38:
4995:
1045:
2975:
3355:
significant digit (10) is "dropped": 10 1 0 11 <- Digits of 0xA10B --------------- 10 Then we multiply the bottom number from the source base (16), the product is placed under the next digit of the source value, and then add: 10 1 0 11 160 --------------- 10 161 Repeat until the final addition is performed: 10 1 0 11 160 2576 41216 --------------- 10 161 2576 41227 and that is 41227 in decimal.
6015:(base-16) bases are most commonly used. Computers, at the most basic level, deal only with sequences of conventional zeroes and ones, thus it is easier in this sense to deal with powers of two. The hexadecimal system is used as "shorthand" for binary—every 4 binary digits (bits) relate to one and only one hexadecimal digit. In hexadecimal, the six digits after 9 are denoted by A, B, C, D, E, and F (and sometimes a, b, c, d, e, and f).
1127:
4594:
931:
3967:
1251:. For example, for the decimal system the radix (and base) is ten, because it uses the ten digits from 0 through 9. When a number "hits" 9, the next number will not be another different symbol, but a "1" followed by a "0". In binary, the radix is two, since after it hits "1", instead of "2" or another written symbol, it jumps straight to "10", followed by "11" and "100".
6131:". Although mostly historical, it is occasionally used colloquially. Verse 10 of Psalm 90 in the King James Version of the Bible starts: "The days of our years are threescore years and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow". The Gettysburg Address starts: "Four score and seven years ago".
4990:{\displaystyle {\begin{array}{l}1\times 3^{0\,\,\,}+{}\\1\times 3^{-1\,\,}+2\times 3^{-2\,\,\,}+{}\\1\times 3^{-3\,\,}+1\times 3^{-4\,\,\,}+2\times 3^{-5\,\,\,}+{}\\1\times 3^{-6\,\,}+1\times 3^{-7\,\,\,}+1\times 3^{-8\,\,\,}+2\times 3^{-9\,\,\,}+{}\\1\times 3^{-10}+1\times 3^{-11}+1\times 3^{-12}+1\times 3^{-13}+2\times 3^{-14}+\cdots \end{array}}}
6382:, which requires finding a minimal set of known counter-weights to determine an unknown weight. Weights of 1, 3, 9, ..., 3 known units can be used to determine any unknown weight up to 1 + 3 + ... + 3 units. A weight can be used on either side of the balance or not at all. Weights used on the balance pan with the unknown weight are designated with
7155:
3777:
2601:
241 in base 8: 2 groups of 8 (64) 4 groups of 8 1 group of 1 oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo + + o oooooooo oooooooo oooooooo oooooooo oooooooo oooooooo
1641:
1112:
only a small advance was required to establish the complete system of decimal positional fractions, and this step was taken promptly by a number of writers ... next to Stevin the most important figure in this development was
Regiomontanus." Dijksterhuis noted that "gives full credit to Regiomontanus
5986:
introduced a modern notational system for
Babylonian and Hellenistic numbers that substitutes modern decimal notation from 0 to 59 in each position, while using a semicolon (;) to separate the integral and fractional portions of the number and using a comma (,) to separate the positions within each
6637:
were positional, but in each position were groups of two kinds of wedges representing ones and tens (a narrow vertical wedge | for the one and an open left pointing wedge ⟨ for the ten) — up to 5+9=14 symbols per position (i.e. 5 tens ⟨⟨⟨⟨⟨ and 9 ones ||||||||| grouped into one or two near squares
951:
to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical
3342:
can be used for base conversion using repeated multiplications, with the same computational complexity as repeated divisions. A number in positional notation can be thought of as a polynomial, where each digit is a coefficient. Coefficients can be larger than one digit, so an efficient way to
3354:
Convert 0xA10B to 41227 A10B = (10*16^3) + (1*16^2) + (0*16^1) + (11*16^0) Lookup table: 0x0 = 0 0x1 = 1 ... 0x9 = 9 0xA = 10 0xB = 11 0xC = 12 0xD = 13 0xE = 14 0xF = 15 Therefore 0xA10B's decimal digits are 10, 1, 0, and 11. Lay out the digits out like this. The most
5903:). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one rotation around a circle is 360°, two rotations are 720°, etc.), and both time and angles use decimal fractions of a second. This contrasts with the numbers used by Hellenistic and
3497:
5000:
Since a complete infinite string of digits cannot be explicitly written, the trailing ellipsis (...) designates the omitted digits, which may or may not follow a pattern of some kind. One common pattern is when a finite sequence of digits repeats infinitely. This is designated by drawing a
6638:
containing up to three tiers of symbols, or a place holder (\\) for the lack of a position). Hellenistic astronomers used one or two alphabetic Greek numerals for each position (one chosen from 5 letters representing 10–50 and/or one chosen from 9 letters representing 1–9, or a
2597:
241 in base 5: 2 groups of 5 (25) 4 groups of 5 1 group of 1 ooooo ooooo ooooo ooooo ooooo ooooo ooooo ooooo + + o ooooo ooooo ooooo ooooo ooooo ooooo
850:
1927:
795:, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. Today, the Hindu–Arabic numeral system (
5978:
Using a digit set of digits with upper and lowercase letters allows short notation for sexagesimal numbers, e.g. 10:25:59 becomes 'ARz' (by omitting I and O, but not i and o), which is useful for use in URLs, etc., but it is not very intelligible to humans.
1099:
so great an integral value that all occurring quantities could be expressed with sufficient accuracy by integers. ¶ The first to apply this method was the German astronomer
Regiomontanus. To the extent that he expressed goniometrical line-segments in a unit
6022:
numbering system is also used as another way to represent binary numbers. In this case the base is 8 and therefore only digits 0, 1, 2, 3, 4, 5, 6, and 7 are used. When converting from binary to octal every 3 bits relate to one and only one octal digit.
1454:
1090:
fractions ... This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ ... Mathematicians sought to avoid fractions by taking the radius
2381:
788:, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.
2511:
2737:
numeral system, but we remove two digits, uppercase "I" and uppercase "O", to reduce confusion with digits "1" and "0". We are left with a base-60, or sexagesimal numeral system utilizing 60 of the 62 standard alphanumerics. (But see
3962:{\displaystyle \mathbb {Z} _{S}:=\left\{x\in \mathbb {Q} \left|\,\exists \mu _{i}\in \mathbb {Z} :x\prod _{i=1}^{n}{p_{i}}^{\mu _{i}}\in \mathbb {Z} \right.\right\}=b^{\mathbb {Z} }\,\mathbb {Z} ={\langle S\rangle }^{-1}\mathbb {Z} }
6359:
Interesting properties exist when the base is not fixed or positive and when the digit symbol sets denote negative values. There are many more variations. These systems are of practical and theoretic value to computer scientists.
4570:
2826:
is 0–9 A–F, where the ten numerics retain their usual meaning, and the alphabetics correspond to values 10–15, for a total of sixteen digits. The numeral "10" is binary numeral "2", octal numeral "8", or hexadecimal numeral "16".
2720:. We could increase the number base again and assign "B" to 11, and so on (but there is also a possible encryption between number and digit in the number-digit-numeral hierarchy). A three-digit numeral "ZZZ" in base-60 could mean
2711:
In certain applications when a numeral with a fixed number of positions needs to represent a greater number, a higher number-base with more digits per position can be used. A three-digit, decimal numeral can represent only up to
1470:
6305:
Many ancient counting systems use five as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some
1254:
The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use.
884:
rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them.
3376:
7105:
5436:
5061:
3576:
2609:
and every real number has at least one representation. The representations of rational numbers are those representations that are finite, use the bar notation, or end with an infinitely repeating cycle of digits.
3326:
fraction's denominator has a prime factor other than any of the base's prime factor(s) to convert to. For example, 0.1 in decimal (1/10) is 0b1/0b1010 in binary, by dividing this in that radix, the result is
2942:
1258:
The radix is an integer that is greater than 1, since a radix of zero would not have any digits, and a radix of 1 would only have the zero digit. Negative bases are rarely used. In a system with more than
6949:
there is virtually only one positional-notation numeral system for each base below 10, and this extends with few, if insignificant, variations on the choice of alphabetic digits for those bases above 10.
6944:
The digit will retain its meaning in other number bases, in general, because a higher number base would normally be a notational extension of the lower number base in any systematic organization. In the
5165:
3358:
Convert 0b11111001 to 249 Lookup table: 0b0 = 0 0b1 = 1 Result: 1 1 1 1 1 0 0 1 <- Digits of 0b11111001 2 6 14 30 62 124 248 ------------------------- 1 3 7 15 31 62 124 249
6279:) has been used in many cultures for counting. Plainly it is based on the number of digits on a human hand. It may also be regarded as a sub-base of other bases, such as base-10, base-20, and base-60.
5271:
3709:
2262:
6172:, particularly for the age of people, dates and in common phrases. 15 is also important, with 16–19 being "one on 15", "two on 15" etc. 18 is normally "two nines". A decimal system is commonly used.
6432:
puzzle configuration as a counting system. The configuration of the towers can be put into 1-to-1 correspondence with the decimal count of the step at which the configuration occurs and vice versa.
5356:
For a given base, any number that can be represented by a finite number of digits (without using the bar notation) will have multiple representations, including one or two infinite representations:
6326:). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as
6294:(from whom most European and Indic languages descend) might have replaced a base-8 system (or a system which could only count up to 8) with a base-10 system. The evidence is that the word for 9,
5544:
4065:
3654:
5208:
1734:
5492:
1726:
6290:
of
Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age
5600:
2605:
The notation can be further augmented by allowing a leading minus sign. This allows the representation of negative numbers. For a given base, every representation corresponds to exactly one
880:. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals. It was a
6055:
or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtraction being just as easy. Twelve is a useful base because it has many
4584:
The representation of non-integers can be extended to allow an infinite string of digits beyond the point. For example, 1.12112111211112 ... base-3 represents the sum of the infinite
4468:
6124:(literally, four twenty twelve). In Old French, forty was expressed as two twenties and sixty was three twenties, so that fifty-three was expressed as two twenties thirteen, and so on.
3347:, removing the need for expensive division or modulus operations; and multiplication by x becomes right-shifting. However, other polynomial evaluation algorithms would work as well, like
1975:
3769:
7067:
3303:
0b11111001/10 = 0b11000 R: 0b1001 (0b1001 = "9" for ones place) 0b11000/10 = 0b10 R: 0b100 (0b100 = "4" for tens) 0b10/10 = 0b0 R: 0b10 (0b10 = "2" for hundreds)
2101:). In books and articles, when using initially the written abbreviations of number bases, the base is not subsequently printed: it is assumed that binary 1111011 is the same as 1111011
7422:
5442:
The last non-zero digit can be reduced by one and an infinite string of digits, each corresponding to one less than the base, are appended (or replace any following zero digits):
4337:
2032:
4372:
6067:. The standard 12-hour clock and common use of 12 in English units emphasize the utility of the base. In addition, prior to its conversion to decimal, the old British currency
3996:
4428:
4149:
1324:
5663:
4270:
4248:
4222:
4091:
6116:(literally, "sixty fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty. For example, eighty-two is
3243:
0xA10B/10 = 0x101A R: 7 (ones place) 0x101A/10 = 0x19C R: 2 (tens place) 0x19C/10 = 0x29 R: 2 (hundreds place) 0x29/10 = 0x4 R: 1 ... 4
6097:), as did several North American tribes (two being in southern California). Evidence of base-20 counting systems is also found in the languages of central and western
2273:
5699:
5299:
5109:
4302:
784:, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the
5742:
4022:
3190:
3160:
3097:
3298:
3271:
3223:
3124:
3063:
3030:
2809:
1245:
6059:. It is the smallest common multiple of one, two, three, four and six. There is still a special word for "dozen" in English, and by analogy with the word for 10,
2403:
1287:
1086:
the Arabs, the idea of positional value for integers, neglected to extend this idea to fractions. For some centuries they confined themselves to using common and
5872:
for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional.
2716:. But if the number-base is increased to 11, say, by adding the digit "A", then the same three positions, maximized to "AAA", can represent a number as great as
1292:
It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be
1113:
for his prior contribution, saying that the trigonometric tables of the German astronomer actually contain the whole theory of 'numbers of the tenth progress'."
1028:
introduced fractions to
Islamic countries in the early 9th century; his fraction presentation was similar to the traditional Chinese mathematical fractions from
776:
in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early
6075:
used base-12; there were 12 pence (d) in a shilling (s), 20 shillings in a pound (£), and therefore 240 pence in a pound. Hence the term LSD or, more properly,
4511:
4491:
4399:
4193:
4169:
4112:
3600:
2782:
2762:
2962:, here »−«, is added to the numeral system. In the usual notation it is prepended to the string of digits representing the otherwise non-negative number.
1636:{\displaystyle 14\mathrm {B} 9_{\mathrm {hex} }=(1\times 16^{3})+(4\times 16^{2})+(\mathrm {B} \times 16^{1})+(9\times 16^{0})\qquad (=5305_{\mathrm {dec} }),}
7293:
4516:
6847:: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.
2634:"0", "1", "2", "3", "4", "5", "6", "7", "8", and "9". The distinction between a digit and a numeral is most pronounced in the context of a number base.
5070:(to which there does not exist a single universally accepted notation or phrasing). For base 10 it is called a repeating decimal or recurring decimal.
