3964:
4228:
149:
169:
4186:
2298:). The number to be multiplied is the "multiplicand", and the number by which it is multiplied is the "multiplier". Usually, the multiplier is placed first, and the multiplicand is placed second; however, sometimes the first factor is the multiplicand and the second the multiplier. Also, as the result of multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some
185:
6116:
3523:
177:
157:
50:
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2448:
413:
766:
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299:
643:
688:
3517:
528:
5958:
2713:
916:
831:
7507:. Here identity 1 is had, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, zero must be excluded because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1. In this example, an
4265:
The
Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but
2223:
for five times two). This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the
3989:
23958233 Γ 5830 βββββββββββββββ 00000000 ( = 23,958,233 Γ 0) 71874699 ( = 23,958,233 Γ 30) 191665864 ( = 23,958,233 Γ 800) + 119791165 ( = 23,958,233 Γ 5,000) βββββββββββββββ 139676498390 ( = 139,676,498,390 )
7319:
4290:, or the box method, is used in primary schools in England and Wales and in some areas of the United States to help teach an understanding of how multiple digit multiplication works. An example of multiplying 34 by 13 would be to lay the numbers out in a grid as follows:
408:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
8514:
2799:
571:
4799:
given by the superscript. The product is obtained by multiplying together all factors obtained by substituting the multiplication index for an integer between the lower and the upper values (the bounds included) in the expression that follows the product operator.
7640:(3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as
4948:
5352:
2545:
761:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
2927:
9026:
3312:
456:
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Product of 45 and 256. Note the order of the numerals in 45 is reversed down the left column. The carry step of the multiplication can be performed at the final stage of the calculation (in bold), returning the final product of
7451:
4448:
The algorithm, also based on the fast
Fourier transform, is conjectured to be asymptotically optimal. The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than
1553:
are all examples where this can be seen. These more advanced constructs tend to affect the basic properties in their own ways, such as becoming noncommutative in matrices and some forms of vector multiplication or changing the
3638:
5458:
1984:
Historically, in the United
Kingdom and Ireland, the middle dot was sometimes used for the decimal to prevent it from disappearing in the ruled line, and the period/full stop was used for multiplication. However, since the
5099:
2722:
An integer can be either zero, a nonzero natural number, or minus a nonzero natural number. The product of zero and another integer is always zero. The product of two nonzero integers is determined by the product of their
4467:
One can only meaningfully add or subtract quantities of the same type, but quantities of different types can be multiplied or divided without problems. For example, four bags with three marbles each can be thought of as:
4406:
was replaced by a function that increases much slower, though still not constant. In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of
1083:
3750:
1538:. For example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Indeed, multiplication by 3, followed by division by 3, yields the original number. The division of a number other than 0 by itself equals 1.
997:
1313:
7525:
Another fact worth noticing is that the integers under multiplication do not form a groupβeven if zero is excluded. This is easily seen by the nonexistence of an inverse for all elements other than 1 and β1.
3253:
As changing the signs transforms least upper bounds into greatest lower bounds, the simplest way to deal with a multiplication involving one or two negative numbers, is to use the rule of signs described above in
5194:
5595:
2732:
4710:
2489:
The product of two numbers or the multiplication between two numbers can be defined for common special cases: natural numbers, integers, rational numbers, real numbers, complex numbers, and quaternions.
7089:
7130:
8865:{\displaystyle z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}\times b_{2}i)+(b_{1}\times a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.}
7618:
3244:
7781:
1900:
638:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
6741:
6295:
7891:
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941:
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438:
10452:
5212:
2306:. Therefore, in some sources, the term "multiplicand" is regarded as a synonym for "factor". In algebra, a number that is the multiplier of a variable or expression (e.g., the 3 in
1484:
9984:
Pletser, Vladimir (2012-04-04). "Does the
Ishango Bone Indicate Knowledge of the Base 12? An Interpretation of a Prehistoric Discovery, the First Mathematical Tool of Humankind".
2538:
1395:
6213:
3175:
9188:, which for example includes matrix multiplication. A very general, and abstract, concept of multiplication is as the "multiplicatively denoted" (second) binary operation in a
8400:
8236:
7484:
how to extend this to multiplying arbitrary integers, and then arbitrary rational numbers. The product of real numbers is defined in terms of products of rational numbers; see
6642:
4556:
3512:{\displaystyle {\begin{aligned}(a+b\,i)\cdot (c+d\,i)&=a\cdot c+a\cdot d\,i+b\,i\cdot c+b\cdot d\cdot i^{2}\\&=(a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)\,i\end{aligned}}}
3130:
6543:
4585:
523:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
2966:
2461:
defining multiplication is not straightforward and different proposals have been made over the centuries, with competing ideas (e.g. recursive vs. non-recursive definitions).
1525:. For example, multiplying the lengths (in meters or feet) of the two sides of a rectangle gives its area (in square meters or square feet). Such a product is the subject of
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Systematic generalizations of this basic definition define the multiplication of integers (including negative numbers), rational numbers (fractions), and real numbers.
1342:
7529:
Multiplication in group theory is typically notated either by a dot or by juxtaposition (the omission of an operation symbol between elements). So multiplying element
5953:{\displaystyle \prod _{i=-\infty }^{\infty }x_{i}=\left(\lim _{m\to -\infty }\prod _{i=m}^{0}x_{i}\right)\cdot \left(\lim _{n\to \infty }\prod _{i=1}^{n}x_{i}\right),}
3947:
3921:
3304:
2708:{\displaystyle r\cdot s\equiv \sum _{i=1}^{s}r=\underbrace {r+r+\cdots +r} _{s{\text{ times}}}\equiv \sum _{j=1}^{r}s=\underbrace {s+s+\cdots +s} _{r{\text{ times}}}.}
1945:
6838:
Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for
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1711:
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2221:
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911:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}
826:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
10234:
2100:
2037:
8509:
6001:
3997:, the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier:
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2077:
2057:
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9846:
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3649:
2434:. A product of integers is a multiple of each factor; for example, 15 is the product of 3 and 5 and is both a multiple of 3 and a multiple of 5.
1135:
271:
7567:. When referring to a group via the indication of the set and operation, the dot is used. For example, our first example could be indicated by
4476:
When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by
962:
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6862:
9930:
5118:
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and the square root, logarithmic, and trigonometric functions can be followed by their arguments as when working with pencil and paper.
6129:
Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.
4593:
4116:. Thus, Babylonian multiplication was very similar to modern decimal multiplication. Because of the relative difficulty of remembering
4076:, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining
10743:
9747:
3986:
algorithm, does not. The example below illustrates "long multiplication" (the "standard algorithm", "grade-school multiplication"):
7124:
are treated as integers. Thus both (0,1) and (1,2) are equivalent to β1. The multiplication axiom for integers defined this way is
10569:
7314:{\displaystyle (x_{p},\,x_{m})\times (y_{p},\,y_{m})=(x_{p}\times y_{p}+x_{m}\times y_{m},\;x_{p}\times y_{m}+x_{m}\times y_{p}).}
5983:. For instance, the product of three factors of two (2Γ2Γ2) is "two raised to the third power", and is denoted by 2, a two with a
7002:
5995:. In general, the exponent (or superscript) indicates how many times the base appears in the expression, so that the expression
10201:
6308:
Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:
7570:
10526:
10386:
9940:
9886:
4203:
3180:
114:
10226:
7709:
6870:
proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
4011:
allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical
10184:
4000:
23958233 Β· 5830 βββββββββββββββ 119791165 191665864 71874699 00000000 βββββββββββββββ 139676498390
2794:{\displaystyle {\begin{array}{|c|c c|}\hline \times &+&-\\\hline +&+&-\\-&-&+\\\hline \end{array}}}
1855:
1223:
86:
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6988:. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including
3023:
1128:
264:
9470:
10851:
10815:
4372:
have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the
3982:
of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the
534:
93:
10486:
9499:
6313:
3246:
In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the
2469:
1801:
1511:. The area of a rectangle does not depend on which side is measured firstβa consequence of the commutative property.
