Positional notation

4995: 38: 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.

1045: 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). 2975: 4594: 1127: 3967: 931: 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

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

850: 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. 736: 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

6684: 6354: 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

1300: 456: 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:

7246: 6829: 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.

1148: 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, 7458: 6659: 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

289: 6272:) to represent the four cardinal directions. Mesoamericans tended to add a second base-5 system to create a modified base-20 system. 7401: 7377: 7354: 7230: 7149: 7000: 6775: 2870: 1174: 5115: 6674: 5779: 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

765: 80: 6898: 6234: 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.

7463: 7139: 6031: 1152: 900:

lamented how science might have progressed had Archimedes only made the leap to something akin to the modern decimal system.

523: 7441: 6634: 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})} 729: 304: 6800: 3605: 6763: 822:

base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers.

649: 5174: 659: 6932: 5447: 1660: 476: 5555: 2814:

The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). In

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

4433: 3348: 1033: 1013: 536: 7297: 1020:

used decimal fractions around 1350, but did not develop any notation to represent them. The Persian mathematician

7192: 6421: 6269: 1935: 632: 401: 3716: 1037: 1021: 7026: 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 1137: 1012:

J. Lennart Berggren notes that positional decimal fractions were used for the first time by Arab mathematician

722: 49: 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 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

6654: 6413: 6039: 6027: 5313: 4375: 1204:, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a 1156: 1141: 947:

and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or

712: 496: 93: 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.

5769: 5327: 4307: 4119: 4068: 3315: 3307: 2704:. The numeral "23" then, in this case, corresponds to the set of base-10 numbers {11, 13, 15, 17, 19, 21, 2066: 1980: 1055: 1051: 881: 830: 396: 312: 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})} 6425: 6219: 6008: 5748: 4342: 4225: 2815: 2059: 2047: 800: 514: 6977: 3975: 4404: 4125: 6946: 6379: 4585: 3323: 2073:

subscript after the number that is being represented (this notation is used in this article). 1111011

1304: 897: 609: 470: 463: 344: 6699: 6689: 6291: 5861: 5821: 916: 869: 792: 691: 556: 507: 319: 251: 106: 37: 5633: 4253: 4231: 4205: 4074: 7330: 6966:

digits "l" and lowercase "o", for in most fonts they are discernible from the digits "1" and "0".

6924: 4172: 3066: 2953: 2376:{\displaystyle 4\times 10^{2}+6\times 10^{1}+5\times 10^{0}=4\times 100+6\times 10+5\times 1=465} 1067: 901: 812: 604: 357: 194: 189: 136: 3339: 1104:/10, Regiomontanus may be called an anticipator of the doctrine of decimal positional fractions. 5782:, each position starting from the right is a higher power of 10. The first position represents 7397: 7373: 7350: 7226: 7145: 6996: 6825: 6771: 6307: 6205: 6188: 5813: 5618:

A (real) irrational number has an infinite non-repeating representation in all integer bases.

5613: 5074: 5067: 5002: 3332: 3319: 979: 893: 808: 686: 676: 664: 644: 599: 594: 530: 362: 334: 241: 174: 164: 151: 116: 111: 31: 6428:

enumerations. This system effectively enumerates permutations. A derivative of this uses the

5671: 5277: 5087: 4275: 2147:. (In all cases, one or more digits is not in the set of allowed digits for the given base.) 963:, showing possible usages of positional-numbers in the 7th century. Khmer numerals and other 7322: 6988: 6374:" has an equivalent value of −1. The negation of a number is easily formed by switching the 6363: 6342: 6338: 6184: 6105: 5721: 5712: 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} 2082: 1017: 987: 589: 483: 236: 224: 169: 159: 126: 101: 7010: 3276: 3249: 3201: 3102: 3041: 3008: 2787: 1214: 7427: 7131: 7006: 6804: 6429: 6319: 6194: 6176: 5992: 5983: 5078: 4200: 2959: 2631: 1201: 972: 968: 964: 701: 671: 614: 584: 569: 329: 297: 269: 246: 229: 88: 3310:

part, conversion can be done by taking digits after the radix point (the numerator), and

1262: 681: 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:

