B 2934:A 2924:A 2920:B 2913:B 2902:B 2898:A 2894:B 2890:B 2886:A 2780:= 2777:1 2774:+ 2771:1 2751:1 2748:= 2745:1 2739:1 2653:2 2645:1 2641:1 2618:2 2610:2 2608:0 2598:n 2594:n 2590:n 2586:n 2582:n 2554:2 2550:2 2546:2 2542:2 2534:2 2526:2 2427:1 2267:1 2175:1 2125:1 2092:1 2082:1 2071:1 2044:2 2040:2 2036:2 2032:2 2024:5 2020:2 1984:2 1974:2 1965:2 1952:1 1946:1 1944:0 1939:1 1932:1 1921:1 1917:0 1880:1 1868:1 1866:0 1853:0 1845:9 1841:0 1766:9 1758:8 1750:7 1742:6 1734:5 1726:4 1718:3 1710:2 1705:1 1702:1 1697:0 1694:0 1642:) 1633:2 1622:) 1605:1 1601:0 1539:y 1536:n 1533:y 1530:y 1527:n 1524:n 1521:y 1518:n 1515:y 1507:☒ 1504:☐ 1501:☒ 1498:☒ 1495:☐ 1492:☐ 1489:☒ 1486:☐ 1483:☒ 1475:| 1472:― 1469:| 1466:| 1463:― 1460:― 1457:| 1454:― 1451:| 1443:1 1440:0 1437:1 1434:1 1431:0 1428:0 1425:1 1422:0 1419:1 1351:K 1182:" 1034:. 788:2 714:e 707:t 700:v 509:) 507:φ 505:( 496:) 493:i 491:2 489:( 475:) 471:( 462:) 458:( 449:) 447:1 445:( 426:) 422:( 393:8 388:6 383:5 378:4 373:3 368:2

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