1089:
133:
9711:
2267:) is still the most common notation. This is due to the fact that most computers historically were limited to small
10280:
9806:
7485:
4943:{\displaystyle \prod _{i=m}^{n}x_{i}=x_{m}\cdot x_{m+1}\cdot x_{m+2}\cdot \,\,\cdots \,\,\cdot x_{n-1}\cdot x_{n},}
4480:. This analysis is routinely applied in physics, but it also has applications in finance and other applied fields.
2978:
1993:(SI) standard has since been widely adopted, this usage is now found only in the more traditional journals such as
650:
9402:
9364:
9234:
7098:
typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (
5347:{\displaystyle \prod _{i=1}^{n}{x_{i}y_{i}}=\left(\prod _{i=1}^{n}x_{i}\right)\left(\prod _{i=1}^{n}y_{i}\right)}
4245:
4067:
3983:
922:
67:
31:
6909:
4721:
2235:, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a
1003:
100:
10938:
10933:
10582:
9824:
8028:
wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.
7788:
7522:) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.
2405:
1121:
419:
257:
71:
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1990:
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Several mathematical concepts expand upon the fundamental idea of multiplication. The product of a sequence,
9903:
4211:
involving place value addition, subtraction, multiplication, and division. The
Chinese were already using a
4207:, multiplication calculations were written out in words, although the early Chinese mathematicians employed
10907:
10902:
9464:
7500:
structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
4212:
4180:
2984:
A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by
1433:
82:
10104:
7473:
2509:
1350:
6386:
The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the
6177:
3135:
2992:
of a set of rational numbers. In particular, every positive real number is the least upper bound of the
1226:; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the
9763:
9605:
With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first)
9021:{\displaystyle z_{1}=r_{1}(\cos \phi _{1}+i\sin \phi _{1}),z_{2}=r_{2}(\cos \phi _{2}+i\sin \phi _{2})}
8365:
8201:
6597:
6131:
Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a
4539:
4287:
4088:. The full product could then be found by adding the appropriate terms found in the doubling sequence:
4073:
4007:
were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The
3093:
1489:
Thus, the designation of multiplier and multiplicand does not affect the result of the multiplication.
10655:
9269:. Multiplication for some types of "numbers" may have corresponding division, without inverses; in an
6498:
4568:
10634:
10128:
Harvey, David; van der Hoeven, Joris; Lecerf, GrΓ©goire (2016). "Even faster integer multiplication".
9529:
9494:
1427:, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
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8004:
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17:
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The geometric meaning of complex multiplication can be understood by rewriting complex numbers in
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1724:
38:
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9647:
Why aren't we using the multiplication sign? | Introduction to algebra | Algebra I | Khan
Academy
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There are many sets that, under the operation of multiplication, satisfy the axioms that define
6103:
are to be multiplied together. This notation can be used whenever multiplication is known to be
4003:
Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone.
2934:
10861:
10825:
10696:
10691:
9585:
9544:
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7518:. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the
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6548:
4781:
4237:
2997:
2351:
2009:
1986:
1534:
1321:
1206:
676:
9196:(polynomials can be added and multiplied, but polynomials are not numbers in any usual sense).
7514:
To see this, consider the set of invertible square matrices of a given dimension over a given
3926:
3900:
3273:
1924:
10897:
10605:
9868:
9549:
6303:
3871:{\displaystyle (a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)i=r\cdot s\cdot e^{i(\varphi +\psi )}.}
3262:
is often preferred in order to avoid consideration of the four possible sign configurations.
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1906:
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The
Egyptian method of multiplication of integers and fractions, which is documented in the
4050:, dated to about 18,000 to 20,000 BC, may hint at a knowledge of multiplication in the
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107:
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A fundamental property of real numbers is that rational approximations are compatible with
2816:
A positive number multiplied by a positive number is positive (product of natural numbers),
2256:
2248:
2240:
1550:
1526:
1424:
9734:
6226:
Expressions solely involving multiplication or addition are invariant with respect to the
6085:{\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}=\prod _{i=1}^{n}a}
3003:
2385:
2162:
2133:
8:
10676:
10080:
9458:
9166:{\textstyle z_{1}z_{2}=r_{1}r_{2}(\cos(\phi _{1}+\phi _{2})+i\sin(\phi _{1}+\phi _{2})).}
8355:{\displaystyle (a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})}
7515:
7497:
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6104:
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4253:
4016:
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The geometric meaning is that the magnitudes are multiplied and the arguments are added.
2303:
2225:
2013:
1978:
1969:, is now standard in the United States and other countries where the period is used as a
1685:
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1152:
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In this case, the hour units cancel out, leaving the product with only kilometer units.
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2019:
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7967:{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}={\frac {(A\times C)}{(B\times D)}}}
7477:
5766:{\displaystyle \prod _{i=m}^{\infty }x_{i}=\lim _{n\to \infty }\prod _{i=m}^{n}x_{i}.}
4587:
is derived from the Greek letter Ξ£ (sigma)). The meaning of this notation is given by
10522:
10515:
10482:
10382:
10167:
10155:
10055:
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These place value decimal arithmetic algorithms were introduced to Arab countries by
4113:
4051:
3531:
2989:
1970:
1917:, multiplication is also denoted by dot signs, usually a middle-position dot (rarely
168:
10305:
7446:{\displaystyle (0,1)\times (0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0).}
2922:{\displaystyle {\frac {z}{n}}\cdot {\frac {z'}{n'}}={\frac {z\cdot z'}{n\cdot n'}},}
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For example: set the monkey's feet to 4 and 9, and get the productβ36βin its hands.
3633:{\displaystyle a+b\,i=r\cdot (\cos(\varphi )+i\sin(\varphi ))=r\cdot e^{i\varphi }}
10560:
4252:. Brahmagupta gave rules for addition, subtraction, multiplication, and division.
4092:
13 Γ 21 = (1 + 4 + 8) Γ 21 = (1 Γ 21) + (4 Γ 21) + (8 Γ 21) = 21 + 84 + 168 = 273.
2834:
Two fractions can be multiplied by multiplying their numerators and denominators:
10871:
10188:
9769:(NB. The TI-88 only existed as a prototype and was never released to the public.)
9620:
Intro to multiplication | Multiplication and division | Arithmetic | Khan
Academy
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do not have an ordering that is compatible with both addition and multiplication.
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10181:
5453:{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{a}=\prod _{i=1}^{n}x_{i}^{a}}
4227:
3270:
Two complex numbers can be multiplied by the distributive law and the fact that
1981:
as a decimal mark, either the period or a middle dot is used for multiplication.
180:
4 Γ 5 = 20. The large rectangle is made up of 20 squares, each 1 unit by 1 unit.
10781:
10752:
8047:
6867:
6831:
6152:
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5109:
5094:{\displaystyle \prod _{i=1}^{n}x_{i}=x_{1}\cdot x_{2}\cdot \ldots \cdot x_{n}.}
4559:
4352:
Multiplication algorithm Β§ Fast multiplication algorithms for large inputs
4216:
4055:
2802:
2724:
2287:), while the asterisk appeared on every keyboard. This usage originated in the
1715:
1546:
1344:
and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
1219:
838:
176:
10151:
10046:
9878:
9645:
9618:
4124:. These tables consisted of a list of the first twenty multiples of a certain
1913:
To reduce confusion between the multiplication sign Γ and the common variable
1078:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
10922:
10701:
10256:
10159:
9712:"The Lancet β Formatting guidelines for electronic submission of manuscripts"
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Many common methods for multiplying numbers using pencil and paper require a
2295:
2268:
2236:
10505:
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The product of a sequence of factors can be written with the product symbol
148:
10564:
10510:
9581:
9511:
6855:
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6748:
4272:
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4185:
4047:
3745:{\displaystyle c+d\,i=s\cdot (\cos(\psi )+i\sin(\psi ))=s\cdot e^{i\psi },}
2977:
There are several equivalent ways to define formally the real numbers; see
1519:
161:
7472:
The product of non-negative integers can be defined with set theory using
5645:
One may also consider products of infinitely many terms; these are called
4023:
and calculators have greatly reduced the need for multiplication by hand.
