Binary number - Knowledge (XXG)

6815:: "The Chinese lost the meaning of the Cova or Lineations of Fuxi, perhaps more than a thousand years ago, and they have written commentaries on the subject in which they have sought I know not what far out meanings, so that their true explanation now has to come from Europeans. Here is how: It was scarcely more than two years ago that I sent to Reverend Father Bouvet, the celebrated French Jesuit who lives in Peking, my method of counting by 0 and 1, and nothing more was required to make him recognize that this was the key to the figures of Fuxi. Writing to me on 14 November 1701, he sent me this philosophical prince's grand figure, which goes up to 64, and leaves no further room to doubt the truth of our interpretation, such that it can be said that this Father has deciphered the enigma of Fuxi, with the help of what I had communicated to him. And as these figures are perhaps the most ancient monument of science which exists in the world, this restitution of their meaning, after such a great interval of time, will seem all the more curious." 1165: 1911: 5945: 4442: 852: 907: 4181: 1305: 1560: 2478: 2526:. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary: 1180:, in 1679, Leibniz introduced conversion between decimal and binary, along with algorithms for performing basic arithmetic operations such as addition, subtraction, multiplication, and division using binary numbers. He also developed a form of binary algebra to calculate the square of a six-digit number and to extract square roots.. 3245:. The quotient is again divided by two; its remainder becomes the next least significant bit. This process repeats until a quotient of one is reached. The sequence of remainders (including the final quotient of one) forms the binary value, as each remainder must be either zero or one when dividing by two. For example, (357) 4437:{\displaystyle {\begin{aligned}x&=&1100&.1{\overline {01110}}\ldots \\x\times 2^{6}&=&1100101110&.{\overline {01110}}\ldots \\x\times 2&=&11001&.{\overline {01110}}\ldots \\x\times (2^{6}-2)&=&1100010101\\x&=&1100010101/111110\\x&=&(789/62)_{10}\end{aligned}}} 3260:

Conversion from base-2 to base-10 simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be

2661:

The top row shows the carry bits used. Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series. The "used" numbers must be crossed off, since

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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

1969:

In the binary system, each bit represents an increasing power of 2, with the rightmost bit representing 2, the next representing 2, then 2, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" bit. For example, the binary number 100101 is converted to decimal

893:

is also closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it; the order in which these steps are to be performed

1123:

had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or "Ars generalis" based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing

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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

4175:

The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base. For example:

2848:. The principle is the same as for carrying. When the result of a subtraction is less than 0, the least possible value of a digit, the procedure is to "borrow" the deficit divided by the radix (that is, 10/10) from the left, subtracting it from the next positional value. 5874: 2618:

Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations. In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1

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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

4168:... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 1/10 + ... + 1/10 (addition of 10 numbers) differs from 1 in binary 1364:

itchen", where he had assembled it), which calculated using binary addition. Bell Labs authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer, completed 8 January 1940, was able to calculate

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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

4542:. Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10 and added to the second converted piece, where 1027:. In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of 2614:

Binary Decimal 1 1 1 1 1 likewise 9 9 9 9 9 + 1 + 1 ——————————— ——————————— 1 0 0 0 0 0 1 0 0 0 0 0

5254:

of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 2, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.

5784: 1424:(binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. Any of the following rows of symbols can be interpreted as the binary numeric value of 667: 1228:

he admired. Of this parallel invention, Liebniz wrote in his "Explanation Of Binary Arithmetic" that "this restitution of their meaning, after such a great interval of time, will seem all the more curious."

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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

2670:). In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort. 3675:, and vice versa. Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part. 3994: 3875: 3933: 3814: 1064: 4186: 3646: 5690: 1132:

at all: "provided those objects be capable of a twofold difference only; as by Bells, by Trumpets, by Lights and Torches, by the report of Muskets, and any instruments of like nature". (See

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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:

2057:+ ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, the zeros and ones in the binary representation of 1/3 alternate forever. 5350:(namely, 2, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of 2658:

1 and cross out the digit that was added to it ——————————————————————— —————————————————————— = 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1

817:, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. 5671: 3716: 2877:

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

1674:, to make its binary nature explicit and for purposes of correctness. Since the binary numeral 100 represents the value four, it would be confusing to refer to the numeral as 874:, although this has been disputed). Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a 1870:), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows: 1341: 1189:"Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of 5926: 3756: 3673: 3593: 961:(1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the 2772: 1266:

of his own religious beliefs as a Christian. Binary numerals were central to Leibniz's theology. He believed that binary numbers were symbolic of the Christian idea of

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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

2985:+ 0 0 0 . 0 0 0 + 1 0 1 1 . 0 1 + 1 0 1 1 0 . 1 --------------------------- = 1 0 0 0 1 1 . 0 0 1 0 1 (35.15625 in decimal) 2740: 1287: 4528: 4508: 4488: 4468: 3222: 1666:

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

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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

723: 2555:. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 10 1381:. It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were 3561:

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

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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:

1678:(a word that represents a completely different value, or amount). Alternatively, the binary numeral 100 can be read out as "four" (the correct 870:

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

7197: 7285: 7151: 6918: 6891: 6864: 6765: 6473: 6439: 6370: 6345: 6237: 6207: 6179: 1934:

are available. Thus, after a bit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next bit to the left:

1918:

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

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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.

4172:. In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which 1/10 is not. 2844:

Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as

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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.

2989: 7280: 7181: 1004:. He described meters in the form of short and long syllables (the latter equal in length to two short syllables). They were known as 276: 1195:. Leibniz's system uses 0 and 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 6834: 6394: 6149: 6122: 6096: 3158: 1339:

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

837:. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. 5882:. Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 1019:(8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to 6408: 1164: 67: 7078: 6018: 4538:

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

1858:. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the 1108: 890: 5869:{\displaystyle {\frac {12_{10}}{17_{10}}}={\frac {1100_{2}}{10001_{2}}}=0.1011010010110100{\overline {10110100}}\ldots \,_{2}} 3944: 3825: 7275: 6289: 5998: 5901:

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

510: 3886: 3767: 7245: 6622: 3179:

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

2870: 2815: 1370: 716: 291: 6957: 1220:

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

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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.

1233: 883: 463: 3598: 2563:. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100 1603:", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use. 3157:

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

7265: 3234: 2854:* (starred columns are borrowed from) 1 0 1 1 1 1 1 – 1 0 1 0 1 1 ---------------- = 0 1 1 0 1 0 0 895: 806: 523: 2977:(6.25 in decimal) ------------------- 1 . 0 1 1 0 1 ← Corresponds to a 'one' in 6910:

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

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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

2851:* * * * (starred columns are borrowed from) 1 1 0 1 1 1 0 − 1 0 1 1 1 ---------------- = 1 0 1 0 1 1 1 2038: 1409: 1263: 1153: 1091:

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

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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)

6913:(Macmillan, Dover Publications, reprinted with corrections ed.). New York: Cambridge University Press. 1016: 1325: 699: 483: 80: 6698: 1887:

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

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

5883: 5288: 3093: 1245: 879: 383: 299: 6013: 5775:{\displaystyle {\frac {1_{10}}{3_{10}}}={\frac {1_{2}}{11_{2}}}=0.01010101{\overline {01}}\ldots \,_{2}} 5642: 3242: 3196: 2642:

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

1618:. When written, binary numerals are often subscripted, prefixed, or suffixed to indicate their base, or 974: 962: 501: 6468:. Contemporary ethnography (1st ed.). Philadelphia: University of Pennsylvania Press. p. 25. 6053: 5944: 3681: 2638:

Traditional Carry Method Long Carry Method vs.

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Arbeitsbuch Informatik – eine praxisorientierte Einführung in die Datenverarbeitung mit Projektaufgabe

5324:= (12 × 16) + (0 × 16) + (14 × 16) + (7 × 16) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,383 3469:. The first Prior Value of 0 is simply an initial decimal value. This method is an application of the 1862:. When the available symbols for this position are exhausted, the least significant digit is reset to 6671: 6614: 5978: 2603:

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

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

3192: 3188: 3041: 3037: 2874: 2819: 2719: 2715: 2486: 2472: 2457: 1650: 1144: 1024: 790: 678: 543: 494: 306: 238: 93: 54: 3208:

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

6721: 6594: 5950: 5907: 3180: 3073: 1328:. His logical calculus was to become instrumental in the design of digital electronic circuitry. 943: 928: 591: 344: 181: 176: 123: 3734: 3651: 3571: 2041:. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 6854: 6648: 3221: 2745: 1659:

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

946:, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the 7226: 7177: 7173:

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

7147: 6914: 6887: 6881: 6860: 6830: 6761: 6586: 6555: 6469: 6435: 6390: 6366: 6341: 6318: 6285: 6279: 6233: 6203: 6175: 6145: 6139: 6118: 6092: 6086: 6023: 5968: 5894: 3184: 3174: 2014: 1596: 1374: 1268: 1133: 863: 834: 673: 663: 651: 631: 586: 581: 517: 349: 321: 228: 161: 151: 138: 103: 98: 6227: 6169: 6112: 3199:

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

3145:. In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. 7139: 7070: 6939: 6753: 6713: 6679: 6545: 6535: 5963: 5929: 5887: 5678: 3648:, etc. So if there is a 1 in the first place after the decimal, then the number is at least 3205: 3047: 2777: 2579: 1630: 1382: 1080: 1001: 576: 470: 223: 211: 156: 146: 113: 88: 6501: 4446:

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

2725: 2635:), using the traditional carry method on the left, and the long carry method on the right: 2463:. Addition, subtraction, multiplication, and division can be performed on binary numerals. 1187:(published in 1703). The full title of Leibniz's article is translated into English as the 782:

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

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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

2862: 2482: 1607: 1366: 1346: 779: 688: 658: 601: 571: 556: 316: 284: 256: 233: 216: 75: 7104: 2559:

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

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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

6550: 6523: 6425: 6082: 6078: 4513: 4493: 4473: 4453: 2892: 2866: 2460: 1866:, and the next digit of higher significance (one position to the left) is incremented ( 1589: 1390: 1353: 1332: 1255: 1213: 1148: 826: 763: 750: 683: 626: 606: 561: 434: 166: 133: 118: 44: 6405: 6229:

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

2551:). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 10 851: 7259: 7049: 7006:"Pioneers – The people and ideas that made a difference – George Stibitz (1904–1995)" 6800: 6725: 6644: 6003: 5983: 5530: 4539: 3470: 3101: 3078: 2599:

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

1405: 1397: 1237: 1128: 1056: 489: 311: 251: 186: 128: 108: 17: 7053: 7239: 7235: 7230: 6757: 5347: 2963: 2951:) --------- 0 0 0 0 ← Corresponds to the rightmost 'zero' in 2575: 1592: 1564: 1386: 1313: 1241: 954: 947: 935: 867: 856: 641: 566: 6812: 6340:. Blackwell ancient religions (1. publ ed.). Malden, Mass.: Wiley-Blackwell. 1662:#b100101 (a prefix indicating binary format, common in Lisp programming languages) 1304: 906: 7171: 7133: 6790: 6429: 5250:

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

5973: 5526: 5434: 5351: 5218: 5176: 5134: 5092: 5046: 4555: 4447: 3141:, as shown on the top line, while the remainder, shown on the bottom line, is 10 2824: 2018: 1576: 1401: 1378: 1225: 1147:

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

1140: 1120: 1100: 1028: 882:, approximately 2400 BC, and its fully developed hieroglyphic form dates to the 611: 476: 428: 418: 1906:

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

7143: 6961: 5940: 2453: 2022: 1600: 810: 413: 171: 7005: 6590: 6322: 3183:. When a string of binary symbols is manipulated in this way, it is called a 2915:

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

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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

2010: 1902: 1357: 1088: 1076: 958: 423: 6986:. Math & Computer Science Department, Denison University. 30 April 2004 6717: 6559: 5904:

1.0110101000001001111001100110011111110... is the binary representation of

1559: 1236:, a popular idea that would be followed closely by his successors such as 1232:

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

7176:(in German). Vol. 2 (5 ed.). Vieweg, reprint: Springer-Verlag. 6908: 5993: 3200: 3126: 2940:

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

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

1104: 978: 814: 6943: 6598: 6574: 6306: 5685:, with a finite sequence of digits repeating indefinitely. For instance 6033: 5879: 3085: 2477: 2066: 1847: 1836: 1585: 1317: 1250: 1095:

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

1084: 997: 916: 783: 408: 393: 7074: 6983: 6683: 3261:

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

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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

1060: 985: 939: 759: 741: 398: 3074:

Division algorithm § Integer division (unsigned) with remainder

1096: 1040: 805:, or binary digit. Because of its straightforward implementation in 1055:

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

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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

2476: 2025:. As a result, 1/10 does not have a finite binary representation ( 1909: 1901: 1619: 1558: 1321: 1303: 1190: 1163: 875: 871: 850: 794: 746: 403: 365: 326: 6255:"Yijing hexagram sequences: The Shao Yong square (Fuxi sequence)" 5513:= (1 × 8) + (2 × 8) + (7 × 8) = (1 × 64) + (2 × 8) + (7 × 1) = 87 3100:, or 27 in decimal. The procedure is the same as that of decimal 1360:, completed a relay-based computer he dubbed the "Model K" (for " 767: 6750:

Handbook of the History and Philosophy of Mathematical Practice

2959:+ 0 0 0 0 + 1 0 1 1 --------------- = 1 1 0 1 1 1 0 1606:

In keeping with the customary representation of numerals using

1945:, (rightmost bit starts over, and the next bit is incremented) 1421: 1336: 802: 771: 7030:. Computer History Association of California. 6 February 1995 6856:

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

6662:

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

6524:"Mangarevan invention of binary steps for easier calculation" 5681:

have binary representation, but instead of terminating, they

2895:

in binary is similar to its decimal counterpart. Two numbers

2535:

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

1258:, who visited China in 1685 as a missionary. Leibniz saw the 1000:(c. 2nd century BC) developed a binary system for describing 6938:(Thesis). Cambridge: Massachusetts Institute of Technology. 5533:

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

2489:, which adds two bits together, producing sum and carry bits 2037:). This causes 10 × 1/10 not to precisely equal 1 in binary 1065:

Masterpieces of the Oral and Intangible Heritage of Humanity

5893:

Binary numerals that neither terminate nor recur represent

5004: 2981:+ 0 0 . 0 0 0 0 ← Corresponds to a 'zero' in 6752:, Cham: Springer International Publishing, pp. 1–31, 6612: 6575:"Diversity in the Numeral Systems of Australian Languages" 5346:

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

4559: 1649:%100101 (a prefix indicating binary format; also known as 1629:

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

7105:"Introducing binary – Revision 1 – GCSE Computer Science" 7054:"Konrad Zuse's Legacy: The Architecture of the Z1 and Z3" 6307:"Mapping the Entrails: The Practice of Greek Hepatoscopy" 4962: 4920: 4874: 4832: 4790: 4748: 4702: 4660: 4618: 4574: 3989:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1} 3870:{\textstyle {\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1} 2903:

can be multiplied by partial products: for each digit in

2532:

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

798: 7165: 7163: 7127: 7125: 2962:

Binary numbers can also be multiplied with bits after a

1610:, binary numbers are commonly written using the symbols 3928:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1} 3809:{\textstyle {\frac {1}{3}}\times 2={\frac {2}{3}}<1} 3081:

in binary is again similar to its decimal counterpart.

2607:

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

7138:(in German). Vieweg-Verlag, reprint: Springer-Verlag. 6085:, eds. (2009), "Myth No. 2: the Horus eye fractions", 3947: 3889: 3828: 3770: 3737: 3684: 3654: 3601: 3574: 2574:

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

2518:

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

2515:

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

1682:), but this does not make its binary nature explicit. 6958:"National Inventors Hall of Fame – George R. Stibitz" 6522:

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

6284:. Stuttgart: Franz Steiner Verlag. pp. 165–170. 5910: 5787: 5693: 5645: 4516: 4496: 4476: 4456: 4184: 4008:... is equivalent to the repeating binary fraction 0. 3050: 2955:+ 1 0 1 1 ← Corresponds to the next 'one' in 2780: 2748: 2728: 2640:

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

1276:

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

6168:

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

5932:, another irrational. It has no discernible pattern. 2973:(5.625 in decimal) × 1 1 0 . 0 1 2506:

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

7170:Küveler, Gerd; Schwoch, Dietrich (4 October 2007). 6936:

A symbolic analysis of relay and switching circuits

6744:Strickland, Lloyd (2020), Sriraman, Bharath (ed.), 6706:

Mitteilungen der deutschen Mathematiker-Vereinigung

3564:

In a fractional binary number such as 0.11010110101

3036:The binary multiplication table is the same as the 1342:

A Symbolic Analysis of Relay and Switching Circuits

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

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Index