6379:
3492:{\displaystyle {\frac {\mathbb {N} _{0}}{b^{\mathbb {N} _{0}}}}:=\left\{mb^{-\nu }\mid m\in \mathbb {N} _{0}\wedge \nu \in \mathbb {N} _{0}\right\}.}
6738:
5824:. Digits to the right of it are multiplied by 10 raised to a negative power or exponent. The first position to the right of the separator indicates
3343:
convert bases is to convert each digit, then evaluate the polynomial via Horner's method within the target base. Converting each digit is a simple
2574:
This concept can be demonstrated using a diagram. One object represents one unit. When the number of objects is equal to or greater than the base
2845:. Thereby the so-called radix point, mostly ».«, is used as separator of the positions with non-negative from those with negative exponent.
971:
of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived
5081:
has a finite representation or requires an infinite repeating representation depends on the base. For example, one third can be represented by:
5365:
5011:
3505:
841:
are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe.
2708:, ..., 121, 123} while its digits "2" and "3" always retain their original meaning: the "2" means "two of", and the "3" means "three of".
7144:, American Oriental Series, vol. 29, New Haven: American Oriental Society and the American Schools of Oriental Research, p. 2,
5879:. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be
7417:
6222:
was used in the
Egyptian Old Kingdom, 3000 BC to 2050 BC. It was cursive by rounding off rational numbers smaller than 1 to
2159:
of the base. A digit's value is the digit multiplied by the value of its place. Place values are the number of the base raised to the
6272:) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system.
736:
2870:
6684:
6354:
5115:
3315:
1300:
456:
2958:
If the base and all the digits in the set of digits are non-negative, negative numbers cannot be expressed. To overcome this, a
5221:
3660:
2196:
1032:. This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by 10th century
982:(1789–1799), the new French government promoted the extension of the decimal system. Some of those pro-decimal efforts—such as
6108:
base-20 system also exist in French, as seen today in the names of the numbers from 60 through 99. For example, sixty-five is
5755:
and the sole way to write them down with a finite number of symbols is to give them a symbol or a finite sequence of symbols.
7246:
7139:
6829:
900:
lamented how science might have progressed had
Archimedes only made the leap to something akin to the modern decimal system.
7441:
1148:
5498:
4027:
1922:{\displaystyle (a_{3}a_{2}a_{1}a_{0})_{b}=(a_{3}\times b^{3})+(a_{2}\times b^{2})+(a_{1}\times b^{1})+(a_{0}\times b^{0})}
7458:
6800:
6659:
3605:
822:
base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers.
5174:
289:
5447:
4068:
1660:
7401:
7377:
7354:
7230:
7149:
7000:
6775:
5555:
2814:
The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In
1174:
2179:
is positive or zero; if the digit is on the right hand side of the radix point (i.e., its value is fractional) then
1156:
4433:
7297:
3332:
1020:
used decimal fractions around 1350, but did not develop any notation to represent them. The
Persian mathematician
7192:
6898:
6674:
6234:
5779:
1935:
765:
80:
17:
3716:
7463:
7026:
6031:
2641:
with more than one digit position will mean a different number in a different number base, but in general, the
2131:−1} is called the standard set of digits. Thus, binary numbers have digits {0, 1}; decimal numbers have digits
1152:
1012:
J. Lennart
Berggren notes that positional decimal fractions were used for the first time by Arab mathematician
523:
3319:
2571:. Note that the last "16" is indicated to be in base 10. The base makes no difference for one-digit numerals.
868:, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the
6634:
6390:
is balanced with 3 (3) on its pan and 1 and 27 (3 and 3) on the other, then its weight in decimal is 25 or 10
2649:
contains two digits, "2" and "3", and with a base number (subscripted) "8". When converted to base-10, the 23
729:
304:
1204:, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a
947:
and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or
6763:
5995:
is 29;31,50,8,20 days, and the angle used in the example above would be written 10;25,59,23,31,12 degrees.
649:
4307:
2986:
2704:. The numeral "23" then, in this case, corresponds to the set of base-10 numbers {11, 13, 15, 17, 19, 21,
1980:
6932:
6742:
2116:". So binary numbers are "base-2"; octal numbers are "base-8"; decimal numbers are "base-10"; and so on.
1449:{\displaystyle 5305_{\mathrm {dec} }=(5\times 10^{3})+(3\times 10^{2})+(0\times 10^{1})+(5\times 10^{0})}
476:
4342:
659:
6977:
3975:
3348:
1033:
1013:
536:
4404:
4125:
6421:
6269:
2073:
subscript after the number that is being represented (this notation is used in this article). 1111011
632:
401:
1137:
722:
49:
5633:
4253:
4231:
4205:
4074:
1037:
1021:
6966:
digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".
6654:
6413:
6039:
6027:
5313:
4375:
2376:{\displaystyle 4\times 10^{2}+6\times 10^{1}+5\times 10^{0}=4\times 100+6\times 10+5\times 1=465}
1141:
712:
496:
93:
6639:
1104:/10, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions.
357:
6169:
5782:, each position starting from the right is a higher power of 10. The first position represents
5769:
5327:
4119:
3307:
2066:
1055:
1051:
881:
830:
396:
312:
5618:
A (real) irrational number has an infinite non-repeating representation in all integer bases.
5067:
6428:
enumerations. This system effectively enumerates permutations. A derivative of this uses the
6425:
6219:
6008:
5748:
5671:
5277:
5087:
4275:
4225:
2815:
2147:. (In all cases, one or more digits is not in the set of allowed digits for the given base.)
2059:
2047:
963:, showing possible usages of positional-numbers in the 7th century. Khmer numerals and other
800:
514:
6374:" has an equivalent value of −1. The negation of a number is easily formed by switching the
5721:
4001:
3331:(because one of the prime factors of 10 is 5). For more general fractions and bases see the
3246:
When converting to a larger base (such as from binary to decimal), the remainder represents
3165:
3135:
3072:
2506:{\displaystyle 4\times 7^{2}+6\times 7^{1}+5\times 7^{0}=4\times 49+6\times 7+5\times 1=243}
7010:
6946:
4585:
3323:
3276:
3249:
3201:
3102:
3041:
3008:
2787:
1304:
1214:
897:
609:
470:
463:
344:
7427:
6194:
8:
6699:
6689:
6291:
5861:
5821:
3310:
part, conversion can be done by taking digits after the radix point (the numerator), and
1262:
916:
869:
792:
691:
681:
556:
507:
319:
251:
106:
2680:
number. Then "23" could likely be any base, from base-4 up. In base-4, the "23" means 11
2594:
objects; and so on. Thus the same number in different bases will have different values:
146:
7330:
6983:. In Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.).
6924:
6903:
6302:, suggesting that the number 9 had been recently invented and called the "new number".
6227:
4565:{\displaystyle b^{\mathbb {Z} }\,\mathbb {Z} \subseteq c^{\mathbb {Z} }\,\mathbb {Z} .}
4496:
4476:
4384:
4178:
4172:
4154:
4097:
3585:
3339:
3066:
2953:
2767:
2747:
1067:
901:
812:
604:
194:
189:
136:
6208:
of New
Zealand also has evidence of an underlying base-20 system as seen in the terms
2856:. For every position behind this point (and thus after the units digit), the exponent
141:
7397:
7373:
7350:
7226:
7145:
6996:
6825:
6771:
6721:
6307:
6188:
5813:
5613:
5074:
5002:
2864:
decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:
2171:. If a given digit is on the left hand side of the radix point (i.e. its value is an
979:
893:
808:
686:
676:
664:
644:
599:
594:
530:
362:
334:
241:
174:
164:
151:
116:
111:
31:
7334:
7219:
6205:
909:
579:
7322:
6988:
6363:
6342:
6338:
6184:
6105:
5712:
2082:
1017:
987:
589:
483:
236:
224:
169:
159:
126:
101:
7368:
The Universal History of Numbers: From Prehistory to the Invention of the Computer
6026:
Hexadecimal, decimal, octal, and a wide variety of other bases have been used for
2190:(which must be at least base 7 because the highest digit in it is 6) is equal to:
37:
7391:
7366:
7131:
7006:
6992:
6804:
6429:
6319:
6176:
5992:
5983:
5078:
4200:
2959:
2631:
1201:
972:
968:
964:
701:
671:
614:
584:
569:
329:
297:
269:
246:
229:
88:
6386:, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight
3162:
the second right-most digit is the remainder of the division of the quotient by
184:
7387:
6323:
6311:
6162:
6135:
6068:
5912:
5900:
5896:
5892:
5876:
5869:
5816:, which can vary in different locations. Usually this separator is a period or
2744:
below.) In general, the number of possible values that can be represented by a
2156:
2035:
1209:
1193:
1029:
998:
of weights and measures—spread widely out of France to almost the whole world.
991:
960:
923:
were used, and accountants in ancient Rome and during the Middle Ages used the
920:
819:
781:
777:
773:
696:
639:
619:
574:
447:
179:
131:
57:
7326:
7080:
6797:
7452:
7135:
6844:
6315:
6238:
6087:
6083:
5988:
5908:
5799:
5077:
has an infinite non-repeating representation in all integer bases. Whether a
3579:
3311:
2730:
1315:
1307:, the definition of the base or the allowed digits deviates from the above.)
1205:
1071:
944:
502:
391:
324:
264:
199:
121:
7313:
Kadvany, John (December 2007). "Positional Value and Linguistic Recursion".
6822:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
7431:
7342:
6876:
6859:
5342:
4379:
3344:
1059:
1025:
983:
956:
935:
905:
654:
7177:
6987:. Computing Supplementa. Vol. 4. Vienna: Springer. pp. 189–220.
6727:. Oxford: Oxford University Press. pp. 11–12 – via archive.org.
5991:
used by both Babylonian and Hellenistic astronomers and still used in the
1095:
equal to a number of units of length of the form 10 and then assuming for
1044:
6679:
6128:
6090:
6012:
5904:
5865:
5857:
5752:
4599:
3318:
in the target radix. Approximation may be needed due to a possibility of
2853:
2836:
2823:
2606:
2168:
2098:
1464:), there are the sixteen hexadecimal digits (0–9 and A–F) and the number
1461:
1289:
unique digits, numbers may have many different possible representations.
1087:
995:
892:(ca. 287–212 BC) invented a decimal positional system based on 10 in his
834:
826:
624:
489:
441:
431:
6862:, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic",
2974:
6287:
6052:
5860:
or base-60 system was used for the integral and fractional portions of
5431:{\displaystyle 3.46_{7}=3.460_{7}=3.460000_{7}=3.46{\overline {0}}_{7}}
5056:{\displaystyle 2.42{\overline {314}}_{5}=2.42314314314314314\dots _{5}}
1109:
1063:
955:
The oldest extant positional notation system is either that of Chinese
889:
838:
426:
3571:{\displaystyle p_{1}^{\nu _{1}}\cdot \ldots \cdot p_{n}^{\nu _{n}}:=b}
6417:
6198:
6166:
6094:
6004:
5817:
4196:
3192:
and so on. The left-most digit is the last quotient. In general, the
2841:
The notation can be extended into the negative exponents of the base
1293:
915:
Before positional notation became standard, simple additive systems (
908:
astronomers used a base-60 system based on the Babylonian model (see
436:
7436:
6879:. "A Chinese Genesis, Rewriting the history of our numeral system".
1126:
7423:
The Development of Hindu Arabic and Traditional Chinese Arithmetics
6669:
5622:
5547:
5211:
3368:
804:
796:
6154:. A remnant of this system may be seen in the modern word for 40,
1024:
made the same discovery of decimal fractions in the 15th century.
6796:
L. F. Menabrea. Translated by Ada Augusta, Countess of Lovelace.
6276:
6180:
6056:
5809:
5775:
2849:
2172:
1311:
1075:
1007:
873:
864:) system, which is presumably motivated by counting with the ten
861:
853:
785:
769:
421:
406:
3300:. For example: converting 0b11111001 (binary) to 249 (decimal):
2077:
implies that the number 1111011 is a base-2 number, equal to 123
2069:
2. Another common way of expressing the base is writing it as a
6237:
employ binary or binary-like counting systems. For example, in
6212:
referring to a war party (literally "the seven 20s of Tu") and
6098:
5835:
As an example, the number 2674 in a base-10 numeral system is:
2627:
2055:
948:
930:
924:
865:
411:
6820:
Berggren, J. Lennart (2007). "Mathematics in Medieval Islam".
6076:
4151:
contains if reduced to lowest terms only prime factors out of
2937:{\displaystyle 2\times 10^{0}+3\times 10^{-1}+5\times 10^{-2}}
2186:
As an example of usage, the number 465 in its respective base
990:—were unsuccessful. Other French pro-decimal efforts—currency
6798:"Sketch of The Analytical Engine Invented by Charles Babbage"
6331:
6283:
6019:
5829:
5825:
2819:
2673:
is part of the numeral, but this may not always be the case.
2630:
with positional notation. Today's most common digits are the
2135:
and so on. Therefore, the following are notational errors: 52
2090:
1108:
In the estimation of Dijksterhuis, "after the publication of
761:
416:
378:
339:
30:"Positional system" redirects here. For the voting rule, see
6216:, referring to a great warrior ("the one man equal to 20").