1044: 7387: 6983:. In Buchberger, Bruno; Collins, George Edwin; Loos, Rüdiger; Albrecht, Rudolf (eds.). 6903: 6639: 6323: 6311: 6302:, suggesting that the number 9 had been recently invented and called the "new number". 6169: 6162: 6135: 6068: 5912: 5900: 5896: 5892: 5876: 5869: 4565:{\displaystyle b^{\mathbb {Z} }\,\mathbb {Z} \subseteq c^{\mathbb {Z} }\,\mathbb {Z} .} 4496: 4476: 4384: 4178: 4154: 4097: 3585: 2767: 2747: 2156: 2035: 1209: 1193: 1029: 991: 960: 920: 909: 819: 781: 777: 773: 696: 639: 619: 574: 447: 179: 146: 131: 57: 6208:

of New Zealand also has evidence of an underlying base-20 system as seen in the terms

2974: 2856:. For every position behind this point (and thus after the units digit), the exponent 7452: 7135: 6844: 6721: 6315: 6238: 6087: 6083: 5988: 5908: 5799: 3579: 3311: 2864:

decreases by 1 and the power approaches 0. For example, the number 2.35 is equal to:

2730: 2171:. If a given digit is on the left hand side of the radix point (i.e. its value is an 1315: 1205: 1071: 944: 502: 391: 324: 264: 199: 141: 121: 7334: 7219: 7431: 7342: 6876: 6859: 6227: 5342: 4379: 3344: 1059: 1025: 983: 956: 935: 905: 654: 579: 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: 7391: 7366: 6992: 6679: 6386:, with 1 if used on the empty pan, and with 0 if not used. If an unknown weight 6128: 6090: 6012: 5904: 5865: 5857: 5752: 3162:

the second right-most digit is the remainder of the division of the quotient by

2853: 2836: 2823: 2606: 2168: 2098: 1461: 1126: 1087: 995: 834: 826: 624: 489: 441: 431: 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

998:

of weights and measures—spread widely out of France to almost the whole world.

923:

were used, and accountants in ancient Rome and during the Middle Ages used the

7326: 7080: 6797: 6287: 6052: 1109: 1063: 889: 838: 426: 184: 5077:

has an infinite non-repeating representation in all integer bases. Whether a

1307:, the definition of the base or the allowed digits deviates from the above.) 7313:

Kadvany, John (December 2007). "Positional Value and Linguistic Recursion".

6822:

The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook

6417: 6198: 6166: 6094: 6004: 5817: 4196: 1293: 436: 17: 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

930: 6669: 5622: 5547: 5211: 4599: 3368: 3318:

in the target radix. Approximation may be needed due to a possibility of

1464:), there are the sixteen hexadecimal digits (0–9 and A–F) and the number 1289:

unique digits, numbers may have many different possible representations.

892:(ca. 287–212 BC) invented a decimal positional system based on 10 in his 804: 796: 6862:, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", 6276: 6180: 6056: 5860:

or base-60 system was used for the integral and fractional portions of

5809: 5775: 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}} 2849: 2172: 1311: 1075: 1007: 955:

The oldest extant positional notation system is either that of Chinese

873: 861: 853: 785: 769: 421: 406: 3571:{\displaystyle p_{1}^{\nu _{1}}\cdot \ldots \cdot p_{n}^{\nu _{n}}:=b} 6098: 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

2627: 2055: 948: 924: 915:

Before positional notation became standard, simple additive systems (

908:

astronomers used a base-60 system based on the Babylonian model (see

865: 411: 7436: 6879:. "A Chinese Genesis, Rewriting the history of our numeral system". 6076: 849: 7423:

The Development of Hindu Arabic and Traditional Chinese Arithmetics

6331: 6283: 6154:. A remnant of this system may be seen in the modern word for 40, 6019: 5829: 5825: 2819: 2090: 1024:

made the same discovery of decimal fractions in the 15th century.

929: 848: 761: 416: 378: 339: 6796:

L. F. Menabrea. Translated by Ada Augusta, Countess of Lovelace.

864:) system, which is presumably motivated by counting with the ten 5791: 5787: 5783: 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

877: 6237:

employ binary or binary-like counting systems. For example, in

6212:

referring to a war party (literally "the seven 20s of Tu") and

5835:

As an example, the number 2674 in a base-10 numeral system is:

6820:

Berggren, J. Lennart (2007). "Mathematics in Medieval Islam".