10856:
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10650:
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are integers or expressions that evaluate to integers. In the case where
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2340:
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1202:
992:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}
444:
10574:
9192:. An example of a ring that is not any of the above number systems is a
7696:
are positive whole numbers. This gives the number of things in an array
4019:, automated multiplication of up to 10-digit numbers. Modern electronic
2981:. The definition of multiplication is a part of all these definitions.
1308:{\displaystyle a\times b=\underbrace {b+\cdots +b} _{a{\text{ times}}}.}
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The numbers to be multiplied are generally called the "factors" (as in
1995:
1974:
1681:
1644:
1194:
184:
164:. Here, 2 is being multiplied by 3 using scaling, giving 6 as a result.
152:
Four bags with three marbles per bag gives twelve marbles (4 Γ 3 = 12).
4275:
in the early 9th century and popularized in the
Western world by
2988:. A standard way for expressing this is that every real number is the
2126:. The notation can also be used for quantities that are surrounded by
30:
This article is about the mathematical operation. For other uses, see
10866:
10791:
10786:
10086:
The Number System of
Algebra β Treated Theoretically and Historically
9696:
9671:
9524:
5977:
When multiplication is repeated, the resulting operation is known as
5189:{\displaystyle \prod _{i=1}^{n}x=x\cdot x\cdot \ldots \cdot x=x^{n}.}
4276:
1918:
1659:
1500:
1028:
4994:
whose value is 1βregardless of the expression for the factors.
4971:, the value of the product is the same as that of the single factor
2243:, yielding a vector as its result, while the dot denotes taking the
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is 0. In this process, information is lost and cannot be regained.
5590:{\displaystyle \prod _{i=1}^{n}x^{a_{i}}=x^{\sum _{i=1}^{n}a_{i}}}
4791:, called the multiplication index, that runs from the lower value
10403:
7893:
is by multiplying the numerators and denominators, respectively:
7481:
7095:
6156:
4788:
4705:{\displaystyle \prod _{i=1}^{4}(i+1)=(1+1)\,(2+1)\,(3+1)\,(4+1),}
3994:
3968:
2288:
2005:
1582:
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6172:
The order in which two numbers are multiplied does not matter:
3082:, and, in particular, with multiplication. This means that, if
2276:
1508:
697:
5657:β. The product of such an infinite sequence is defined as the
2825:
A negative number multiplied by a negative number is positive.
2822:
A negative number multiplied by a positive number is negative,
2819:
A positive number multiplied by a negative number is negative,
1989:
ruled to use the period as the decimal point in 1968, and the
9827:. Algebra, Arithmetic / Ambiguity, PEMDAS. The Math Doctors.
4484:
4031:
Methods of multiplication were documented in the writings of
2272:
10105:"How modern mathematics emerged from a lost Islamic library"
6119:
Multiplication of numbers 0β10. Line labels = multiplicand.
7084:{\displaystyle x\times 1=x\times S(0)=(x\times 0)+x=0+x=x.}
1504:
856:
819:
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577:
462:
305:
10127:
4506:
Other examples of multiplication involving units include:
2727:, combined with the sign derived from the following rule:
4483:
A common example in physics is the fact that multiplying
1973:. When the dot operator character is not accessible, the
8040:
can be defined in terms of sequences of rational numbers
7613:{\displaystyle \left(\mathbb {Q} /\{0\},\,\cdot \right)}
6996:(0), denoted by 1, is a multiplicative identity because
6794:
Multiplication by a negative number reverses the order:
7843:
The same sign rules apply to rational and real numbers.
7704:
high. Generalization to negative numbers can be done by
4120:
different products, Babylonian mathematicians employed
3239:{\displaystyle a\cdot b=\sup _{x\in A,y\in B}x\cdot y.}
1209:. The result of a multiplication operation is called a
9870:
FORTRAN Programming: A Supplement for Calculus Courses
9034:
7776:{\displaystyle N\times (-M)=(-N)\times M=-(N\times M)}
6438:
Any number multiplied by 0 is 0. This is known as the
4572:
4558:, which derives from the capital letter Ξ (pi) in the
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10570:
Modern Chinese Multiplication Techniques on an Abacus
10202:"Mathematicians Discover the Perfect Way to Multiply"
10034:"Ancient times table hidden in Chinese bamboo strips"
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1895:{\displaystyle 2\times 2\times 2\times 2\times 2=32.}
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6736:{\displaystyle x\cdot \left({\frac {1}{x}}\right)=1}
6290:{\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z).}
4997:
2356:
of the other or of the product of the others. Thus,
10434:. UNSW Sydney, School of Mathematics and Statistics
9807:"Implied Multiplication 1: Not as Bad as You Think"
9181:
7886:{\displaystyle {\frac {A}{B}}\times {\frac {C}{D}}}
7324:The rule that β1 Γ β1 = 1 can then be deduced from
3971:dated 1918, used as a multiplication "calculator".
3069:{\displaystyle \{3,\;3.1,\;3.14,\;3.141,\ldots \}.}
74:. Unsourced material may be challenged and removed.
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6754:Multiplication by a positive number preserves the
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4550:
4516:4.5 residents per house Γ 20 houses = 90 residents
4499:50 kilometers per hour Γ 3 hours = 150 kilometers.
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2350:. When one factor is an integer, the product is a
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9809:. Algebra / Ambiguity, PEMDAS. The Math Doctors.
9762:Now, implied multiplication is recognized by the
7491:
7467:
4244:The modern method of multiplication based on the
4159:. Then to compute any sexagesimal product, say 53
3265:
2493:
1977: (Β·) is used. In other countries that use a
1318:For example, 4 multiplied by 3, often written as
10920:
9825:"Implied Multiplication 2: Is There a Standard?"
9361:may be ambiguous in non-commutative rings since
5896:
5833:
5714:
3197:
3146:
3104:
548:{\displaystyle \scriptstyle {\text{difference}}}
10561:Arithmetic Operations In Various Number Systems
9781:"Order of Operations: Implicit Multiplication?"
9231:, is the same as multiplication by an inverse,
7456:Multiplication is extended in a similar way to
6368:{\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z.}
3258:. The construction of the real numbers through
2805:of multiplication over addition, and is not an
1844:{\displaystyle 2\times 3\times 5=6\times 5=30,}
1518:) is a new type of measurement, usually with a
1103:{\displaystyle \scriptstyle {\text{logarithm}}}
10479:Introduction to group theory with applications
9843:"Implied Multiplication 3: You Can't Prove It"
5987:three. In this example, the number two is the
4795:indicated in the subscript to the upper value
2972:
10737:
10590:
9932:Advance Brain Stimulation by Psychoconduction
6863:Arithmetices principia, nova methodo exposita
4510:2.5 meters Γ 4.5 meters = 11.25 square meters
3884:
2457:needs attention from an expert in Mathematics
2279:) that lacked a multiplication sign (such as
1129:
664:{\displaystyle \scriptstyle {\text{product}}}
265:
10541:: CS1 maint: multiple names: authors list (
9798:
9772:
9643:
9616:
9430:{\displaystyle \left({\frac {1}{y}}\right)x}
9392:{\displaystyle x\left({\frac {1}{y}}\right)}
9262:{\displaystyle x\left({\frac {1}{y}}\right)}
7644:) or do not look much like numbers (such as
7624:Multiplication of different kinds of numbers
7595:
7589:
3255:
3060:
3027:
1684:, multiplication is often written using the
39:Interpunct Β§ In mathematics and science
10180:David Harvey, Joris Van Der Hoeven (2019).