6127:
In English the same base-20 counting appears in the use of "
3198:
th digit from the right is the remainder of the division by
2677:
2397:
If however, the number were in base 7, then it would equal:
2167:
is the number of other digits between a given digit and the
5791:
5787:
5783:
5160:{\displaystyle 0.{\overline {3}}_{10}=0.3333333\dots _{10}}
3901:
1082:
European mathematicians, when taking over from the Hindus,
877:
849:
6348:
6298:, is suggested by some to derive from the word for "new",
6226:, with a 1/64 term thrown away (the system was called the
2586:, then a group of these groups of objects is created with
799:) is the most commonly used system globally. However, the
7106:"Irrational Numbers: Definition, Examples and Properties"
5266:{\displaystyle 0.{\overline {01}}_{2}=0.010101\dots _{2}}
3704:{\displaystyle \nu _{1},\ldots ,\nu _{n}\in \mathbb {N} }
2257:{\displaystyle 4\times b^{2}+6\times b^{1}+5\times b^{0}}
1646:
where B represents the number eleven as a single symbol.
3367:
The numbers which have a finite representation form the
2622:
is a symbol that is used for positional notation, and a
856:(the number represented in the picture is 6,302,715,408)
41:
Glossary of terms used in the positional numeral systems
5360:
A finite or infinite number of zeroes can be appended:
2626:
consists of one or more digits used for representing a
2267:
If the number 465 was in base-10, then it would equal:
1016:
as early as the 10th century. The Jewish mathematician
959:, used from at least the early 8th century, or perhaps
872:, credited as the first positional numeral system, was
6975:
6345:, is notable for possessing a base-27 numeral system.
4228:
for the usual (Archimedean) metric is the same as for
2645:
will mean the same. For example, the base-8 numeral 23
7291:
7029:
6897:
6633:
Each position does not need to be positional itself.
5724:
5674:
5636:
5558:
5501:
5450:
5368:
5280:
5224:
5177:
5118:
5090:
5014:
4597:
4519:
4499:
4479:
4436:
4407:
4387:
4345:
4310:
4278:
4256:
4234:
4208:
4181:
4157:
4128:
4100:
4077:
4030:
4004:
3978:
3780:
3719:
3663:
3608:
3588:
3508:
3379:
3279:
3252:
3204:
3168:
3138:
3105:
3075:
3044:
3011:
2873:
2790:
2770:
2750:
2406:
2276:
2199:
1983:
1938:
1737:
1663:
1473:
1327:
1265:
1217:
7130:
6985:
Computer Algebra: Symbolic and Algebraic Computation
6929:
Simon Stevin: Science in the Netherlands around 1600
6905:
A History of Algebra. From Khwarizmi to Emmy Noether
5875:
Modern time separates each position by a colon or a
5539:{\displaystyle 1_{10}=0.{\overline {9}}_{10}\qquad }
4060:{\displaystyle {\langle S\rangle }^{-1}\mathbb {Z} }
2818:
only digits "0" and "1" are in the numerals. In the
7292:O'Connor, John; Robertson, Edmund (December 2000).
6768:
Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl
6310:
the word for five is the same as "hand" or "fist" (
6120:(literally, four twenty two), while ninety-two is
3649:{\displaystyle p_{1},\ldots ,p_{n}\in \mathbb {P} }
837:with arbitrary accuracy. With positional notation,
7365:
7218:
7061:
6976:Collins, G. E.; Mignotte, M.; Winkler, F. (1983).
6902:
6720:
6112:(literally, "sixty five"), while seventy-five is
5915:, etc. for finer increments. Where we might write
5736:
5693:
5657:
5594:
5538:
5486:
5430:
5293:
5265:
5202:
5159:
5103:
5055:
4989:
4564:
4505:
4485:
4462:
4422:
4393:
4366:
4331:
4296:
4264:
4242:
4216:
4187:
4163:
4143:
4106:
4085:
4059:
4016:
3990:
3961:
3763:
3703:
3648:
3594:
3570:
3491:
3292:
3265:
3217:
3184:
3154:
3118:
3091:
3057:
3024:
2936:
2803:
2776:
2756:
2582:objects. When the number of these groups exceeds
2505:
2375:
2256:
2026:
1969:
1921:
1720:
1635:
1448:
1281:
1239:
1001:
7349:. Vol. 2. Addison-Wesley. pp. 195–213.
5203:{\displaystyle 0.{\overline {3}}=0.3333333\dots }
811:because it is easier to implement efficiently in
7450:
7069:does not matter. They only have to be ≥ 1.
6378:on the 1s. This system can be used to solve the
6268:North and Central American natives used base-4 (
6046:
6038:For a list of bases and their applications, see
5832:(0.01), and so on for each successive position.
5487:{\displaystyle 3.46_{7}=3.45{\overline {6}}_{7}}
1721:{\displaystyle \{d_{1},d_{2},\dotsb ,d_{b}\}=:D}
1074:contributed to the European adoption of general
829:(decimal point in base ten), extends to include
5595:{\displaystyle 220_{5}=214.{\overline {4}}_{5}}
6723:The Nothing That Is: A Natural History of Zero
6731:
4463:{\displaystyle T=\mathbb {P} \setminus \{p\}}
1187:
730:
7221:Pi in the sky: counting, thinking, and being
7176:Bartley, Wm. Clark (January–February 1997).
6770:, Vandenhoeck und Ruprecht, 3rd. ed., 1979,
6138:also used base-20 in the past, twenty being
5847:(2 × 1000) + (6 × 100) + (7 × 10) + (4 × 1).
4457:
4451:
4324:
4318:
4291:
4285:
4039:
4033:
3985:
3979:
3941:
3935:
3758:
3726:
3713:then with the non-empty set of denominators
1709:
1664:
772:). More generally, a positional system is a
7396:. Courier Dover Publications. p. 176.
6824:. Princeton University Press. p. 518.
6628:
4574:
2692:. In base-60, the "23" means the number 123
1970:{\displaystyle \forall k\colon a_{k}\in D.}
1155:. Unsourced material may be challenged and
27:Method for representing or encoding numbers
7437:Learn to count other bases on your fingers
6815:
6813:
3764:{\displaystyle S:=\{p_{1},\ldots ,p_{n}\}}
2578:, then a group of objects is created with
2112:may also be indicated by the phrase "base-
737:
723:
7442:Online Arbitrary Precision Base Converter
7062:{\displaystyle \nu _{1},\ldots ,\nu _{n}}
5839:(2 × 10) + (6 × 10) + (7 × 10) + (4 × 10)
4858:
4857:
4856:
4833:
4832:
4831:
4808:
4807:
4806:
4783:
4782:
4753:
4752:
4751:
4728:
4727:
4726:
4703:
4702:
4673:
4672:
4671:
4648:
4647:
4618:
4617:
4616:
4555:
4553:
4547:
4534:
4532:
4526:
4444:
4410:
4348:
4313:
4258:
4236:
4210:
4131:
4079:
4053:
3955:
3926:
3924:
3918:
3896:
3835:
3817:
3808:
3783:
3697:
3642:
3471:
3450:
3401:
3384:
2729:. If we use the entire collection of our
1175:Learn how and when to remove this message
6819:
6712:
3362:
929:
848:
36:
7386:
7312:
7175:
6978:"Arithmetic in basic algebraic domains"
6810:
6685:Non-standard positional numeral systems
6366:uses a base of 3 but the digit set is {
6355:Non-standard positional numeral systems
6349:Non-standard positional numeral systems
6330:. It is found in many languages of the
2740:
2665:. In our notation here, the subscript "
2602:oooooooo oooooooo oooooooo oooooooo
1301:non-standard positional numeral systems
760:) usually denotes the extension to any
14:
7451:
7216:
6920:
6918:
6916:
6718:
3351:for single or sparse digits. Example:
2613:
2155:Positional numeral systems work using
7393:Handbook of the Indians of California
7363:
7341:
7247:Encyclopedia of Indo-European Culture
7078:
6881:Archive for History of Exact Sciences
6855:
6853:
6420:as place values; they are related to
6407:= 1 × 3 + 0 × 3 − 1 × 3 + 1 × 3 = 25.
5851:
5607:
4175:of all terminating fractions to base
3273:as a single digit, using digits from
2034:represents a sequence of digits, not
1314:) positional notation, there are ten
7169:
6875:
6660:Category: Positional numeral systems
6224:1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64
6063:, commerce developed a word for 12,
5337:has a finite representation in base
4332:{\displaystyle \mathbb {Z} _{\{p\}}}
3126:is the remainder of the division of
2969:
2822:numerals, are the eight digits 0–7.
2027:{\displaystyle a_{3}a_{2}a_{1}a_{0}}
1153:adding citations to reliable sources
1120:
927:or stone counters to do arithmetic.
896:; 19th century German mathematician
7158:from the original on 1 October 2016
6913:
6891:
5864:and other Mesopotamian systems, by
5751:. The number of transcendentals is
4579:
2676:Imagine the numeral "23" as having
24:
6866:, 1996, p. 38, Kurt Vogel notation
6850:
6241:, the numbers one through six are
4367:{\displaystyle \mathbb {Z} _{(p)}}
3818:
2965:
1939:
1621:
1618:
1615:
1557:
1494:
1491:
1488:
1478:
1340:
1337:
1334:
975:, recorded from the 10th century.
25:
7475:
7428:Implementation of Base Conversion
7411:
7198:from the original on 25 June 2013
5763:
4448:
3991:{\displaystyle \langle S\rangle }
2733:we could ultimately serve a base-
2150:
803:(base two) is used in almost all
4423:{\displaystyle \mathbb {Z} _{T}}
4144:{\displaystyle \mathbb {Z} _{S}}
2973:
1200:is usually the number of unique
1125:
1043:
7347:The art of Computer Programming
7272:
7263:
7251:
7239:
7225:, Clarendon Press, p. 38,
7210:
7124:
7098:
7072:
7017:
6969:
6952:
6938:
6635:Babylonian sexagesimal numerals
6370:,0,1} instead of {0,1,2}. The "
6235:Australian Aboriginal languages
6183:counting system. Students from
5987:portion. For example, the mean
5758:
5535:
3065:can be done by a succession of
1602:
1002:History of positional fractions
6869:
6838:
6790:
6781:
6757:
6032:arbitrary-precision arithmetic
5621:Examples are the non-solvable
4359:
4353:
3998:is the group generated by the
2830:
2123:the set of digits {0, 1, ...,
1916:
1890:
1884:
1858:
1852:
1826:
1820:
1794:
1782:
1738:
1627:
1603:
1599:
1580:
1574:
1553:
1547:
1528:
1522:
1503:
1443:
1424:
1418:
1399:
1393:
1374:
1368:
1349:
1275:
1267:
1233:
1225:
1116:
13:
1:
7285:
6416:uses a varying radix, giving
6047:Other bases in human language
6011:(base-2), octal (base-8) and
5798:or 100), the fourth position
3099:the right-most digit in base
7315:Journal of Indian Philosophy
7141:Mathematical Cuneiform Texts
6993:10.1007/978-3-7091-7551-4_13
5998:
5658:{\displaystyle y={\sqrt{x}}}
5581:
5524:
5473:
5417:
5234:
5186:
5128:
5024:
5005:across the repeating block:
4339:has not to be confused with
4265:{\displaystyle \mathbb {R} }
4243:{\displaystyle \mathbb {Q} }
4217:{\displaystyle \mathbb {Q} }
4086:{\displaystyle \mathbb {Z} }
3333:algorithm for positive bases
3236:For example: converting A10B
2058:for this concept, so, for a
1194:mathematical numeral systems
1054:of numbers less than one, a
876:. However, it lacked a real
833:and allows representing any
818:Systems with negative base,
7:
7245:(Mallory & Adams 1997)
6933:Martinus Nijhoff Publishers
6675:Hindu–Arabic numeral system
6645:
6434:
6086:and other civilizations of
5828:(0.1), the second position
5780:Hindu–Arabic numeral system
5711:, numbers which are called
2041:
766:Hindu–Arabic numeral system
10:
7480:
7459:Positional numeral systems
7257:
7178:"Making the Old Way Count"
7138:; Götze, Albrecht (1945),
6909:. Berlin: Springer-Verlag.
6588:
6547:
6478:
6437:
6352:
6034:, and other applications.
5927:, they would have written
5812:values are indicated by a
5767:
5611:
5349:is also a prime factor of
5169:or, with the base implied:
5068:repeating decimal notation
4250:, namely the real numbers
2951:
2834:
1460:In standard base-sixteen (
1188:Base of the numeral system
1040:'s work "Arithmetic Key".
1005:
910:Greek numerals § Zero
844:
457:Non-standard radices/bases
29:
7327:10.1007/s10781-007-9025-5
6803:15 September 2008 at the
6422:Chinese remainder theorem
5790:(10), the third position
5786:(1), the second position
3005:The conversion to a base
940:Lower row horizontal form
938:; Upper row vertical form
870:Babylonian numeral system
839:arithmetical computations
793:Babylonian numeral system
758:positional numeral system
7418:Accurate Base Conversion
7217:Barrow, John D. (1992),
6705:
6629:Non-positional positions
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
4575:Infinite representations
2046:When describing base in
6719:Kaplan, Robert (2000).
6655:List of numeral systems
6414:factorial number system
6394:1 in balanced base-3.
6040:list of numeral systems
6028:binary-to-text encoding
5694:{\displaystyle y^{n}=x}
5294:{\displaystyle 0.2_{6}}
5104:{\displaystyle 0.1_{3}}
4376:discrete valuation ring
4297:{\displaystyle S=\{p\}}
2947:
2054:is generally used as a
1058:, is often credited to
1034:Abu'l-Hasan al-Uqlidisi
1014:Abu'l-Hasan al-Uqlidisi
713:List of numeral systems
7364:Ifrah, George (2000).