4151:

contains if reduced to lowest terms only prime factors out of

2969: 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

1120: 990:—were unsuccessful. Other French pro-decimal efforts—currency 6798:"Sketch of The Analytical Engine Invented by Charles Babbage" 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

1108:

In the estimation of Dijksterhuis, "after the publication of

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

5160:{\displaystyle 0.{\overline {3}}_{10}=0.3333333\dots _{10}} 3901: 1082:

European mathematicians, when taking over from the Hindus,

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:

2986: 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 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

7029: 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: 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

6434: 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: 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 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 ( 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 4463:{\displaystyle T=\mathbb {P} \setminus \{p\}} 730: 8: 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. 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: 73: 44: 7442:Online Arbitrary Precision Base Converter 7062:{\displaystyle \nu _{1},\ldots ,\nu _{n}} 7053: 7034: 7028: 5839:(2 × 10) + (6 × 10) + (7 × 10) + (4 × 10) 5723: 5679: 5673: 5648: 5643: 5635: 5586: 5576: 5563: 5557: 5529: 5519: 5506: 5500: 5478: 5468: 5455: 5449: 5422: 5412: 5399: 5386: 5373: 5367: 5285: 5279: 5257: 5239: 5229: 5223: 5181: 5176: 5151: 5133: 5123: 5117: 5095: 5089: 5047: 5029: 5019: 5013: 4968: 4946: 4924: 4902: 4880: 4864: 4858: 4857: 4856: 4849: 4833: 4832: 4831: 4824: 4808: 4807: 4806: 4799: 4783: 4782: 4775: 4759: 4753: 4752: 4751: 4744: 4728: 4727: 4726: 4719: 4703: 4702: 4695: 4679: 4673: 4672: 4671: 4664: 4648: 4647: 4640: 4624: 4618: 4617: 4616: 4612: 4598: 4596: 4555: 4554: 4553: 4547: 4546: 4545: 4534: 4533: 4532: 4526: 4525: 4524: 4518: 4498: 4478: 4444: 4443: 4435: 4414: 4410: 4409: 4406: 4386: 4352: 4348: 4347: 4344: 4317: 4313: 4312: 4309: 4277: 4258: 4257: 4255: 4236: 4235: 4233: 4210: 4209: 4207: 4180: 4156: 4135: 4131: 4130: 4127: 4099: 4079: 4078: 4076: 4053: 4052: 4043: 4032: 4029: 4003: 3977: 3955: 3954: 3945: 3934: 3926: 3925: 3924: 3918: 3917: 3916: 3896: 3895: 3884: 3879: 3872: 3867: 3860: 3849: 3835: 3834: 3825: 3817: 3808: 3807: 3787: 3783: 3782: 3779: 3752: 3733: 3718: 3697: 3696: 3687: 3668: 3662: 3642: 3641: 3632: 3613: 3607: 3587: 3554: 3549: 3544: 3523: 3518: 3513: 3507: 3475: 3471: 3470: 3454: 3450: 3449: 3430: 3405: 3401: 3400: 3398: 3388: 3384: 3383: 3380: 3378: 3284: 3278: 3257: 3251: 3209: 3203: 3173: 3167: 3143: 3137: 3110: 3104: 3080: 3074: 3049: 3043: 3016: 3010: 2925: 2903: 2884: 2872: 2795: 2789: 2769: 2749: 2729:. If we use the entire collection of our 2455: 2436: 2417: 2405: 2325: 2306: 2287: 2275: 2248: 2229: 2210: 2198: 2018: 2008: 1998: 1988: 1982: 1952: 1937: 1910: 1897: 1878: 1865: 1846: 1833: 1814: 1801: 1785: 1775: 1765: 1755: 1745: 1736: 1703: 1684: 1671: 1662: 1614: 1613: 1593: 1568: 1556: 1541: 1516: 1487: 1486: 1477: 1472: 1437: 1412: 1387: 1362: 1333: 1332: 1326: 1274: 1266: 1264: 1232: 1224: 1216: 1175:Learn how and when to remove this message 36: 6978:"Arithmetic in basic algebraic domains" 6711: 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 4448: 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 56: 6920: 6918: 6916: 3351:for single or sparse digits. Example: 2155:Positional numeral systems work using 7393:Handbook of the Indians of California 7247:Encyclopedia of Indo-European Culture 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. 