5649:. Notationally, this consists in replacing
5104:If all factors are identical, a product of
4803:More generally, the notation is defined as
4456:
2829:
2346:The result of a multiplication is called a
1532:The inverse operation of multiplication is
1507:of a rectangle whose sides have some given
936:{\displaystyle \scriptstyle {\text{power}}}
10744:
10730:
10597:
10583:
7255:
6958:{\displaystyle x\times S(y)=(x\times y)+x}
4770:{\displaystyle \prod _{i=1}^{4}(i+1)=120.}
3050:
3043:
3036:
2717:
1136:
1122:
1017:{\displaystyle \scriptstyle {\text{root}}}
272:
258:
172:Animation for the multiplication 2 Γ 3 = 6
10604:
10182:Integer multiplication in time O(n log n)
10141:
10045:
9989:
9695:
8874:Alternatively, in trigonometric form, if
7834:{\displaystyle (-N)\times (-M)=N\times M}
7601:
7580:
7511:is had, but that is not always the case.
7397:
7183:
7150:
6163:, multiplication has certain properties:
4904:
4903:
4899:
4898:
4683:
4667:
4651:
4532:Iterated binary operation Β§ Notation
4513:11 meters/seconds Γ 9 seconds = 99 meters
3662:
3553:
3501:
3399:
3389:
3354:
3332:
2526:
2427:{\displaystyle 5133\times 486\times \pi }
1495:Multiplication can also be visualized as
1073:
1069:
987:
983:
906:
902:
756:
752:
633:
629:
614:
610:
593:
589:
518:
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499:
495:
478:
474:
433:{\displaystyle \scriptstyle {\text{sum}}}
403:
399:
384:
380:
363:
359:
342:
338:
321:
317:
160:Multiplication can also be thought of as
134:Learn how and when to remove this message
10199:
9840:
9822:
9804:
9778:
9576:
9574:
9572:
7503:A simple example is the set of non-zero
6984:; i.e., the natural number that follows
6114:
4520:
4356:The classical method of multiplying two
4226:
4184:
4026:
3962:
3521:
2497:
1232:, as the quantity of the other one, the
183:
175:
167:
155:
147:
37:"β
" redirects here. For the symbol, see
10381:. Oxford University Press. p. 25.
9983:
4300:
14:
10921:
10022:
9928:
9873:. Universitext. Springer. p. 10.
9866:
9860:
9845:. Algebra / PEMDAS. The Math Doctors.
9783:. Algebra / PEMDAS. The Math Doctors.
9727:
9580:
6151:numbers, which includes, for example,
4525:
4345:
4303:
4297:
4201:, dated prior to 300 BC, and the
2472:may be able to help recruit an expert.
1503:(for whole numbers) or as finding the
10725:
10578:
10476:
10401:
10376:
10372:
10370:
10368:
10366:
10341:
10339:
10337:
10335:
10333:
10331:
10329:
10327:
10325:
10303:
10254:
10227:"Multiplication Hits the Speed Limit"
10224:
9569:
7974:. This gives the area of a rectangle
5467:is a non-negative integer, or if all
4256:, then a professor of mathematics at
4204:Nine Chapters on the Mathematical Art
3526:A complex number in polar coordinates
1479:{\displaystyle 4\times 3=3+3+3+3=12.}
1238:; both numbers can be referred to as
10102:
10079:
6487:β1 times any number is equal to the
5780:with negative infinity, and define:
5634:
3090:are positive real numbers such that
2441:
1514:The product of two measurements (or
72:adding citations to reliable sources
43:
10751:
10028:
9590:Mathematical Association of America
2801:(This rule is a consequence of the
2533:{\displaystyle r,s\in \mathbb {N} }
2506:The product of two natural numbers
1390:{\displaystyle 3\times 4=4+4+4=12.}
24:
10499:
10429:"ORDERING COMPLEX NUMBERS... NOT*"
10426:
10363:
10322:
9906:. Crewton Ramone's House of Math.
9901:
9736:Announcing the TI Programmable 88!
6208:{\displaystyle x\cdot y=y\cdot x.}
5906:
5846:
5809:
5804:
5724:
5695:
3250:of their decimal representations.
3170:{\displaystyle b=\sup _{y\in B}y,}
25:
10950:
10550:
10453:"10.2: Building the Real Numbers"
9586:"What Exactly is Multiplication?"
8395:{\displaystyle a_{1}\times a_{2}}
8231:{\displaystyle z_{1}\times z_{2}}
8106:as ordered pairs of real numbers
6637:{\displaystyle (-1)\cdot (-1)=1.}
5966:
5627:are non-negative integers, or if
4998:Properties of capital pi notation
4551:{\displaystyle \textstyle \prod }
4222:
4143:; followed by the multiples of 10
3125:{\displaystyle a=\sup _{x\in A}x}
1714:) between the terms (that is, in
10281:"Summation and Product Notation"
9490:Booth's multiplication algorithm
8362:. This is the same as for reals
8038:Real numbers and their products
7486:construction of the real numbers
6538:{\displaystyle (-1)\cdot x=(-x)}
4580:{\displaystyle \textstyle \sum }
2979:Construction of the real numbers
2446:
48:
10470:
10445:
10420:
10395:
10297:
10273:
10248:
10237:from the original on 2020-10-31
10218:
10193:
10174:
10121:
10096:
10073:
10062:from the original on 2014-01-22
9998:
9977:
9953:
9935:. Trafford. pp. 2β3, 5β6.
9922:
9910:from the original on 2015-10-26
9895:
9849:from the original on 2023-09-24
9831:from the original on 2023-09-24
9813:from the original on 2023-09-24
9787:from the original on 2023-09-24
9753:from the original on 2017-08-03
9654:from the original on 2017-03-27
9627:from the original on 2017-03-24
9596:from the original on 2017-05-27
4068:Ancient Egyptian multiplication
3893:can be found in the article on
2247:of two vectors, resulting in a
1420:
59:needs additional citations for
32:Multiplication (disambiguation)
10200:Hartnett, Kevin (2019-04-11).
9704:
9664:
9637:
9610:
9354:{\displaystyle {\frac {x}{y}}}
9323:{\displaystyle {\frac {x}{y}}}
9296:{\displaystyle {\frac {1}{x}}}
9224:{\displaystyle {\frac {x}{y}}}
9182:Multiplication in group theory
9157:
9154:
9128:
9113:
9087:
9078:
9015:
8974:
8945:
8904:
8853:
8807:
8801:
8755:
8749:
8713:
8707:
8678:
8672:
8643:
8637:
8611:
8605:
8576:
8573:
8544:
8349:
8245:
8185:
8159:
8139:
8113:
8021:{\displaystyle {\frac {C}{D}}}
7994:{\displaystyle {\frac {A}{B}}}
7958:
7946:
7941:
7929:
7816:
7807:
7801:
7792:
7770:
7758:
7743:
7734:
7728:
7719:
7492:Multiplication in group theory
7468:Multiplication with set theory
7437:
7425:
7419:
7370:
7364:
7352:
7346:
7334:
7305:
7200:
7194:
7167:
7161:
7134:
7051:
7039:
7033:
7027:
6946:
6934:
6928:
6922:
6689:{\displaystyle {\frac {1}{x}}}
6625:
6616:
6610:
6601:
6561:
6552:
6532:
6523:
6511:
6502:
6335:
6323:
6281:
6269:
6251:
6239:
5903:
5840:
5721:
5669:grows without bound. That is,
4758:
4746:
4696:
4684:
4680:
4668:
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4652:
4648:
4636:
4630:
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4432:
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4282:
4112:, analogous to the modern-day
4096:
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3789:
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3714:
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3705:
3690:
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3596:
3581:
3575:
3566:
3498:
3474:
3468:
3444:
3358:
3342:
3336:
3320:
3266:Product of two complex numbers
3256:Β§ Product of two integers
2494:Product of two natural numbers
2437:
2210:
2204:
2201:
2195:
2172:
2166:
2146:
2140:
1066:
1058:
13:
1:
10893:Conway chained arrow notation
9904:"Multiplicand and Multiplier"
9841:Peterson, Dave (2023-09-01).