7063:
7023:The exact size of the
6189:base-20 numeral system
5907:astronomers, who used
5770:Decimal representation
5738:
5737:{\displaystyle \pi ,e}
5695:
5659:
5596:
5540:
5488:
5432:
5295:
5267:
5204:
5161:
5105:
5057:
4991:
4566:
4507:
4487:
4464:
4424:
4395:
4368:
4333:
4298:
4266:
4244:
4218:
4189:
4165:
4145:
4108:
4087:
4061:
4018:
4017:{\displaystyle p\in S}
3992:
3963:
3865:
3765:
3705:
3650:
3596:
3572:
3493:
3320:non-terminating digits
3294:
3267:
3219:
3186:
3185:{\displaystyle b_{2},}
3156:
3155:{\displaystyle b_{2};}
3120:
3093:
3092:{\displaystyle b_{2}:}
3059:
3026:
2938:
2852:use places beyond the
2805:
2778:
2758:
2507:
2377:
2258:
2028:
1971:
1923:
1722:
1637:
1450:
1310:In standard base-ten (
1283:
1241:
1066:; but both Stevin and
1052:decimal representation
941:
857:
42:
7464:Mathematical notation
7294:"Babylonian Numerals"
7269:Ifrah, pages 326, 379
7136:Sachs, Abraham Joseph
7085:mathworld.wolfram.com
7064:
6947:mathematical sciences
6935:, Dutch original 1943
6899:B. L. van der Waerden
6426:residue number system
6286:) was devised by the
6030:, implementations of
5806:or 1000), and so on.
5739:
5696:
5660:
5597:
5541:
5489:
5433:
5296:
5268:
5205:
5162:
5106:
5058:
4992:
4567:
4508:
4488:
4465:
4425:
4396:
4369:
4334:
4299:
4267:
4245:
4219:
4190:
4166:
4146:
4109:
4088:
4062:
4019:
3993:
3964:
3845:
3766:
3706:
3651:
3597:
3573:
3494:
3363:Terminating fractions
3295:
3293:{\displaystyle b_{1}}
3268:
3266:{\displaystyle b_{2}}
3220:
3218:{\displaystyle b_{2}}
3187:
3157:
3121:
3119:{\displaystyle b_{2}}
3094:
3060:
3058:{\displaystyle b_{1}}
3027:
3025:{\displaystyle b_{2}}
2939:
2848:Numbers that are not
2806:
2804:{\displaystyle r^{d}}
2779:
2764:digit number in base
2759:
2508:
2378:
2259:
2133:{0, 1, 2, ..., 8, 9};
2048:mathematical notation
2029:
1972:
1924:
1723:
1638:
1451:
1284:
1242:
1240:{\displaystyle r=|b|}
1062:through his textbook
933:
852:
801:binary numeral system
81:Hindu–Arabic numerals
40:
7300:on 11 September 2014
7278:Ifrah, pages 261–264
7185:Sharing Our Pathways
7027:
6438:Decimal equivalents
6292:Proto-Indo Europeans
6263:ukasar-ukasar-ukasar
6259:ukasar-ukasar-urapon
5722:
5672:
5634:
5556:
5499:
5448:
5366:
5341:if and only if each
5278:
5222:
5175:
5116:
5088:
5012:
4595:
4517:
4497:
4477:
4434:
4405:
4401:, which is equal to
4385:
4343:
4308:
4276:
4254:
4232:
4206:
4179:
4155:
4126:
4098:
4075:
4028:
4002:
3976:
3778:
3717:
3661:
3606:
3586:
3506:
3502:More explicitly, if
3377:
3277:
3250:
3240:to decimal (41227):
3202:
3166:
3136:
3103:
3073:
3042:
3038:represented in base
3009:
2871:
2788:
2768:
2748:
2404:
2274:
2197:
2085:representation), 173
1981:
1936:
1735:
1661:
1649:In general, in base-
1471:
1325:
1305:bijective numeration
1263:
1215:
1149:improve this section
1050:The adoption of the
860:Today, the base-10 (
754:place-value notation
610:Prehistoric counting
386:Common radices/bases
68:Place-value notation
7079:Weisstein, Eric W.
6962:usually remove the
6745:on 26 November 2016
6700:Significant figures
6690:Scientific notation
6165:continues to use a
5862:Babylonian numerals
5039:2.42314314314314314
3561:
3530:
3316:implied denominator
3067:Euclidean divisions
2669:" of the numeral 23
2653:is equivalent to 19
2614:Digits and numerals
1282:{\displaystyle |b|}
1208:, the radix is the
967:originate with the
917:sign-value notation
813:electronic circuits
750:Positional notation
557:Sign-value notation
7059:
6925:E. J. Dijksterhuis
6197:display a similar
6122:quatre-vingt-douze
5868:astronomers using
5852:Sexagesimal system
5734:
5715:, or numbers like
5691:
5655:
5608:Irrational numbers
5592:
5536:
5484:
5428:
5291:
5263:
5200:
5157:
5101:
5053:
4987:
4985:
4562:
4503:
4483:
4460:
4420:
4391:
4364:
4329:
4294:
4262:
4240:
4214:
4185:
4161:
4141:
4104:
4083:
4057:
4014:
3988:
3959:
3761:
3701:
3646:
3592:
3568:
3540:
3509:
3489:
3290:
3263:
3215:
3182:
3152:
3116:
3089:
3055:
3022:
2985:. You can help by
2954:Sign (mathematics)
2934:
2801:
2774:
2754:
2741:Sexagesimal system
2503:
2373:
2254:
2024:
1967:
1919:
1718:
1633:
1446:
1279:
1237:
1068:E. J. Dijksterhuis
952:results quickly.
942:
858:
809:electronic devices
213:East Asian systems
43:
6831:978-0-691-11485-9
6626:
6625:
6308:African languages
6282:A base-8 system (
6275:A base-5 system (
6220:The binary system
6210:Te Hokowhitu a Tu
6118:quatre-vingt-deux
6084:Maya civilization
6051:Base-12 systems (
5653:
5614:irrational number
5584:
5527:
5476:
5420:
5237:
5189:
5131:
5075:irrational number
5027:
4506:{\displaystyle c}
4486:{\displaystyle b}
4394:{\displaystyle p}
4188:{\displaystyle b}
4164:{\displaystyle S}
4122:of an element of
4107:{\displaystyle S}
4067:is the so-called
3595:{\displaystyle b}
3413:
3349:repeated squaring
3003:
3002:
2777:{\displaystyle r}
2757:{\displaystyle d}
2678:an ambiguous base
2555:. For example, 10
2119:To a given radix
1185:
1184:
1177:
1036:and 15th century
980:French Revolution
747:
746:
546:
545:
32:positional voting
16:(Redirected from
7471:
7407:
7383:
7371:
7360:
7338:
7321:(5–6): 487–520.
7309:
7307:
7305:
7296:. Archived from
7279:
7276:
7270:
7267:
7261:
7255:
7249:
7243:
7237:
7235:
7224:
7214:
7208:
7207:
7205:
7203:
7197:
7182:
7173:
7167:
7166:
7165:
7163:
7132:Neugebauer, Otto
7128:
7122:
7121:
7119:
7117:
7102:
7096:
7095:
7093:
7091:
7076:
7070:
7068:
7066:
7065:
7060:
7058:
7057:
7039:
7038:
7021:
7015:
7014:
6982:
6973:
6967:
6956:
6950:
6942:
6936:
6922:
6911:
6910:
6908:
6895:
6889:
6888:
6873:
6867:
6857:
6848:
6842:
6836:
6835:
6817:
6808:
6794:
6788:
6785:
6779:
6761:
6755:
6754:
6752:
6750:
6741:. Archived from
6739:"Greek numerals"
6735:
6729:
6728:
6726:
6716:
6665:Related topics:
6543:
6536:
6529:
6523:
6520:
6508:
6496:
6490:
6484:
6479:Balanced base 3
6435:
6408:
6402:
6393:
6385:
6377:
6373:
6369:
6364:Balanced ternary
6343:Papua New Guinea
6339:Telefol language
6328:quinquavigesimal
6225:
6185:Kaktovik, Alaska
5974:
5972:
5968:
5964:
5960:
5956:
5950:
5948:
5944:
5940:
5936:
5932:
5926:
5924:
5920:
5890:
5888:
5884:
5805:
5797:
5743:
5741:
5740:
5735:
5710:
5700:
5698:
5697:
5692:
5684:
5683:
5664:
5662:
5661:
5656:
5654:
5652:
5644:
5601:
5599:
5598:
5593:
5591:
5590:
5585:
5577:
5568:
5567:
5545:
5543:
5542:
5537:
5534:
5533:
5528:
5520:
5511:
5510:
5493:
5491:
5490:
5485:
5483:
5482:
5477:
5469:
5460:
5459:
5437:
5435:
5434:
5429:
5427:
5426:
5421:
5413:
5404:
5403:
5391:
5390:
5378:
5377:
5300:
5298:
5297:
5292:
5290:
5289:
5272:
5270:
5269:
5264:
5262:
5261:
5244:
5243:
5238:
5230:
5209:
5207:
5206:
5201:
5190:
5182:
5166:
5164:
5163:
5158:
5156:
5155:
5138:
5137:
5132:
5124:
5110:
5108:
5107:
5102:
5100:
5099:
5062:
5060:
5059:
5054:
5052:
5051:
5034:
5033:
5028:
5020:
4996:
4994:
4993:
4988:
4986:
4976:
4975:
4954:
4953:
4932:
4931:
4910:
4909:
4888:
4887:
4865:
4860:
4859:
4835:
4834:
4810:
4809:
4785:
4784:
4760:
4755:
4754:
4730:
4729:
4705:
4704:
4680:
4675:
4674:
4650:
4649:
4625:
4620:
4619:
4580:Rational numbers
4571:
4569:
4568:
4563:
4558:
4552:
4551:
4550:
4537:
4531:
4530:
4529:
4512:
4510:
4509:
4504:
4492:
4490:
4489:
4484:
4469:
4467:
4466:
4461:
4447:
4429:
4427:
4426:
4421:
4419:
4418:
4413:
4400:
4398:
4397:
4392:
4373:
4371:
4370:
4365:
4363:
4362:
4351:
4338:
4336:
4335:
4330:
4328:
4327:
4316:
4303:
4301:
4300:
4295:
4271:
4269:
4268:
4263:
4261:
4249:
4247:
4246:
4241:
4239:
4223:
4221:
4220:
4215:
4213:
4201:rational numbers
4199:in the field of
4194:
4192:
4191:
4186:
4170:
4168:
4167:
4162:
4150:
4148:
4147:
4142:
4140:
4139:
4134:
4115:
4113:
4111:
4110:
4105:
4093:with respect to
4092:
4090:
4089:
4084:
4082:
4066:
4064:
4063:
4058:
4056:
4051:
4050:
4042:
4023:
4021:
4020:
4015:
3997:
3995:
3994:
3989:
3968:
3966:
3965:
3960:
3958:
3953:
3952:
3944:
3929:
3923:
3922:
3921:
3908:
3904:
3903:
3900:
3899:
3891:
3890:
3889:
3888:
3878:
3877:
3876:
3864:
3859:
3838:
3830:
3829:
3811:
3792:
3791:
3786:
3770:
3768:
3767:
3762:
3757:
3756:
3738:
3737:
3712:
3710:
3708:
3707:
3702:
3700:
3692:
3691:
3673:
3672:
3655:
3653:
3652:
3647:
3645:
3637:
3636:
3618:
3617:
3602:into the primes
3601:
3599:
3598:
3593:
3577:
3575:
3574:
3569:
3560:
3559:
3558:
3548:
3529:
3528:
3527:
3517:
3498:
3496:
3495:
3490:
3485:
3481:
3480:
3479:
3474:
3459:
3458:
3453:
3438:
3437:
3414:
3412:
3411:
3410:
3409:
3404:
3393:
3392:
3387:
3381:
3330:
3299:
3297:
3296:
3291:
3289:
3288:
3272:
3270:
3269:
3264:
3262:
3261:
3232:
3224:
3222:
3221:
3216:
3214:
3213:
3197:
3191:
3189:
3188:
3183:
3178:
3177:
3161:
3159:
3158:
3153:
3148:
3147:
3131:
3125:
3123:
3122:
3117:
3115:
3114:
3098:
3096:
3095:
3090:
3085:
3084:
3064:
3062:
3061:
3056:
3054:
3053:
3037:
3031:
3029:
3028:
3023:
3021:
3020:
2998:
2995:
2977:
2970:
2943:
2941:
2940:
2935:
2933:
2932:
2911:
2910:
2889:
2888:
2810:
2808:
2807:
2802:
2800:
2799:
2783:
2781:
2780:
2775:
2763:
2761:
2760:
2755:
2727:
2726:
2512:
2510:
2509:
2504:
2460:
2459:
2441:
2440:
2422:
2421:
2382:
2380:
2379:
2374:
2330:
2329:
2311:
2310:
2292:
2291:
2263:
2261:
2260:
2255:
2253:
2252:
2234:
2233:
2215:
2214:
2163:th power, where
2134:
2083:decimal notation
2033:
2031:
2030:
2025:
2023:
2022:
2013:
2012:
2003:
2002:
1993:
1992:
1976:
1974:
1973:
1968:
1957:
1956:
1928:
1926:
1925:
1920:
1915:
1914:
1902:
1901:
1883:
1882:
1870:
1869:
1851:
1850:
1838:
1837:
1819:
1818:
1806:
1805:
1790:
1789:
1780:
1779:
1770:
1769:
1760:
1759:
1750:
1749:
1727:
1725:
1724:
1719:
1708:
1707:
1689:
1688:
1676:
1675:
1642:
1640:
1639:
1634:
1626:
1625:
1624:
1598:
1597:
1573:
1572:
1560:
1546:
1545:
1521:
1520:
1499:
1498:
1497:
1481:
1455:
1453:
1452:
1447:
1442:
1441:
1417:
1416:
1392:
1391:
1367:
1366:
1345:
1344:
1343:
1288:
1286:
1285:
1280:
1278:
1270:
1250:
1246:
1244:
1243:
1238:
1236:
1228:
1199:
1180:
1173:
1169:
1166:
1160:
1129:
1121:
1047:
1038:Jamshīd al-Kāshī
1022:Jamshīd al-Kāshī
1018:Immanuel Bonfils
988:decimal calendar
739:
732:
725:
528:
512:
494:
484:balanced ternary
481:
468:
74:
73:
45:
44:
21:
7479:
7478:
7474:
7473:
7472:
7470:
7469:
7468:
7449:
7448:
7414:
7404:
7388:Kroeber, Alfred
7380:
7357:
7303:
7301:
7288:
7283:
7282:
7277:
7273:
7268:
7264:
7260:, pages 195–213
7256:
7252:
7244:
7240:
7233:
7215:
7211:
7201:
7199:
7195:
7180:
7174:
7170:
7161:
7159:
7152:
7129:
7125:
7115:
7113:
7112:. 10 April 2024
7104:
7103:
7099:
7089:
7087:
7077:
7073:
7053:
7049:
7034:
7030:
7028:
7025:
7024:
7022:
7018:
7003:
6980:
6974:
6970:
6957:
6953:
6943:
6939:
6923:
6914:
6896:
6892:
6874:
6870:
6864:Chinese Science
6858:
6851:
6843:
6839:
6832:
6818:
6811:
6805:Wayback Machine
6795:
6791:
6787:Ifrah, page 187
6786:
6782:
6764:Menninger, Karl
6762:
6758:
6748:
6746:
6737:
6736:
6732:
6717:
6713:
6708:
6648:
6631:
6541:
6534:
6527:
6521:
6518:
6506:
6494:
6488:
6482:
6430:Towers of Hanoi
6406:
6400:
6398:
6391:
6383:
6380:balance problem
6375:
6371:
6367:
6357:
6351:
6223:
6195:Danish numerals
6177:Inuit languages
6170:counting system
6152:ceithre fhichid
6114:soixante-quinze
6049:
6001:
5993:Hebrew calendar
5984:Otto Neugebauer
5970:
5966:
5962:
5958:
5954:
5952:
5946:
5942:
5938:
5934:
5930:
5928:
5922:
5918:
5916:
5886:
5882:
5880:
5854:
5803:
5795:
5772:
5766:
5761:
5723:
5720:
5719:
5702:
5679:
5675:
5673:
5670:
5669:
5648:
5643:
5635:
5632:
5631:
5616:
5610:
5586:
5576:
5575:
5563:
5559:
5557:
5554:
5553:
5529:
5519:
5518:
5506:
5502:
5500:
5497:
5496:
5478:
5468:
5467:
5455:
5451:
5449:
5446:
5445:
5422:
5412:
5411:
5399:
5395:
5386:
5382:
5373:
5369:
5367:
5364:
5363:
5285:
5281:
5279:
5276:
5275:
5257:
5254:
5239:
5229:
5228:
5223:
5220:
5219:
5181:
5176:
5173:
5172:
5151:
5148:
5133:
5123:
5122:
5117:
5114:
5113:
5095:
5091:
5089:
5086:
5085:
5079:rational number
5047:
5044:
5029:
5019:
5018:
5013:
5010:
5009:
4984:
4983:
4968:
4964:
4946:
4942:
4924:
4920:
4902:
4898:
4880:
4876:
4867:
4866:
4864:
4849:
4845:
4824:
4820:
4799:
4795:
4775:
4771:
4762:
4761:
4759:
4744:
4740:
4719:
4715:
4695:
4691:
4682:
4681:
4679:
4664:
4660:
4640:
4636:
4627:
4626:
4624:
4612:
4608:
4598:
4596:
4593:
4592:
4582:
4577:
4554:
4546:
4545:
4541:
4533:
4525:
4524:
4520:
4518:
4515:
4514:
4498:
4495:
4494:
4478:
4475:
4474:
4443:
4435:
4432:
4431:
4414:
4409:
4408:
4406:
4403:
4402:
4386:
4383:
4382:
4352:
4347:
4346:
4344:
4341:
4340:
4317:
4312:
4311:
4309:
4306:
4305:
4277:
4274:
4273:
4257:
4255:
4252:
4251:
4235:
4233:
4230:
4229:
4209:
4207:
4204:
4203:
4180:
4177:
4176:
4156:
4153:
4152:
4135:
4130:
4129:
4127:
4124:
4123:
4099:
4096:
4095:
4094:
4078:
4076:
4073:
4072:
4052:
4043:
4032:
4031:
4029:
4026:
4025:
4003:
4000:
3999:
3977:
3974:
3973:
3954:
3945:
3934:
3933:
3925:
3917:
3916:
3912:
3895:
3884:
3880:
3879:
3872:
3868:
3867:
3866:
3860:
3849:
3834:
3825:
3821:
3816:
3812:
3807:
3800:
3796:
3787:
3782:
3781:
3779:
3776:
3775:
3752:
3748:
3733:
3729:
3718:
3715:
3714:
3696:
3687:
3683:
3668:
3664:
3662:
3659:
3658:
3657:
3656:with exponents
3641:
3632:
3628:
3613:
3609:
3607:
3604:
3603:
3587:
3584:
3583:
3554:
3550:
3549:
3544:
3523:
3519:
3518:
3513:
3507:
3504:
3503:
3475:
3470:
3469:
3454:
3449:
3448:
3430:
3426:
3422:
3418:
3405:
3400:
3399:
3398:
3394:
3388:
3383:
3382:
3380:
3378:
3375:
3374:
3365:
3360:
3356:
3340:Horner's method
3338:Alternatively,
3328:
3304:
3284:
3280:
3278:
3275:
3274:
3257:
3253:
3251:
3248:
3247:
3244:
3239:
3226:
3209:
3205:
3203:
3200:
3199:
3193:
3173:
3169:
3167:
3164:
3163:
3143:
3139:
3137:
3134:
3133:
3127:
3110:
3106:
3104:
3101:
3100:
3080:
3076:
3074:
3071:
3070:
3049:
3045:
3043:
3040:
3039:
3033:
3016:
3012:
3010:
3007:
3006:
2999:
2993:
2990:
2983:needs expansion
2968:
2966:Base conversion
2956:
2950:
2925:
2921:
2903:
2899:
2884:
2880:
2872:
2869:
2868:
2839:
2833:
2795:
2791:
2789:
2786:
2785:
2769:
2766:
2765:
2749:
2746:
2745:
2724:
2722:
2703:
2699:
2695:
2691:
2687:
2683:
2672:
2668:
2664:
2660:
2656:
2652:
2648:
2616:
2603:
2599:
2570:
2566:
2562:
2558:
2546:
2532:
2523:
2519:
2455:
2451:
2436:
2432:
2417:
2413:
2405:
2402:
2401:
2393:
2389:
2325:
2321:
2306:
2302:
2287:
2283:
2275:
2272:
2271:
2248:
2244:
2229:
2225:
2210:
2206:
2198:
2195:
2194:
2153:
2146:
2142:
2138:
2132:
2104:
2096:
2088:
2080:
2076:
2044:
2018:
2014:
2008:
2004:
1998:
1994:
1988:
1984:
1982:
1979:
1978:
1952:
1948:
1937:
1934:
1933:
1910:
1906:
1897:
1893:
1878:
1874:
1865:
1861:
1846:
1842:
1833:
1829:
1814:
1810:
1801:
1797:
1785:
1781:
1775:
1771:
1765:
1761:
1755:
1751:
1745:
1741:
1736:
1733:
1732:
1728:and the number
1703:
1699:
1684:
1680:
1671:
1667:
1662:
1659:
1658:
1614:
1613:
1609:
1593:
1589:
1568:
1564:
1556:
1541:
1537:
1516:
1512:
1487:
1486:
1482:
1477:
1472:
1469:
1468:
1437:
1433:
1412:
1408:
1387:
1383:
1362:
1358:
1333:
1332:
1328:
1326:
1323:
1322:
1318:and the number
1274:
1266:
1264:
1261:
1260:
1248:
1232:
1224:
1216:
1213:
1212:
1197:
1190:
1181:
1170:
1164:
1161:
1146:
1130:
1119:
1048:
1010:
1004:
973:Arabic numerals
969:Brahmi numerals
965:Indian numerals
939:
847:
778:numeral systems
743:
707:
706:
629:
615:Proto-cuneiform
560:
559:
548:
547:
542:
541:
526:
510:
492:
479:
466:
453:
382:
381:
369:
368:
349:
309:
294:
285:
284:
275:
274:
256:
215:
214:
205:
204:
156:
98:
84:
83:
71:
70:
58:Numeral systems
35:
28:
23:
22:
18:Base conversion
15:
12:
11:
5:
7477:
7467:
7466:
7461:
7445:
7444:
7439:
7434:
7425:
7420:
7413:
7412:External links
7410:
7409:
7408:
7402:
7384:
7378:
7361:
7355:
7339:
7310:
7287:
7284:
7281:
7280:
7271:
7262:
7250:
7238:
7231:
7209:
7168:
7150:
7123:
7097:
7071:
7056:
7052:
7048:
7045:
7042:
7037:
7033:
7016:
7001:
6968:
6951:
6937:
6912:
6890:
6868:
6849:
6837:
6830:
6809:
6789:
6780:
6756:
6730:
6710:
6709:
6707:
6704:
6703:
6702:
6693:
6692:
6687:
6682:
6677:
6672:
6663:
6662:
6657:
6647:
6644:
6630:
6627:
6624:
6623:
6620:
6617:
6614:
6611:
6608:
6605:
6602:
6599:
6596:
6594:
6592:
6590:
6586:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6555:
6552:
6549:
6545:
6544:
6538:
6531:
6524:
6515:
6512:
6509:
6503:
6500:
6497:
6492:
6486:
6480:
6476:
6475:
6472:
6469:
6466:
6463:
6460:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6410:
6409:
6404:
6353:Main article:
6350:
6347:
6324:Central Africa
6320:Banda language
6312:Dyola language
6206:Māori language
6163:Welsh language
6136:Irish language
6104:Remnants of a
6093:used base-20 (
6069:Pound Sterling
6048:
6045:
6000:
5997:
5982:In the 1930s,
5870:Greek numerals
5853:
5850:
5849:
5848:
5841:
5840:
5768:Main article:
5765:
5764:Decimal system
5762:
5760:
5757:
5749:transcendental
5745:
5744:
5733:
5730:
5727:
5690:
5687:
5682:
5678:
5666:
5665:
5651:
5647:
5642:
5639:
5612:Main article:
5609:
5606:
5605:
5604:
5603:
5602:
5589:
5583:
5580:
5574:
5571:
5566:
5562:
5551:
5532:
5526:
5523:
5517:
5514:
5509:
5505:
5494:
5481:
5475:
5472:
5466:
5463:
5458:
5454:
5440:
5439:
5438:
5425:
5419:
5416:
5410:
5407:
5402:
5398:
5394:
5389:
5385:
5381:
5376:
5372:
5302:
5301:
5288:
5284:
5273:
5260:
5256:
5253:
5250:
5247:
5242:
5236:
5233:
5227:
5217:
5216:
5215:
5199:
5196:
5193:
5188:
5185:
5180:
5170:
5154:
5150:
5147:
5144:
5141:
5136:
5130:
5127:
5121:
5111:
5098:
5094:
5064:
5063:
5050:
5046:
5043:
5040:
5037:
5032:
5026:
5023:
5017:
4998:
4997:
4982:
4979:
4974:
4971:
4967:
4963:
4960:
4957:
4952:
4949:
4945:
4941:
4938:
4935:
4930:
4927:
4923:
4919:
4916:
4913:
4908:
4905:
4901:
4897:
4894:
4891:
4886:
4883:
4879:
4875:
4872:
4869:
4868:
4863:
4855:
4852:
4848:
4844:
4841:
4838:
4830:
4827:
4823:
4819:
4816:
4813:
4805:
4802:
4798:
4794:
4791:
4788:
4781:
4778:
4774:
4770:
4767:
4764:
4763:
4758:
4750:
4747:
4743:
4739:
4736:
4733:
4725:
4722:
4718:
4714:
4711:
4708:
4701:
4698:
4694:
4690:
4687:
4684:
4683:
4678:
4670:
4667:
4663:
4659:
4656:
4653:
4646:
4643:
4639:
4635:
4632:
4629:
4628:
4623:
4615:
4611:
4607:
4604:
4601:
4600:
4581:
4578:
4576:
4573:
4561:
4557:
4549:
4544:
4540:
4536:
4528:
4523:
4502:
4482:
4459:
4456:
4453:
4450:
4446:
4442:
4439:
4417:
4412:
4390:
4361:
4358:
4355:
4350:
4326:
4323:
4320:
4315:
4293:
4290:
4287:
4284:
4281:
4260:
4238:
4212:
4184:
4160:
4138:
4133:
4103:
4081:
4055:
4049:
4046:
4041:
4038:
4035:
4013:
4010:
4007:
3987:
3984:
3981:
3970:
3969:
3957:
3951:
3948:
3943:
3940:
3937:
3932:
3928:
3920:
3915:
3911:
3907:
3902:
3898:
3894:
3887:
3883:
3875:
3871:
3863:
3858:
3855:
3852:
3848:
3844:
3841:
3837:
3833:
3828:
3824:
3820:
3815:
3810:
3806:
3803:
3799:
3795:
3790:
3785:
3760:
3755:
3751:
3747:
3744:
3741:
3736:
3732:
3728:
3725:
3722:
3699:
3695:
3690:
3686:
3682:
3679:
3676:
3671:
3667:
3644:
3640:
3635:
3631:
3627:
3624:
3621:
3616:
3612:
3591:
3567:
3564:
3557:
3553:
3547:
3543:
3539:
3536:
3533:
3526:
3522:
3516:
3512:
3500:
3499:
3488:
3484:
3478:
3473:
3468:
3465:
3462:
3457:
3452:
3447:
3444:
3441:
3436:
3433:
3429:
3425:
3421:
3417:
3408:
3403:
3397:
3391:
3386:
3364:
3361:
3357:
3353:
3302:
3287:
3283:
3260:
3256:
3242:
3237:
3212:
3208:
3181:
3176:
3172:
3151:
3146:
3142:
3113:
3109:
3088:
3083:
3079:
3052:
3048:
3032:of an integer
3019:
3015:
3001:
3000:
2980:
2978:
2967:
2964:
2952:Main article:
2949:
2946:
2945:
2944:
2931:
2928:
2924:
2920:
2917:
2914:
2909:
2906:
2902:
2898:
2895:
2892:
2887:
2883:
2879:
2876:
2835:Main article:
2832:
2829:
2798:
2794:
2773:
2753:
2701:
2697:
2693:
2689:
2685:
2681:
2670:
2666:
2662:
2658:
2654:
2650:
2646:
2632:decimal digits
2615:
2612:
2600:
2596:
2568:
2564:
2560:
2556:
2542:
2528:
2521:
2517:
2514:
2513:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2458:
2454:
2450:
2447:
2444:
2439:
2435:
2431:
2428:
2425:
2420:
2416:
2412:
2409:
2391:
2387:
2384:
2383:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2328:
2324:
2320:
2317:
2314:
2309:
2305:
2301:
2298:
2295:
2290:
2286:
2282:
2279:
2265:
2264:
2251:
2247:
2243:
2240:
2237:
2232:
2228:
2224:
2221:
2218:
2213:
2209:
2205:
2202:
2157:exponentiation
2152:
2151:Exponentiation
2149:
2144:
2140:
2136:
2102:
2094:
2086:
2078:
2074:
2043:
2040:
2036:multiplication
2021:
2017:
2011:
2007:
2001:
1997:
1991:
1987:
1966:
1963:
1960:
1955:
1951:
1947:
1944:
1941:
1930:
1929:
1918:
1913:
1909:
1905:
1900:
1896:
1892:
1889:
1886:
1881:
1877:
1873:
1868:
1864:
1860:
1857:
1854:
1849:
1845:
1841:
1836:
1832:
1828:
1825:
1822:
1817:
1813:
1809:
1804:
1800:
1796:
1793:
1788:
1784:
1778:
1774:
1768:
1764:
1758:
1754:
1748:
1744:
1740:
1717:
1714:
1711:
1706:
1702:
1698:
1695:
1692:
1687:
1683:
1679:
1674:
1670:
1666:
1644:
1643:
1632:
1629:
1623:
1620:
1617:
1612:
1608:
1605:
1601:
1596:
1592:
1588:
1585:
1582:
1579:
1576:
1571:
1567:
1563:
1559:
1555:
1552:
1549:
1544:
1540:
1536:
1533:
1530:
1527:
1524:
1519:
1515:
1511:
1508:
1505:
1502:
1496:
1493:
1490:
1485:
1480:
1476:
1458:
1457:
1445:
1440:
1436:
1432:
1429:
1426:
1423:
1420:
1415:
1411:
1407:
1404:
1401:
1398:
1395:
1390:
1386:
1382:
1379:
1376:
1373:
1370:
1365:
1361:
1357:
1354:
1351:
1348:
1342:
1339:
1336:
1331:
1316:decimal digits
1277:
1273:
1269:
1235:
1231:
1227:
1223:
1220:
1210:absolute value
1189:
1186:
1183:
1182:
1133:
1131:
1124:
1118:
1115:
1106:
1105:
1070:indicate that
1042:
1030:Sunzi Suanjing
1006:Main article:
1003:
1000:
992:decimalisation
961:Khmer numerals
921:Roman numerals
846:
843:
786:decimal system
782:Roman numerals
774:numeral system
770:decimal system
745:
744:
742:
741:
734:
727:
719:
716:
715:
709:
708:
705:
704:
699:
694:
689:
684:
679:
674:
669:
668:
667:
662:
657:
647:
642:
636:
635:
628:
627:
622:
617:
612:
607:
602:
597:
592:
587:
582:
577:
572:
566:
565:
564:Non-alphabetic
561:
555:
554:
553:
550:
549:
544:
543:
540:
539:
534:
521:
505:
500:
487:
474:
460:
459:
452:
451:
444:
439:
434:
429:
424:
419:
414:
409:
404:
399:
394:
388:
387:
383:
376:
375:
374:
371:
370:
367:
366:
360:
354:
353:
348:
347:
342:
337:
332:
327:
322:
316:
315:
313:Post-classical
308:
307:
301:
300:
293:
292:
286:
282:
281:
280:
277:
276:
273:
272:
267:
261:
260:
255:
254:
249:
244:
239:
234:
233:
232:
221:
220:
216:
212:
211:
210:
207:
206:
203:
202:
197:
192:
187:
182:
177:
172:
167:
162:
155:
154:
149:
144:
139:
134:
129:
124:
119:
114:
109:
104:
97:
96:
94:Eastern Arabic
91:
89:Western Arabic
85:
79:
78:
77:
72:
66:
65:
64:
61:
60:
54:
53:
26:
9:
6:
4:
3:
2:
7476:
7465:
7462:
7460:
7457:
7456:
7454:
7447:
7443:
7440:
7438:
7435:
7433:
7429:
7426:
7424:
7421:
7419:
7416:
7415:
7405:
7403:9780486233680
7399:
7395:
7394:
7389:
7385:
7381:
7379:0-471-37568-3
7375:
7370:
7369:
7362:
7358:
7356:0-201-89684-2
7352:
7348:
7344:
7343:Knuth, Donald
7340:
7336:
7332:
7328:
7324:
7320:
7316:
7311:
7299:
7295:
7290:
7289:
7275:
7266:
7259:
7254:
7248:
7242:
7234:
7232:9780198539568
7228:
7223:
7222:
7213:
7194:
7190:
7186:
7179:
7172:
7157:
7153:
7151:9780940490291
7147:
7143:
7142:
7137:
7133:
7127:
7111:
7107:
7101:
7086:
7082:
7075:
7054:
7050:
7046:
7043:
7040:
7035:
7031:
7020:
7012:
7008:
7004:
7002:3-211-81776-X
6998:
6994:
6990:
6986:
6979:
6972:
6965:
6961:
6955:
6948:
6941:
6934:
6930:
6926:
6921:
6919:
6917:
6907:
6906:
6900:
6894:
6886:
6882:
6878:
6877:Lay Yong, Lam
6872:
6865:
6861:
6856:
6854:
6846:
6841:
6833:
6827:
6823:
6816:
6814:
6806:
6802:
6799:
6793:
6784:
6778:, pp. 150–153
6777:
6776:3-525-40725-4
6773:
6769:
6765:
6760:
6744:
6740:
6734:
6725:
6724:
6715:
6711:
6701:
6698:
6697:
6696:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6671:
6668:
6667:
6666:
6661:
6658:
6656:
6653:
6652:
6651:
6643:
6641:
6636:
6621:
6618:
6615:
6612:
6609:
6606:
6603:
6600:
6597:
6595:
6593:
6591:
6587:
6583:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6556:
6553:
6550:
6546:
6539:
6532:
6525:
6516:
6513:
6510:
6504:
6501:
6498:
6493:
6487:
6481:
6477:
6436:
6433:
6431:
6427:
6423:
6419:
6415:
6397:
6396:
6395:
6389:
6381:
6365:
6361:
6356:
6346:
6344:
6340:
6335:
6333:
6329:
6325:
6321:
6317:
6316:Guinea-Bissau
6313:
6309:
6303:
6301:
6297:
6293:
6289:
6285:
6280:
6278:
6273:
6271:
6266:
6264:
6260:
6256:
6255:ukasar-ukasar
6252:
6251:ukasar-urapon
6248:
6244:
6240:
6239:Kala Lagaw Ya
6236:
6231:
6229:
6221:
6217:
6215:
6214:Tama-hokotahi
6211:
6207:
6202:
6200:
6196:
6192:
6190:
6186:
6182:
6178:
6173:
6171:
6168:
6164:
6159:
6157:
6153:
6149:
6145:
6141:
6137:
6132:
6130:
6125:
6123:
6119:
6115:
6111:
6110:soixante-cinq
6107:
6102:
6100:
6096:
6092:
6089:
6088:pre-Columbian
6085:
6080:
6078:
6074:
6070:
6066:
6062:
6058:
6054:
6044:
6043:
6041:
6035:
6033:
6029:
6024:
6021:
6016:
6014:
6010:
6006:
5996:
5994:
5990:
5989:synodic month
5985:
5980:
5976:
5914:
5910:
5906:
5902:
5898:
5894:
5878:
5873:
5871:
5867:
5863:
5859:
5846:
5845:
5844:
5838:
5837:
5836:
5833:
5831:
5827:
5823:
5819:
5815:
5811:
5807:
5801:
5793:
5789:
5785:
5781:
5777:
5771:
5756:
5754:
5750:
5731:
5728:
5725:
5718:
5717:
5716:
5714:
5709:
5705:
5688:
5685:
5680:
5676:
5649:
5645:
5640:
5637:
5630:
5629:
5628:
5627:
5625:
5619:
5615:
5587:
5578:
5572:
5569:
5564:
5560:
5552:
5549:
5530:
5521:
5515:
5512:
5507:
5503:
5495:
5479:
5470:
5464:
5461:
5456:
5452:
5444:
5443:
5441:
5423:
5414:
5408:
5405:
5400:
5396:
5392:
5387:
5383:
5379:
5374:
5370:
5362:
5361:
5359:
5358:
5357:
5354:
5352:
5348:
5344:
5340:
5336:
5332:
5329:
5325:
5321:
5317:
5316:
5311:
5307:
5304:For integers
5286:
5282:
5274:
5258:
5255:
5251:
5248:
5245:
5240:
5231:
5225:
5218:
5213:
5197:
5194:
5191:
5183:
5178:
5171:
5168:
5167:
5152:
5149:
5145:
5142:
5139:
5134:
5125:
5119:
5112:
5096:
5092:
5084:
5083:
5082:
5080:
5076:
5071:
5069:
5048:
5045:
5041:
5038:
5035:
5030:
5021:
5015:
5008:
5007:
5006:
5004:
4980:
4977:
4972:
4969:
4965:
4961:
4958:
4955:
4950:
4947:
4943:
4939:
4936:
4933:
4928:
4925:
4921:
4917:
4914:
4911:
4906:
4903:
4899:
4895:
4892:
4889:
4884:
4881:
4877:
4873:
4870:
4861:
4853:
4850:
4846:
4842:
4839:
4836:
4828:
4825:
4821:
4817:
4814:
4811:
4803:
4800:
4796:
4792:
4789:
4786:
4779:
4776:
4772:
4768:
4765:
4756:
4748:
4745:
4741:
4737:
4734:
4731:
4723:
4720:
4716:
4712:
4709:
4706:
4699:
4696:
4692:
4688:
4685:
4676:
4668:
4665:
4661:
4657:
4654:
4651:
4644:
4641:
4637:
4633:
4630:
4621:
4613:
4609:
4605:
4602:
4591:
4590:
4589:
4587:
4572:
4559:
4542:
4538:
4521:
4500:
4480:
4471:
4454:
4440:
4437:
4415:
4388:
4381:
4377:
4356:
4321:
4288:
4282:
4279:
4227:
4202:
4198:
4182:
4174:
4158:
4136:
4121:
4116:
4101:
4070:
4047:
4044:
4036:
4011:
4008:
4005:
3982:
3949:
3946:
3938:
3930:
3913:
3909:
3905:
3892:
3885:
3881:
3873:
3869:
3861:
3856:
3853:
3850:
3846:
3842:
3839:
3831:
3826:
3822:
3813:
3804:
3801:
3797:
3793:
3788:
3774:
3773:
3772:
3753:
3749:
3745:
3742:
3739:
3734:
3730:
3723:
3720:
3693:
3688:
3684:
3680:
3677:
3674:
3669:
3665:
3638:
3633:
3629:
3625:
3622:
3619:
3614:
3610:
3589:
3581:
3580:factorization
3565:
3562:
3555:
3551:
3545:
3541:
3537:
3534:
3531:
3524:
3520:
3514:
3510:
3486:
3482:
3476:
3466:
3463:
3460:
3455:
3445:
3442:
3439:
3434:
3431:
3427:
3423:
3419:
3415:
3406:
3395:
3389:
3373:
3372:
3371:
3370:
3352:
3350:
3346:
3341:
3336:
3334:
3325:
3321:
3317:
3313:
3309:
3301:
3285:
3281:
3258:
3254:
3241:
3234:
3233:th quotient.