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 7: 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 2822:numerals, are the eight digits 0–7. 2027:{\displaystyle a_{3}a_{2}a_{1}a_{0}} 1153:adding citations to reliable sources 927:or stone counters to do arithmetic. 896:; 19th century German mathematician 7158:from the original on 1 October 2016 5864:and other Mesopotamian systems, by 5751:. The number of transcendentals is 2676:Imagine the numeral "23" as having 6866:, 1996, p. 38, Kurt Vogel notation 6241:, the numbers one through six are 4367:{\displaystyle \mathbb {Z} _{(p)}} 3818: 1939: 1621: 1618: 1615: 1557: 1494: 1491: 1488: 1478: 1340: 1337: 1334: 975:, recorded from the 10th century. 25: 7428:Implementation of Base Conversion 7198:from the original on 25 June 2013 3991:{\displaystyle \langle S\rangle } 2733:we could ultimately serve a base- 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 7225:, Clarendon Press, p. 38, 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 5535: 3065:can be done by a succession of 1602: 1002:History of positional fractions 6032:arbitrary-precision arithmetic 5621:Examples are the non-solvable 4359: 4353: 3998:is the group generated by the 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: 1: 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 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, 7245:(Mallory & Adams 1997) 6933:Martinus Nijhoff Publishers 6675:Hindu–Arabic numeral system 6086:and other civilizations of 5828:(0.1), the second position 5780:Hindu–Arabic numeral system 5711:, numbers which are called 766:Hindu–Arabic numeral system 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 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), 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\}} 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: 45: 21: 7479: 7478: 7474: 7473: 7472: 7470: 7469: 7468: 7449: 7448: 7414: 7404: 7388:Kroeber, Alfred 7386: 7380: 7363: 7357: 7341: 7312: 7303: 7301: 7291: 7288: 7283: 7282: 7277: 7273: 7268: 7264: 7260:, pages 195–213 7256: 7252: 7244: 7240: 7233: 7216: 7215: 7211: 7201: 7199: 7195: 7180: 7175: 7174: 7170: 7161: 7159: 7152: 7130: 7129: 7125: 7115: 7113: 7112:. 10 April 2024 7104: 7103: 7099: 7089: 7087: 7078: 7077: 7073: 7049: 7030: 7025: 7024: 7022: 7018: 7003: 6980: 6975: 6974: 6970: 6957: 6953: 6943: 6939: 6923: 6914: 6897: 6896: 6892: 6875: 6874: 6870: 6864:Chinese Science 6858: 6851: 6843: 6839: 6832: 6819: 6818: 6811: 6805:Wayback Machine 6795: 6791: 6787:Ifrah, page 187 6786: 6782: 6764:Menninger, Karl 6762: 6758: 6748: 6746: 6737: 6736: 6732: 6718: 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: 5720: 5719: 5702: 5675: 5670: 5669: 5632: 5631: 5616: 5610: 5575: 5559: 5554: 5553: 5518: 5502: 5497: 5496: 5467: 5451: 5446: 5445: 5411: 5395: 5382: 5369: 5364: 5363: 5281: 5276: 5275: 5254: 5228: 5220: 5219: 5173: 5172: 5148: 5122: 5114: 5113: 5091: 5086: 5085: 5079:rational number 5044: 5018: 5010: 5009: 4984: 4983: 4964: 4942: 4920: 4898: 4876: 4867: 4866: 4845: 4820: 4795: 4771: 4762: 4761: 4740: 4715: 4691: 4682: 4681: 4660: 4636: 4627: 4626: 4608: 4593: 4592: 4582: 4577: 4541: 4520: 4515: 4514: 4495: 4494: 4475: 4474: 4432: 4431: 4408: 4403: 4402: 4383: 4382: 4346: 4341: 4340: 4311: 4306: 4305: 4274: 4273: 4252: 4251: 4230: 4229: 4204: 4203: 4177: 4176: 4153: 4152: 4129: 4124: 4123: 4096: 4095: 4094: 4073: 4072: 4031: 4026: 4025: 4000: 3999: 3974: 3973: 3933: 3912: 3880: 3868: 3866: 3821: 3816: 3812: 3800: 3796: 3781: 3776: 3775: 3748: 3729: 3715: 3714: 3683: 3664: 3659: 3658: 3657: 3656:with exponents 3628: 3609: 3604: 3603: 3584: 3583: 3550: 3519: 3504: 3503: 3469: 3448: 3426: 3422: 3418: 3399: 3394: 3382: 3375: 3374: 3365: 3360: 3356: 3340:Horner's method 3338:Alternatively, 3328: 3304: 3280: 3275: 3274: 