9823:Peterson, Dave (2023-08-25).
9805:Peterson, Dave (2023-08-18).
9779:Peterson, Dave (2019-10-14).
9562:
9500:Multiplyβaccumulate operation
8191:{\displaystyle (a_{2},b_{2})}
8145:{\displaystyle (a_{1},b_{1})}
6110:
4266:division they did cumbrously.
1991:International System of Units
1790:{\displaystyle 3\times 4=12,}
1161:
10521:. John Wiley and Sons, Inc.
10481:. New York: Academic Press.
9471:SchΓΆnhageβStrassen algorithm
8484:{\displaystyle {\sqrt {-1}}}
8052:Considering complex numbers
7853:Generalization to fractions
5963:provided both limits exist.
5661:of the product of the first
4562:(much like the same way the
4213:decimal multiplication table
4181:Chinese multiplication table
4061:
3020:is the least upper bound of
2375:{\displaystyle 2\times \pi }
1749:{\displaystyle 2\times 3=6,}
1197:, with the other ones being
7:
10351:Encyclopedia of Mathematics
10092:(2nd ed.). p. 90.
9965:mathematische-basteleien.de
9867:Fuller, William R. (1977).
9644:Khan Academy (2012-09-06),
9617:Khan Academy (2015-08-14),
9441:
6898:{\displaystyle x\times 0=0}
6469:{\displaystyle x\cdot 0=0.}
6420:{\displaystyle x\cdot 1=x.}
5631:is a positive real number.
4441:{\displaystyle O(n\log n).}
4246:HinduβArabic numeral system
4163:, one only needed to add 50
4058:, but this is speculative.
3897:. Note, in this case, that
2973:Product of two real numbers
2459:. The specific problem is:
2008:, multiplication involving
1952:The middle dot notation or
1561:
10:
10955:
10845:Inverse for right argument
6853:
5970:
5776:One can similarly replace
5638:
4529:
4460:
4374:discrete Fourier transform
4349:
4342:and then add the entries.
4288:Grid method multiplication
4178:
4174:
4074:Rhind Mathematical Papyrus
4065:
3993:In some countries such as
3956:
3949:are in general different.
3885:Product of two quaternions
2961:{\displaystyle n,n'\neq 0}
1673:
1667:
36:
29:
10903:Knuth's up-arrow notation
10880:
10844:
10805:
10759:
10612:
10377:Biggs, Norman L. (2002).
10152:10.1016/j.jco.2016.03.001
10047:10.1038/nature.2014.14482
9879:10.1007/978-1-4612-9938-7
9495:Floating-point arithmetic
9486:, how computers multiply
7673:{\displaystyle N\times M}
7636:(the 3rd apple), or
6849:
6579:{\displaystyle (-x)+x=0.}
6123: axis = multiplier.
5108:factors is equivalent to
4370:Multiplication algorithms
4323:
4308:
4195:In the mathematical text
4171:computed from the table.
2300:multiplication algorithms
1630:
1625:
1580:
1573:
1568:
1423:of multiplication is the
1337:{\displaystyle 3\times 4}
1035:
1027:
957:
946:
845:
837:
683:
675:
566:
558:
451:
443:
294:
286:
10908:SteinhausβMoser notation
10404:"Multiplicative Inverse"
10006:"Peasant Multiplication"
9929:Litvin, Chester (2012).
9467:, for very large numbers
9465:ToomβCook multiplication
9454:Multiplication algorithm
9399:need not be the same as
9334:there are inverses, but
5206:of multiplication imply
4780:In such a notation, the
4457:Products of measurements
4378:computational complexity
4362:-digit numbers requires
4110:positional number system
3959:Multiplication algorithm
3942:{\displaystyle b\cdot a}
3916:{\displaystyle a\cdot b}
3299:{\displaystyle i^{2}=-1}
2830:Product of two fractions
1956:, encoded in Unicode as
1940:{\displaystyle 5\cdot 2}
1676:Multiplier (linguistics)
9177:Further generalizations
8464:Equivalently, denoting
4368:digit multiplications.
4260:, wrote the following:
4248:was first described by
4236:. This is a variant of
3755:from which one obtains
2718:Product of two integers
2470:WikiProject Mathematics
2332:{\displaystyle 3xy^{2}}
1706:{\displaystyle \times }
1191:mathematical operations
10517:History of Mathematics
10477:Burns, Gerald (1977).
10457:Mathematics LibreTexts
10285:math.illinoisstate.edu
9545:Peasant multiplication
9519:Multiplicative inverse
9431:
9393:
9355:
9324:
9297:
9263:
9225:
9167:
9022:
8866:
8505:
8485:
8453:
8426:
8396:
8356:
8232:
8192:
8146:
8100:
8073:
8022:
7995:
7968:
7887:
7835:
7777:
7674:
7614:
7554:
7553:{\displaystyle \cdot }
7447:
7315:
7085:
6959:
6899:
6737:
6690:
6666:multiplicative inverse
6638:
6580:
6539:
6470:
6421:
6369:
6291:
6209:
6140:
6086:
6078:
5954:
5931:
5871:
5813:
5767:
5749:
5699:
5621:
5591:
5574:
5528:
5488:
5454:
5434:
5389:
5348:
5328:
5287:
5236:
5190:
5142:
5095:
5032:
4944:
4833:
4771:
4745:
4706:
4617:
4581:
4552:
4442:
4399:. In 2016, the factor
4241:
4238:Lattice multiplication
4192:
4086:8 Γ 21 = 2 Γ 84 = 168
3984:peasant multiplication
3975:
3967:The Educated Monkeyβa
3943:
3917:
3872:
3746:
3634:
3527:
3513:
3300:
3240:
3171:
3126:
3070:
3014:
2998:decimal representation
2962:
2931:which is defined when
2923:
2795:
2709:
2653:
2581:
2534:
2503:
2428:
2396:
2376:
2333:
2291:programming language.
2217:
2216:{\displaystyle (5)(2)}
2182:
2153:
2124:implied multiplication
2116:
2096:
2073:
2053:
2033:
2012:is often written as a
1987:Ministry of Technology
1941:
1907:mathematical notations
1896:
1845:
1791:
1750:
1707:
1499:objects arranged in a
1480:
1391:
1338:
1309:
1218:The multiplication of
1151:(often denoted by the
1104:
1079:
1018:
993:
937:
912:
827:
762:
665:
639:
549:
524:
434:
409:
243:
181:
173:
165:
153:
27:Arithmetical operation
10939:Mathematical notation
10934:Elementary arithmetic
10898:Grzegorczyk hierarchy
10606:Elementary arithmetic
10310:mathworld.wolfram.com
10261:mathworld.wolfram.com
10130:Journal of Complexity
9552:, for generalizations
9550:Product (mathematics)
9530:GenailleβLucas rulers
9432:
9394:
9356:
9330:may be defined. In a
9325:
9298:
9276:may have no inverse "
9264:
9226:
9168:
9023:
8867:
8506:
8486:
8454:
8452:{\displaystyle b_{2}}
8427:
8425:{\displaystyle b_{1}}
8397:
8357:
8233:
8193:
8147:
8101:
8099:{\displaystyle z_{2}}
8074:
8072:{\displaystyle z_{1}}
8023:
7996:
7969:
7888:
7836:
7778:
7675:
7615:
7555:
7448:
7316:
7086:
6960:
6900:
6738:
6691:
6639:
6581:
6540:
6471:
6422:
6370:
6304:Distributive property
6292:
6210:
6127: axis = product.