3230:
3210:
3206:
3196:
3179:
3174:
3170:
3149:
3144:
3140:
3130:
3111:
3107:
3086:
3081:
3077:
3068:
3050:
3046:
3036:
3017:
3013:
2997:
2988:
2984:
2981:This section
2979:
2976:
2972:
2971:
2963:
2961:
2955:
2929:
2926:
2922:
2918:
2915:
2912:
2907:
2904:
2900:
2896:
2893:
2890:
2885:
2881:
2877:
2874:
2867:
2866:
2865:
2863:
2860:of the power
2859:
2855:
2851:
2846:
2844:
2838:
2828:
2825:
2821:
2817:
2812:
2796:
2792:
2771:
2751:
2743:
2742:
2736:
2732:
2731:alphanumerics
2728:
2719:
2715:
2709:
2707:
2679:
2674:
2644:
2640:
2635:
2633:
2629:
2625:
2621:
2611:
2608:
2595:
2593:
2589:
2585:
2581:
2577:
2572:
2554:
2550:
2545:
2540:
2537:for any base
2536:
2531:
2525:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2456:
2452:
2448:
2445:
2442:
2437:
2433:
2429:
2426:
2423:
2418:
2414:
2410:
2407:
2400:
2399:
2398:
2395:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2326:
2322:
2318:
2315:
2312:
2307:
2303:
2299:
2296:
2293:
2288:
2284:
2280:
2277:
2270:
2269:
2268:
2249:
2245:
2241:
2238:
2235:
2230:
2226:
2222:
2219:
2216:
2211:
2207:
2203:
2200:
2193:
2192:
2191:
2189:
2184:
2183:is negative.
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2148:
2130:
2126:
2122:
2117:
2115:
2111:
2106:
2100:
2092:
2084:
2072:
2068:
2065:
2061:
2057:
2053:
2050:, the letter
2049:
2039:
2037:
2019:
2015:
2009:
2005:
1999:
1995:
1989:
1985:
1964:
1961:
1958:
1953:
1949:
1945:
1942:
1911:
1907:
1903:
1898:
1894:
1887:
1879:
1875:
1871:
1866:
1862:
1855:
1847:
1843:
1839:
1834:
1830:
1823:
1815:
1811:
1807:
1802:
1798:
1791:
1786:
1776:
1772:
1766:
1762:
1756:
1752:
1746:
1742:
1731:
1730:
1729:
1715:
1712:
1704:
1700:
1696:
1693:
1690:
1685:
1681:
1677:
1672:
1668:
1656:
1652:
1647:
1630:
1610:
1606:
1594:
1590:
1586:
1583:
1577:
1569:
1565:
1561:
1550:
1542:
1538:
1534:
1531:
1525:
1517:
1513:
1509:
1506:
1500:
1483:
1474:
1467:
1466:
1465:
1463:
1438:
1434:
1430:
1427:
1421:
1413:
1409:
1405:
1402:
1396:
1388:
1384:
1380:
1377:
1371:
1363:
1359:
1355:
1352:
1346:
1329:
1321:
1320:
1319:
1317:
1313:
1308:
1306:
1302:
1297:
1296:in its size.
1295:
1290:
1271:
1256:
1252:
1229:
1221:
1218:
1211:
1207:
1206:negative base
1203:
1195:
1179:
1176:
1168:
1158:
1154:
1150:
1144:
1143:
1139:
1134:This section
1132:
1128:
1123:
1122:
1114:
1111:
1103:
1098:
1094:
1089:
1085:
1081:
1080:
1079:
1077:
1073:
1072:Regiomontanus
1069:
1065:
1061:
1057:
1053:
1046:
1041:
1039:
1035:
1031:
1027:
1023:
1019:
1015:
1009:
999:
997:
993:
989:
985:
981:
976:
974:
970:
966:
962:
958:
953:
950:
946:
945:Counting rods
937:
932:
928:
926:
922:
918:
913:
911:
907:
903:
899:
895:
894:Sand Reckoner
891:
888:The polymath
886:
883:
879:
875:
871:
867:
863:
855:
851:
842:
840:
836:
832:
828:
825:The use of a
823:
821:
816:
814:
810:
806:
802:
798:
794:
789:
787:
783:
779:
775:
771:
767:
763:
759:
755:
751:
740:
735:
733:
728:
726:
721:
720:
718:
717:
714:
711:
710:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
666:
663:
661:
658:
656:
653:
652:
651:
650:Alphasyllabic
648:
646:
643:
641:
638:
637:
634:
631:
630:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
567:
563:
562:
558:
552:
551:
538:
535:
532:
525:
522:
519:
518:
509:
506:
504:
501:
498:
491:
488:
485:
478:
475:
472:
465:
462:
461:
458:
455:
454:
449:
445:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
389:
385:
384:
380:
373:
372:
364:
361:
359:
356:
355:
351:
350:
346:
343:
341:
338:
336:
333:
331:
328:
326:
323:
321:
318:
317:
314:
311:
310:
306:
303:
302:
299:
296:
295:
291:
288:
287:
283:Other systems
279:
278:
271:
268:
266:
265:Counting rods
263:
262:
258:
257:
253:
250:
248:
245:
243:
240:
238:
235:
231:
228:
227:
226:
223:
222:
218:
217:
209:
208:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
157:
153:
150:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
118:
115:
113:
110:
108:
105:
103:
100:
99:
95:
92:
90:
87:
86:
82:
76:
75:
69:
63:
62:
59:
56:
55:
51:
47:
46:
39:
33:
19:
7446:
7432:cut-the-knot
7392:
7367:
7346:
7318:
7314:
7302:. Retrieved
7298:the original
7274:
7265:
7253:
7241:
7220:
7212:
7200:. Retrieved
7191:(1): 12–13.
7188:
7184:
7171:
7162:18 September
7160:, retrieved
7140:
7126:
7114:. Retrieved
7109:
7100:
7088:. Retrieved
7084:
7074:
7019:
6984:
6971:
6963:
6959:
6954:
6940:
6928:
6904:
6893:
6884:
6880:
6871:
6863:
6860:Lam Lay Yong
6840:
6821:
6792:
6783:
6767:
6759:
6747:. Retrieved
6743:the original
6733:
6722:
6714:
6694:
6664:
6649:
6632:
6411:
6387:
6376:
6362:
6358:
6341:, spoken in
6336:
6327:
6304:
6299:
6295:
6281:
6274:
6267:
6262:
6258:
6254:
6250:
6246:
6242:
6233:A number of
6232:
6228:Eye of Horus
6218:
6213:
6209:
6203:
6193:
6174:
6160:
6155:
6151:
6147:
6143:
6139:
6133:
6126:
6121:
6117:
6113:
6109:
6103:
6081:
6072:
6064:
6060:
6050:
6037:
6036:
6025:
6017:
6002:
5981:
5977:
5877:prime symbol
5874:
5855:
5842:
5834:
5808:
5804:10 × 10 × 10
5773:
5759:Applications
5746:
5707:
5703:
5667:
5623:
5620:
5617:
5355:
5350:
5346:
5343:prime factor
5338:
5334:
5330:
5323:
5319:
5314:
5309:
5305:
5303:
5072:
5066:This is the
5065:
4999:
4583:
4472:
4117:
4069:localization
3971:
3501:
3366:
3345:lookup table
3337:
3305:
3245:
3235:
3228:
3194:
3128:
3034:
3004:
2991:
2987:adding to it
2982:
2957:
2861:
2857:
2847:
2842:
2840:
2813:
2739:
2734:
2721:
2717:
2713:
2710:
2705:
2675:
2642:
2638:
2636:
2623:
2619:
2617:
2604:
2591:
2587:
2583:
2579:
2575:
2573:
2552:
2548:
2543:
2538:
2534:
2529:
2526:
2515:
2396:
2385:
2266:
2187:
2185:
2180:
2176:
2164:
2160:
2154:
2128:
2124:
2120:
2118:
2113:
2109:
2107:
2070:
2063:
2051:
2045:
1931:
1654:
1653:, there are
1650:
1648:
1645:
1459:
1309:
1303:, including
1299:(In certain
1298:
1291:
1257:
1253:
1247:of the base
1191:
1171:
1162:
1147:Please help
1135:
1107:
1101:
1096:
1092:
1083:
1060:Simon Stevin
1049:
1026:Al Khwarizmi
1011:
984:decimal time
977:
957:rod numerals
954:
943:
936:rod numerals
914:
887:
859:
824:
817:
790:
757:
753:
749:
748:
516:
477:Signed-digit
352:Contemporary
219:Contemporary
67:
7202:27 February
7110:flamath.com
6680:Mixed radix
6640:zero symbol
6201:structure.
6187:invented a
6150:and eighty
6148:trí fhichid
6144:dhá fhichid
6091:Mesoamerica
6013:hexadecimal
5905:Renaissance
5866:Hellenistic
5858:sexagesimal
5753:uncountable
5326:) = 1, the
4120:denominator
2854:radix point
2837:Radix point
2831:Radix point
2637:A non-zero
2607:real number
2169:radix point
2099:hexadecimal
1462:hexadecimal
1294:logarithmic
1117:Mathematics
1088:sexagesimal
996:metrication
902:Hellenistic
882:placeholder
835:real number
827:radix point
655:Akṣarapallī
625:Tally marks
524:Non-integer
7453:Categories
7286:References
7081:"Vinculum"
6887:: 101–108.
6650:Examples:
6589:Factoroid
6418:factorials
6288:Yuki tribe
6270:quaternary
6053:duodecimal
5810:Fractional
5778:(base-10)
5747:which are
5546:(see also
5210:(see also
4513:, we have
4226:completion
3314:it by the
3308:fractional
2994:March 2017
2960:minus sign
2590:groups of
2541:, since 10
1977:Note that
1196:the radix
1165:March 2013
1110:De Thiende
1064:De Thiende
978:After the
919:) such as
898:Carl Gauss
890:Archimedes
780:, such as
692:Glagolitic
665:Kaṭapayādi
633:Alphabetic
537:Asymmetric
379:radix/base
320:Cistercian
305:Babylonian
252:Vietnamese
107:Devanagari
7390:(1976) .
7372:. Wiley.
7304:21 August
7116:22 August
7090:22 August
7051:ν
7044:…
7032:ν
6964:lowercase
6845:Gandz, S.
6156:daoichead
6095:vigesimal
6073:partially
6005:computing
5999:Computing
5818:full stop
5814:separator
5726:π
5713:algebraic
5582:¯
5525:¯
5474:¯
5418:¯
5252:…
5235:¯
5198:…
5195:0.3333333
5187:¯
5146:…
5143:0.3333333
5129:¯
5042:…
5025:¯
4981:⋯
4970:−
4962:×
4948:−
4940:×
4926:−
4918:×
4904:−
4896:×
4882:−
4874:×
4851:−
4843:×
4826:−
4818:×
4801:−
4793:×
4777:−
4769:×
4746:−
4738:×
4721:−
4713:×
4697:−
4689:×
4666:−
4658:×
4642:−
4634:×
4606:×
4539:⊆
4449:∖
4272:. So, if
4045:−
4040:⟩
4034:⟨
4009:∈
3986:⟩
3980:⟨
3947:−
3942:⟩
3936:⟨
3893:∈
3882:μ
3847:∏
3832:∈
3823:μ
3819:∃
3805:∈
3743:…
3694:∈
3685:ν
3678:…
3666:ν
3639:∈
3623:…
3552:ν
3538:⋅
3535:…
3532:⋅
3521:ν
3467:∈
3464:ν
3461:∧
3446:∈
3440:∣
3435:ν
3432:−
2927:−
2919:×
2905:−
2897:×
2878:×
2696:, i.e. 23
2684:, i.e. 23
2657:, i.e. 23
2492:×
2480:×
2468:×
2449:×
2430:×
2411:×
2362:×
2350:×
2338:×
2319:×
2300:×
2281:×
2242:×
2223:×
2204:×
2108:The base
1959:∈
1946::
1940:∀
1904:×
1872:×
1840:×
1808:×
1694:⋯
1587:×
1562:×
1535:×
1510:×
1431:×
1406:×
1381:×
1356:×
1136:does not
831:fractions
805:computers
660:Āryabhaṭa
605:Kharosthi
497:factorial
464:Bijective
365:(Iñupiaq)
195:Sundanese
190:Mongolian
137:Malayalam
7345:(1997).
7335:52885600
7193:Archived
7156:archived
6901:(1985).
6801:Archived
6670:Algorism
6646:See also
6548:Base −2
6334:region.
6191:in 1994
6146:, sixty
6142:, forty
5626:th roots
5548:0.999...
5397:3.460000
5328:fraction
5249:0.010101
5212:0.999...
5003:vinculum
4493:divides
4378:for the
3771:we have
3369:semiring
3312:dividing
3306:For the
3225:of the
2850:integers
2093:) and 7B
2062:system,
2042:Notation
1076:decimals
1056:fraction
994:and the
986:and the
934:Chinese
797:base ten
687:Georgian
677:Cyrillic
645:Armenian
600:Etruscan
595:Egyptian
503:Negative
363:Kaktovik
358:Cherokee
335:Pentadic
259:Historic
242:Japanese
175:Javanese
165:Balinese
152:Dzongkha
117:Gurmukhi
112:Gujarati
50:a series
48:Part of
7011:0728973
6927:(1970)
6807:. 1842.