3253: 3248: 3247: 3244: 3239: 3226: 3205: 3200: 3199: 3193: 3169: 3164: 3163: 3139: 3134: 3133: 3127: 3106: 3101: 3100: 3076: 3071: 3070: 3045: 3040: 3039: 3033: 3012: 3007: 3006: 2999: 2993: 2990: 2983:needs expansion 2968: 2966:Base conversion 2956: 2950: 2921: 2899: 2880: 2869: 2868: 2839: 2833: 2791: 2786: 2785: 2766: 2765: 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: 2451: 2432: 2413: 2402: 2401: 2393: 2389: 2321: 2302: 2283: 2272: 2271: 2244: 2225: 2206: 2195: 2194: 2153: 2146: 2142: 2138: 2132: 2104: 2096: 2088: 2080: 2076: 2044: 2014: 2004: 1994: 1984: 1979: 1978: 1948: 1934: 1933: 1906: 1893: 1874: 1861: 1842: 1829: 1810: 1797: 1781: 1771: 1761: 1751: 1741: 1733: 1732: 1728:and the number 1699: 1680: 1667: 1659: 1658: 1609: 1589: 1564: 1537: 1512: 1482: 1469: 1468: 1433: 1408: 1383: 1358: 1328: 1323: 1322: 1318:and the number 1261: 1260: 1248: 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: 15: 12: 11: 5: 7477: 7475: 7467: 7466: 7461: 7451: 7450: 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: 24: 14: 13: 10: 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: 7411: 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: 7285: 7275: 7272: 7266: 7263: 7259: 7254: 7251: 7248: 7242: 7239: 7234: 7232:9780198539568 7228: 7223: 7222: 7213: 7210: 7194: 7190: 7186: 7179: 7172: 7169: 7157: 7153: 7151:9780940490291 7147: 7143: 7142: 7137: 7133: 7127: 7124: 7111: 7107: 7101: 7098: 7086: 7082: 7075: 7072: 7054: 7050: 7046: 7043: 7040: 7035: 7031: 7020: 7017: 7012: 7008: 7004: 7002:3-211-81776-X 6998: 6994: 6990: 6986: 6979: 6972: 6969: 6965: 6961: 6955: 6952: 6948: 6941: 6938: 6934: 6930: 6926: 6921: 6919: 6917: 6913: 6907: 6906: 6900: 6894: 6891: 6886: 6882: 6878: 6877:Lay Yong, Lam 6872: 6869: 6865: 6861: 6856: 6854: 6850: 6846: 6841: 6838: 6833: 6827: 6823: 6816: 6814: 6810: 6806: 6802: 6799: 6793: 6790: 6784: 6781: 6778:, pp. 150–153 6777: 6776:3-525-40725-4 6773: 6769: 6765: 6760: 6757: 6744: 6740: 6734: 6731: 6725: 6724: 6715: 6712: 6705: 6701: 6698: 6697: 6696: 6691: 6688: 6686: 6683: 6681: 6678: 6676: 6673: 6671: 6668: 6667: 6666: 6661: 6658: 6656: 6653: 6652: 6651: 6645: 6643: 6641: 6636: 6628: 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: 6348: 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: 6046: 6044: 6043: 6041: 6035: 6033: 6029: 6024: 6021: 6016: 6014: 6010: 6006: 5998: 5996: 5994: 5990: 5989:synodic month 5985: 5980: 5976: 5914: 5910: 5906: 5902: 5898: 5894: 5878: 5873: 5871: 5867: 5863: 5859: 5851: 5846: 5845: 5844: 5838: 5837: 5836: 5833: 5831: 5827: 5823: 5819: 5815: 5811: 5807: 5801: 5793: 5789: 5785: 5781: 5777: 5771: 5763: 5758: 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: 5607: 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: 4579: 4574: 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: 3362: 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: 2965: 2963: 2961: 2955: 2947: 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: 2830: 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: 2613: 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: 2150: 2148: 2130: 2126: 2122: 2117: 2115: 2111: 2106: 2100: 2092: 2084: 2072: 2068: 2065: 2061: 2057: 2053: 2050:, the letter 2049: 2041: 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: 1187: 1179: 1176: 1168: 1158: 1154: 1150: 1144: 1143: 1139: 1134:This section 1132: 1128: 1123: 1122: 1116: 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: 1001: 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: 844: 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: 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 18:Place-value 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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Knowledge

Positional notation

Index