6118:
6087:
6058:
5955:
5911:
5851:
5790:
5768:
5729:
5679:
5622:
5620:{\displaystyle a_{i}}
5592:
5554:
5508:
5489:
5487:{\displaystyle x_{i}}
5455:
5414:
5369:
5349:
5308:
5267:
5216:
5191:
5122:
5096:
5012:
4945:
4813:
4787:represents a varying
4772:
4725:
4707:
4597:
4582:
4553:
4530:Further information:
4521:Product of a sequence
4443:
4279:in the 13th century.
4230:
4188:
4122:multiplication tables
4027:Historical algorithms
3966:
3944:
3918:
3873:
3747:
3635:
3525:
3514:
3301:
3241:
3172:
3127:
3080:arithmetic operations
3071:
3015:
2963:
2924:
2796:
2710:
2633:
2561:
2535:
2501:
2429:
2397:
2377:
2334:
2233:vector multiplication
2218:
2183:
2154:
2117:
2097:
2074:
2054:
2034:
1942:
1897:
1846:
1792:
1751:
1708:
1543:vector multiplication
1481:
1392:
1339:
1310:
1222:may be thought of as
1186:) is one of the four
1105:
1080:
1019:
994:
938:
913:
828:
763:
666:
640:
550:
525:
435:
410:
251:Arithmetic operations
187:
179:
171:
159:
151:
10379:Discrete Mathematics
10103:Bernhard, Adrienne.
9479:Multiplication table
9449:Dimensional analysis
9403:
9365:
9338:
9307:
9280:
9235:
9208:
9186:multiplicative group
9032:
8878:
8515:
8495:
8468:
8436:
8409:
8366:
8242:
8202:
8156:
8110:
8083:
8056:
8005:
7978:
7897:
7857:
7789:
7710:
7658:
7571:
7544:
7537:could be notated as
7331:
7131:
7003:
6910:
6877:
6700:
6673:
6598:
6549:
6499:
6448:
6396:
6314:
6236:
6222:Associative property
6178:
6168:Commutative property
6002:
5787:
5676:
5604:
5505:
5471:
5360:
5213:
5119:
5009:
4990:, the product is an
4810:
4722:
4594:
4569:
4540:
4478:dimensional analysis
4463:Dimensional analysis
4411:
4258:Princeton University
4082:4 Γ 21 = 2 Γ 42 = 84
3980:multiplication table
3927:
3901:
3762:
3650:
3541:
3313:
3274:
3181:
3136:
3094:
3024:
3013:{\displaystyle \pi }
3004:
2935:
2841:
2733:
2546:
2510:
2406:
2395:{\displaystyle \pi }
2386:
2360:
2310:
2257:computer programming
2192:
2181:{\displaystyle (5)2}
2163:
2152:{\displaystyle 5(2)}
2134:
2106:
2083:
2063:
2043:
2020:
1925:
1909:for multiplication:
1856:
1802:
1766:
1725:
1697:
1575:Multiplication signs
1558:of complex numbers.
1527:dimensional analysis
1434:
1425:commutative property
1351:
1322:
1251:
1090:
1041:
1004:
963:
923:
851:
774:
689:
651:
572:
535:
457:
420:
300:
190:4.5m Γ 2.5m = 11.25m
68:improve this article
10872:Super-logarithm (4)
10831:Root extraction (3)
10402:Weisstein, Eric W.
10304:Weisstein, Eric W.
10255:Weisstein, Eric W.
9688:1968Natur.218S.111.
9682:(5137): 111. 1968.
9672:"Victory on Points"
9461:, for large numbers
9459:Karatsuba algorithm
7106:) as equivalent to
6442:of multiplication:
6228:order of operations
6099:copies of the base
5991:, and three is the
5449:
4526:Capital pi notation
4346:Computer algorithms
4254:Henry Burchard Fine
3889:The product of two
2304:long multiplication
2226:order of operations
1686:multiplication sign
1670:Multiplication sign
1597:MULTIPLICATION SIGN
1516:physical quantities
10888:Ackermann function
10782:Exponentiation (3)
10777:Multiplication (2)
10618:
10225:Klarreich, Erica.
10187:2019-04-08 at the
9947:Google Book Search
9505:Fused multiplyβadd
9473:, for huge numbers
9427:
9389:
9351:
9320:
9293:
9259:
9221:
9163:
9018:
8862:
8501:
8481:
8449:
8422:
8392:
8352:
8228:
8188:
8142:
8096:
8069:
8018:
7991:
7964:
7883:
7831:
7773:
7670:
7610:
7550:
7443:
7311:
7081:
6955:
6895:
6733:
6686:
6634:
6592:β1 times β1 is 1:
6576:
6535:
6466:
6417:
6365:
6287:
6205:
6141:
6082:
6054:
6047:
5950:
5910:
5850:
5763:
5728:
5617:
5587:
5484:
5450:
5435:
5344:
5186:
5091:
4940:
4767:
4702:
4577:
4576:
4548:
4547:
4438:
4242:
4215:by the end of the
4193:
3976:
3939:
3913:
3868:
3742:
3630:
3528:
3509:
3507:
3296:
3236:
3223:
3167:
3160:
3122:
3118:
3066:
3010:
2958:
2919:
2791:
2789:
2705:
2701:
2689:
2629:
2617:
2530:
2504:
2424:
2392:
2372:
2329:
2213:
2178:
2149:
2112:
2095:{\displaystyle 5x}
2092:
2069:
2049:
2032:{\displaystyle xy}
2029:
1937:
1892:
1841:
1787:
1756:("two times three
1746:
1703:
1476:
1387:
1334:
1305:
1301:
1289:
1160:, by the mid-line
1100:
1099:
1075:
1074:
1014:
1013:
989:
988:
975:
933:
932:
908:
907:
896:
893:
875:
823:
822:
817:
814:
803:
792:
758:
757:
746:
743:
740:
733:
719:
716:
709:
661:
660:
635:
634:
623:
620:
599:
545:
544:
520:
519:
508:
505:
484:
430:
429:
405:
404:
393:
390:
369:
348:
327:
244:
182:
174:
166:
154:
10916:
10915:
10809:for left argument
10719:
10718:
10714:
10713:
10528:978-0-471-54397-8
10408:Wolfram MathWorld
10388:978-0-19-871369-2
9942:978-1-4669-0152-0
9902:Ramone, Crewton.
9888:978-0-387-90283-8
9744:Texas Instruments
9484:Binary multiplier
9418:
9383:
9349:
9318:
9291:
9253:
9219:
8504:{\displaystyle i}
8479:
8016:
7989:
7962:
7921:
7908:
7881:
7868:
7632:(3 apples),
6976:) represents the
6721:
6684:
6388:identity property
6105:power associative
6020:
6018:
5895:
5832:
5713:
5647:infinite products
5635:Infinite products
4715:which results in
4338:
4337:
4052:Upper Paleolithic
4005:Common logarithms
3974:
3532:polar coordinates
3196:
3145:
3103:
2990:least upper bound
2914:
2875:
2852:
2698:
2662:
2660:
2626:
2590:
2588:
2487:
2486:
2382:is a multiple of
2115:{\displaystyle x}
2072:{\displaystyle y}
2052:{\displaystyle x}
1666:
1665:
1298:
1268:
1266:
1224:repeated addition
1146:
1145:
1113:
1112:
1097:
1064:
1052:
1011:
981:
979:
973:
930:
890:
885:
872:
867:
812:
801:
790:
741:
738:
731:
717:
714:
707:
658:
618:
608:
597:
587:
542:
503:
493:
482:
472:
427:
388:
378:
367:
357:
346:
336:
325:
315:
144:
143:
136:
118:
16:(Redirected from
10946:
10881:Related articles
10746:
10739:
10732:
10723:
10722:
10694:
10669:
10648:
10627:
10615:
10614:
10599:
10592:
10585:
10576:
10575:
10546:
10540:
10532:
10520:
10511:Merzbach, Uta C.