6695:Other:
6277:quinary
6199:base-20
6181:base-20
6167:base-20
6106:Gaulish
6061:hundred
6057:factors
5971:
5967:
5963:
5959:
5955:
5949:12′′′′′
5947:
5943:
5939:
5935:
5931:
5925:59.392″
5923:
5919:
5913:fourths
5901:seconds
5897:minutes
5893:degrees
5887:
5883:
5820:, or a
5796:10 × 10
5776:decimal
5774:In the
4171:. This
3324:reduced
3322:if the
2639:numeral
2624:numeral
2563:= 3; 10
2559:= 2; 10
2175:) then
2173:integer
2071:decimal
1657:digits
1312:decimal
1157:removed
1142:sources
1008:Decimal
874:base-60
866:fingers
862:decimal
854:Suanpan
845:History
820:complex
764:of the
590:Chuvash
508:Complex
298:Ancient
290:History
237:Hokkien
225:Chinese
170:Burmese
160:Tibetan
147:Kannada
127:Sinhala
102:Bengali
7400:
7376:
7353:
7333:
7229:
7148:
7009:
6999:
6958:We do
6828:
6774:
6749:31 May
6584:11000
6581:11011
6578:11010
6247:ukasar
6243:urapon
6179:use a
6140:fichid
6129:scores
6099:Africa
6071:(GBP)
6009:binary
6007:, the
5945:31′′′′
5909:thirds
4586:series
4374:, the
4224:. Its
3972:where
2816:binary
2643:digits
2628:number
2067:equals
2060:binary
2056:symbol
1202:digits
949:abacus
925:abacus
702:Hebrew
672:Coptic
585:Brahmi
570:Aegean
527:
511:
493:
480:
467:
330:Muisca
270:Tangut
247:Korean
230:Suzhou
142:Telugu
7331:S2CID
7258:Knuth
7196:(PDF)
7181:(PDF)
6981:(PDF)
6706:Notes
6622:1100
6551:1101
6332:Sudan
6300:newo-
6284:octal
6065:gross
6020:octal
5941:23′′′
5822:comma
5668:with
5384:3.460
5312:with
4430:with
4380:prime
4304:then
4197:dense
3578:is a
3327:0b0.0
2820:octal
2700:= 123
2620:digit
2520:= 243
2390:= 465
2091:octal
906:Roman
756:, or
697:Greek
682:Geʽez
640:Abjad
620:Roman
580:Aztec
575:Attic
490:Mixed
448:table
340:Quipu
325:Mayan
180:Khmer
132:Tamil
7398:ISBN
7374:ISBN
7351:ISBN
7306:2010
7227:ISBN
7204:2017
7164:2019
7146:ISBN
7118:2024
7092:2024
6997:ISBN
6826:ISBN
6772:ISBN
6751:2016
6619:1010
6616:1000
6575:101
6572:100
6569:111
6566:110
6424:and
6412:The
6337:The
6296:newm
6204:The
6175:The
6161:The
6134:The
6082:The
6018:The
5937:59′′
5891:(10
5856:The
5701:and
5573:214.
5465:3.45
5453:3.46
5409:3.46
5371:3.46
5308:and
5016:2.42
4173:ring
4118:The
4024:and
3329:0011
2948:Sign
2718:1330
2688:= 11
2661:= 19
2567:= 16
2551:+ 0×
2547:= 1×
2516:(465
2386:(465
2143:, 1A
2127:−2,
1932:has
1611:5305
1330:5305
1140:any
1138:cite
904:and
878:zero
807:and
791:The
768:(or
762:base
752:(or
345:Rumi
200:Thai
122:Odia
7430:at
7323:doi
6989:doi
6960:not
6642:).
6613:210
6610:200
6607:110
6604:100
6557:11
6554:10
6514:11
6511:10
6447:−1
6444:−2
6441:−3
6322:of
6314:of
6230:).
6077:£sd
6003:In
5953:10°
5951:or
5933:25′
5929:10°
5921:25′
5917:10°
5899:59
5895:25
5889:59″
5885:25′
5881:10°
5843:or
5561:220
5345:of
5315:gcd
5283:0.2
5093:0.1
5073:An
5022:314
4473:If
4195:is
4071:of
3582:of
3238:Hex
3231:−1)
3132:by
3069:by
2989:.
2824:Hex
2784:is
2725:999
2723:215
2714:999
2501:243
2371:465
2341:100
2139:, 2
2081:(a
1192:In
1151:by
1084:via
912:).
377:By
185:Lao
7455::
7329:.
7319:35
7317:.
7187:.
7183:.
7154:,
7134:;
7108:.
7083:.
7007:MR
7005:.
6995:.
6931:,
6915:^
6885:38
6883:.
6852:^
6812:^
6766::
6601:10
6563:1
6560:0
6540:10
6537:1
6530:0
6502:1
6499:0
6491:1
6485:0
6474:8
6471:7
6468:6
6465:5
6462:4
6459:3
6456:2
6453:1
6450:0
6399:10
6318:,
6265:.
6261:,
6257:,
6253:,
6249:,
6245:,
6158:.
6101:.
6079:.
5975:.
5973:12
5969:31
5965:23
5961:59
5957:25
5911:,
5830:10
5826:10
5800:10
5792:10
5788:10
5784:10
5706:∉
5531:10
5516:0.
5508:10
5353:.
5322:,
5232:01
5226:0.
5179:0.
5153:10
5135:10
5120:0.
4973:14
4951:13
4929:12
4907:11
4885:10
4588::
4470:.
3794::=
3724::=
3563::=
3416::=
3335:.
2923:10
2901:10
2882:10
2811:.
2735:62
2706:23
2702:10
2698:60
2694:10
2690:10
2682:10
2663:10
2655:10
2618:A
2569:10
2565:16
2533:=
2527:10
2524:)
2522:10
2471:49
2394:)
2392:10
2388:10
2353:10
2323:10
2304:10
2285:10
2105:.
2095:16
2079:10
2038:.
1713:=:
1591:16
1566:16
1539:16
1514:16
1475:14
1435:10
1410:10
1385:10
1360:10
1078::
815:.
442:60
437:20
432:16
427:12
422:10
52:on
7406:.
7382:.
7359:.
7337:.
7325::
7308:.
7236:.
7206:.
7189:2
7120:.
7094:.
7055:n
7047:,
7041:,
7036:1
7013:.
6991::
6834:.
6753:.
6598:0
6542:1
6535:1
6533:1
6528:1
6526:1
6522:1
6519:1
6517:1
6507:1
6505:1
6495:1
6489:1
6483:1
6405:3
6403:1
6401:1
6392:1
6388:W
6384:1
6372:1
6368:1
6042:.
5802:(
5794:(
5732:e
5729:,
5708:Q
5704:y
5689:x
5686:=
5681:n
5677:y
5650:n
5646:x
5641:=
5638:y
5624:n
5588:5
5579:4
5570:=
5565:5
5550:)
5522:9
5513:=
5504:1
5480:7
5471:6
5462:=
5457:7
5424:7
5415:0
5406:=
5401:7
5393:=
5388:7
5380:=
5375:7
5351:b
5347:q
5339:b
5335:q
5333:/
5331:p
5324:q
5320:p
5318:(
5310:q
5306:p
5287:6
5259:2
5246:=
5241:2
5214:)
5192:=
5184:3
5140:=
5126:3
5097:3
5049:5
5036:=
5031:5
4978:+
4966:3
4959:2
4956:+
4944:3
4937:1
4934:+
4922:3
4915:1
4912:+
4900:3
4893:1
4890:+
4878:3
4871:1
4862:+
4854:9
4847:3
4840:2
4837:+
4829:8
4822:3
4815:1
4812:+
4804:7
4797:3
4790:1
4787:+
4780:6
4773:3
4766:1
4757:+
4749:5
4742:3
4735:2
4732:+
4724:4
4717:3
4710:1
4707:+
4700:3
4693:3
4686:1
4677:+
4669:2
4662:3
4655:2
4652:+
4645:1
4638:3
4631:1
4622:+
4614:0
4610:3
4603:1
4560:.
4556:Z
4548:Z
4543:c
4535:Z
4527:Z
4522:b
4501:c
4481:b
4458:}
4455:p
4452:{
4445:P
4441:=
4438:T
4416:T
4411:Z
4389:p
4360:)
4357:p
4354:(
4349:Z
4325:}
4322:p
4319:{
4314:Z
4292:}
4289:p
4286:{
4283:=
4280:S
4259:R
4237:Q
4211:Q
4183:b
4159:S
4137:S
4132:Z
4114:.
4102:S
4080:Z
4054:Z
4048:1
4037:S
4012:S
4006:p
3983:S
3956:Z
3950:1
3939:S
3931:=
3927:Z
3919:Z
3914:b
3910:=
3906:}
3897:Z
3886:i
3874:i
3870:p
3862:n
3857:1
3854:=
3851:i
3843:x
3840::
3836:Z
3827:i
3814:|
3809:Q
3802:x
3798:{
3789:S
3784:Z
3759:}
3754:n
3750:p
3746:,
3740:,
3735:1
3731:p
3727:{
3721:S
3711:,
3698:N
3689:n
3681:,
3675:,
3670:1
3643:P
3634:n
3630:p
3626:,
3620:,
3615:1
3611:p
3590:b
3566:b
3556:n
3546:n
3542:p
3525:1
3515:1
3511:p
3487:.
3483:}
3477:0
3472:N
3456:0
3451:N
3443:m
3428:b
3424:m
3420:{
3407:0
3402:N
3396:b
3390:0
3385:N
3286:1
3282:b
3259:2
3255:b
3229:k
3227:(
3211:2
3207:b
3195:k
3180:,
3175:2
3171:b
3150:;
3145:2
3141:b
3129:n
3112:2
3108:b
3087::
3082:2
3078:b
3051:1
3047:b
3035:n
3018:2
3014:b
2996:)
2992:(
2930:2
2916:5
2913:+
2908:1
2894:3
2891:+
2886:0
2875:2
2862:b
2858:n
2843:b
2797:d
2793:r
2772:r
2752:d
2686:4
2671:8
2667:8
2659:8
2651:8
2647:8
2592:b
2588:b
2584:b
2580:b
2576:b
2561:3
2557:2
2553:b
2549:b
2544:b
2539:b
2535:b
2530:b
2518:7
2498:=
2495:1
2489:5
2486:+
2483:7
2477:6
2474:+
2465:4
2462:=
2457:0
2453:7
2446:5
2443:+
2438:1
2434:7
2427:6
2424:+
2419:2
2415:7
2408:4
2368:=
2365:1
2359:5
2356:+
2347:6
2344:+
2335:4
2332:=
2327:0
2316:5
2313:+
2308:1
2297:6
2294:+
2289:2
2278:4
2250:0
2246:b
2239:5
2236:+
2231:1
2227:b
2220:6
2217:+
2212:2
2208:b
2201:4
2188:b
2181:n
2177:n
2165:n
2161:n
2145:9
2141:2
2137:2
2129:b
2125:b
2121:b
2114:b
2110:b
2103:2
2097:(
2089:(
2087:8
2075:2
2064:b
2052:b
2020:0
2016:a
2010:1
2006:a
2000:2
1996:a
1990:3
1986:a
1965:.
1962:D
1954:k
1950:a
1943:k
1917:)
1912:0
1908:b
1899:0
1895:a
1891:(
1888:+
1885:)
1880:1
1876:b
1867:1
1863:a
1859:(
1856:+
1853:)
1848:2
1844:b
1835:2
1831:a
1827:(
1824:+
1821:)
1816:3
1812:b
1803:3
1799:a
1795:(
1792:=
1787:b
1783:)
1777:0
1773:a
1767:1
1763:a
1757:2
1753:a
1747:3
1743:a
1739:(
1716:D
1710:}
1705:b
1701:d
1697:,
1691:,
1686:2
1682:d
1678:,
1673:1
1669:d
1665:{
1655:b
1651:b
1631:,
1628:)
1622:c
1619:e
1616:d
1607:=
1604:(
1600:)
1595:0
1584:9
1581:(
1578:+
1575:)
1570:1
1558:B
1554:(
1551:+
1548:)
1543:2
1532:4
1529:(
1526:+
1523:)
1518:3
1507:1
1504:(
1501:=
1495:x
1492:e
1489:h
1484:9
1479:B
1456:.
1444:)
1439:0
1428:5
1425:(
1422:+
1419:)
1414:1
1403:0
1400:(
1397:+
1394:)
1389:2
1378:3
1375:(
1372:+
1369:)
1364:3
1353:5
1350:(
1347:=
1341:c
1338:e
1335:d
1276:|
1272:b
1268:|
1249:b
1234:|
1230:b
1226:|
1222:=
1219:r
1198:r
1178:)
1172:(
1167:)
1163:(
1159:.
1145:.
1102:R
1097:n
1093:R
738:e
731:t
724:v
533:)
531:φ
529:(
520:)
517:i
515:2
513:(
499:)
495:(
486:)
482:(
473:)
471:1
469:(
450:)
446:(
417:8
412:6
407:5
402:4
397:3
392:2
34:.
20:)
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