10493:
10492:
10474:
10468:
10467:
10465:
10464:
10449:
10443:
10442:
10440:
10439:
10433:
10424:
10418:
10417:
10415:
10414:
10399:
10393:
10392:
10374:
10361:
10360:
10358:
10357:
10347:"Multiplication"
10343:
10320:
10319:
10317:
10316:
10306:"Exponentiation"
10301:
10295:
10294:
10292:
10291:
10277:
10271:
10270:
10268:
10267:
10252:
10246:
10245:
10243:
10242:
10222:
10216:
10215:
10213:
10212:
10197:
10191:
10178:
10172:
10171:
10145:
10125:
10119:
10118:
10116:
10115:
10100:
10094:
10093:
10091:
10077:
10071:
10070:
10068:
10067:
10049:
10026:
10020:
10019:
10017:
10016:
10010:cut-the-knot.org
10002:
9996:
9995:
9993:
9981:
9975:
9974:
9972:
9971:
9961:"Multiplication"
9957:
9951:
9950:
9926:
9920:
9918:
9916:
9915:
9899:
9893:
9892:
9864:
9858:
9857:
9855:
9854:
9838:
9837:
9836:
9820:
9819:
9818:
9802:
9796:
9795:
9793:
9792:
9776:
9770:
9768:
9759:
9758:
9752:
9741:
9731:
9725:
9724:
9722:
9721:
9716:
9708:
9702:
9701:
9699:
9697:10.1038/218111c0
9668:
9662:
9661:
9660:
9659:
9641:
9635:
9634:
9633:
9632:
9614:
9608:
9607:
9602:
9601:
9584:(January 2011).
9578:
9535:Lunar arithmetic
9436:
9434:
9433:
9428:
9423:
9419:
9411:
9398:
9396:
9395:
9390:
9388:
9384:
9376:
9360:
9358:
9357:
9352:
9350:
9342:
9329:
9327:
9326:
9321:
9319:
9311:
9302:
9300:
9299:
9294:
9292:
9284:
9268:
9266:
9265:
9260:
9258:
9254:
9246:
9230:
9228:
9227:
9222:
9220:
9212:
9204:Often division,
9172:
9170:
9169:
9164:
9153:
9152:
9140:
9139:
9112:
9111:
9099:
9098:
9077:
9076:
9067:
9066:
9054:
9053:
9044:
9043:
9027:
9025:
9024:
9019:
9014:
9013:
8992:
8991:
8973:
8972:
8960:
8959:
8944:
8943:
8922:
8921:
8903:
8902:
8890:
8889:
8871:
8869:
8868:
8863:
8852:
8851:
8842:
8841:
8829:
8828:
8819:
8818:
8800:
8799:
8790:
8789:
8777:
8776:
8767:
8766:
8748:
8747:
8738:
8737:
8725:
8724:
8703:
8702:
8690:
8689:
8668:
8667:
8655:
8654:
8636:
8635:
8623:
8622:
8601:
8600:
8588:
8587:
8569:
8568:
8556:
8555:
8540:
8539:
8527:
8526:
8510:
8508:
8507:
8502:
8490:
8488:
8487:
8482:
8480:
8472:
8458:
8456:
8455:
8450:
8448:
8447:
8431:
8429:
8428:
8423:
8421:
8420:
8401:
8399:
8398:
8393:
8391:
8390:
8378:
8377:
8361:
8359:
8358:
8353:
8348:
8347:
8335:
8334:
8322:
8321:
8309:
8308:
8296:
8295:
8283:
8282:
8270:
8269:
8257:
8256:
8237:
8235:
8234:
8229:
8227:
8226:
8214:
8213:
8197:
8195:
8194:
8189:
8184:
8183:
8171:
8170:
8151:
8149:
8148:
8143:
8138:
8137:
8125:
8124:
8105:
8103:
8102:
8097:
8095:
8094:
8078:
8076:
8075:
8070:
8068:
8067:
8027:
8025:
8024:
8019:
8017:
8009:
8000:
7998:
7997:
7992:
7990:
7982:
7973:
7971:
7970:
7965:
7963:
7961:
7944:
7927:
7922:
7914:
7909:
7901:
7892:
7890:
7889:
7884:
7882:
7874:
7869:
7861:
7849:Rational numbers
7840:
7838:
7837:
7832:
7782:
7780:
7779:
7774:
7679:
7677:
7676:
7671:
7619:
7617:
7616:
7611:
7609:
7605:
7588:
7583:
7559:
7557:
7556:
7551:
7505:rational numbers
7474:cardinal numbers
7458:rational numbers
7452:
7450:
7449:
7444:
7320:
7318:
7317:
7312:
7304:
7303:
7291:
7290:
7278:
7277:
7265:
7264:
7251:
7250:
7238:
7237:
7225:
7224:
7212:
7211:
7193:
7192:
7179:
7178:
7160:
7159:
7146:
7145:
7115:
7090:
7088:
7087:
7082:
6992:. For instance,
6964:
6962:
6961:
6956:
6904:
6902:
6901:
6896:
6824:
6814:
6803:
6788:
6778:
6767:
6742:
6740:
6739:
6734:
6726:
6722:
6714:
6695:
6693:
6692:
6687:
6685:
6677:
6643:
6641:
6640:
6635:
6585:
6583:
6582:
6577:
6544:
6542:
6541:
6536:
6493:of that number:
6490:additive inverse
6475:
6473:
6472:
6467:
6426:
6424:
6423:
6418:
6382:Identity element
6374:
6372:
6371:
6366:
6296:
6294:
6293:
6288:
6214:
6212:
6211:
6206:
6091:
6089:
6088:
6083:
6077:
6072:
6053:
6048:
6043:
6014:
6013:
5959:
5957:
5956:
5951:
5946:
5942:
5941:
5940:
5930:
5925:
5909:
5886:
5882:
5881:
5880:
5870:
5865:
5849:
5823:
5822:
5812:
5807:
5772:
5770:
5769:
5764:
5759:
5758:
5748:
5743:
5727:
5709:
5708:
5698:
5693:
5641:Infinite product
5630:
5626:
5624:
5623:
5618:
5616:
5615:
5596:
5594:
5593:
5588:
5586:
5585:
5584:
5583:
5573:
5568:
5545:
5544:
5543:
5542:
5527:
5522:
5493:
5491:
5490:
5485:
5483:
5482:
5466:
5459:
5457:
5456:
5451:
5448:
5443:
5433:
5428:
5410:
5409:
5404:
5400:
5399:
5398:
5388:
5383:
5353:
5351:
5350:
5345:
5343:
5339:
5338:
5337:
5327:
5322:
5302:
5298:
5297:
5296:
5286:
5281:
5258:
5257:
5256:
5247:
5246:
5235:
5230:
5195:
5193:
5192:
5187:
5182:
5181:
5141:
5136:
5107:
5100:
5098:
5097:
5092:
5087:
5086:
5068:
5067:
5055:
5054:
5042:
5041:
5031:
5026:
5002:By definition,
4989:
4970:
4949:
4947:
4946:
4941:
4936:
4935:
4923:
4922:
4894:
4893:
4875:
4874:
4856:
4855:
4843:
4842:
4832:
4827:
4798:
4794:
4786:
4776:
4774:
4773:
4768:
4744:
4739:
4711:
4709:
4708:
4703:
4616:
4611:
4586:
4584:
4583:
4578:
4564:summation symbol
4557:
4555:
4554:
4549:
4472:Γ = 12 marbles.
4452:
4447:
4445:
4444:
4439:
4405:
4398:
4367:
4361:
4295:
4294:
4235:
4234:45 Γ 256 = 11520
4191:
4126:principal number
4119:
4087:
4083:
4079:
4038:
4033:ancient Egyptian
3972:
3948:
3946:
3945:
3940:
3922:
3920:
3919:
3914:
3877:
3875:
3874:
3869:
3864:
3863:
3751:
3749:
3748:
3743:
3738:
3737:
3639:
3637:
3636:
3631:
3629:
3628:
3518:
3516:
3515:
3510:
3508:
3437:
3433:
3432:
3305:
3303:
3302:
3297:
3286:
3285:
3260:Cauchy sequences
3245:
3243:
3242:
3237:
3222:
3176:
3174:
3173:
3168:
3159:
3131:
3129:
3128:
3123:
3117:
3089:
3085:
3075:
3073:
3072:
3067:
3019:
3017:
3016:
3011:
2996:of its infinite
2986:rational numbers
2967:
2965:
2964:
2959:
2951:
2928:
2926:
2925:
2920:
2915:
2913:
2912:
2897:
2896:
2881:
2876:
2874:
2866:
2858:
2853:
2845:
2800:
2798:
2797:
2792:
2790:
2725:positive amounts
2714:
2712:
2711:
2706:
2700:
2699:
2696:
2690:
2685:
2652:
2647:
2628:
2627:
2624:
2618:
2613:
2580:
2575:
2539:
2537:
2536:
2531:
2529:
2482:
2479:
2473:
2450:
2449:
2442:
2433:
2431:
2430:
2425:
2401:
2399:
2398:
2393:
2381:
2379:
2378:
2373:
2338:
2336:
2335:
2330:
2328:
2327:
2286:
2282:
2266:
2222:
2220:
2219:
2214:
2187:
2185:
2184:
2179:
2158:
2156:
2155:
2150:
2121:
2119:
2118:
2113:
2101:
2099:
2098:
2093:
2078:
2076:
2075:
2070:
2058:
2056:
2055:
2050:
2038:
2036:
2035:
2030:
1968:
1965:
1962:
1960:
1946:
1944:
1943:
1938:
1916:
1905:There are other
1901:
1899:
1898:
1893:
1850:
1848:
1847:
1842:
1796:
1794:
1793:
1788:
1755:
1753:
1752:
1747:
1718:). For example,
1713:
1712:
1710:
1709:
1704:
1691:
1662:
1657:
1654:
1652:
1647:
1642:
1639:
1637:
1620:
1616:
1613:
1610:
1608:
1602:
1598:
1595:
1592:
1590:
1566:
1565:
1485:
1483:
1482:
1477:
1419:One of the main
1412:, and 12 is the
1396:
1394:
1393:
1388:
1343:
1341:
1340:
1335:
1314:
1312:
1311:
1306:
1300:
1299:
1296:
1290:
1285:
1185:
1168:
1159:
1138:
1131:
1124:
1109:
1107:
1106:
1101:
1098:
1095:
1084:
1082:
1081:
1076:
1065:
1062:
1054:
1053:
1050:
1023:
1021:
1020:
1015:
1012:
1009:
998:
996:
995:
990:
982:
980:
977:
974:
971:
968:
942:
940:
939:
934:
931:
928:
917:
915:
914:
909:
901:
897:
892:
891:
888:
886:
883:
874:
873:
870:
868:
865:
832:
830:
829:
824:
821:
818:
813:
810:
802:
799:
791:
788:
767:
765:
764:
759:
751:
747:
742:
739:
736:
732:
729:
726:
718:
715:
712:
708:
705:
702:
670:
668:
667:
662:
659:
656:
644:
642:
641:
636:
628:
624:
619:
616:
609:
606:
598:
595:
588:
585:
554:
552:
551:
546:
543:
540:
529:
527:
526:
521:
513:
509:
504:
501:
494:
491:
483:
480:
473:
470:
439:
437:
436:
431:
428:
425:
414:
412:
411:
406:
398:
394:
389:
386:
379:
376:
368:
365:
358:
355:
347:
344:
337:
334:
326:
323:
316:
313:
284:
283:
274:
267:
260:
253:
246:
245:
242:
241:
239:
238:
235:
232:
225:
223:
222:
219:
216:
209:
207:
206:
203:
200:
191:
188:Area of a cloth
139:
132:
128:
125:
119:
117:
83:"Multiplication"
76:
52:
44:
21:
10954:
10953:
10949:
10948:
10947:
10945:
10944:
10943:
10919:
10918:
10917:
10912:
10876:
10857:Subtraction (1)
10852:Predecessor (0)
10840:
10821:Subtraction (1)
10816:Predecessor (0)
10801:
10755:
10753:Hyperoperations
10750:
10720:
10715:
10710:
10699:
10695:
10690:
10685:
10674:
10670:
10665:
10660:
10653:
10649:
10644:
10639:
10632:
10628:
10623:
10608:
10603:
10553:
10534:
10533:
10529:
10502:
10500:Further reading
10497:
10496:
10489:
10475:
10471:
10462:
10460:
10451:
10450:
10446:
10437:
10435:
10431:
10427:Angell, David.
10425:
10421:
10412:
10410:
10400:
10396:
10389:
10375:
10364:
10355:
10353:
10345:
10344:
10323:
10314:
10312:
10302:
10298:
10289:
10287:
10279:
10278:
10274:
10265:
10263:
10253:
10249:
10240:
10238:
10223:
10219:
10210:
10208:
10206:Quanta Magazine
10198:
10194:
10189:Wayback Machine
10179:
10175:
10126:
10122:
10113:
10111:
10101:
10097:
10089:
10078:
10074:
10065:
10063:
10027:
10023:
10014:
10012:
10004:
10003:
9999:
9982:
9978:
9969:
9967:
9959:
9958:
9954:
9943:
9927:
9923:
9913:
9911:
9900:
9896:
9889:
9865:
9861:
9852:
9850:
9834:
9832:
9816:
9814:
9803:
9799:
9790:
9788:
9777:
9773:
9756:
9754:
9750:
9739:
9733:
9732:
9728:
9719:
9717:
9714:
9710:
9709:
9705:
9670:
9669:
9665:
9657:
9655:
9642:
9638:
9630:
9628:
9615:
9611:
9599:
9597:
9579:
9570:
9565:
9560:
9444:
9410:
9406:
9404:
9401:
9400:
9375:
9371:
9366:
9363:
9362:
9341:
9339:
9336:
9335:
9310:
9308:
9305:
9304:
9283:
9281:
9278:
9277:
9271:integral domain
9245:
9241:
9236:
9233:
9232:
9211:
9209:
9206:
9205:
9194:polynomial ring
9148:
9144:
9135:
9131:
9107:
9103:
9094:
9090:
9072:
9068:
9062:
9058:
9049:
9045:
9039:
9035:
9033:
9030:
9029:
9009:
9005:
8987:
8983:
8968:
8964:
8955:
8951:
8939:
8935:
8917:
8913:
8898:
8894:
8885:
8881:
8879:
8876:
8875:
8847:
8843:
8837:
8833:
8824:
8820:
8814:
8810:
8795:
8791:
8785:
8781:
8772:
8768:
8762:
8758:
8743:
8739:
8733:
8729:
8720:
8716:
8698:
8694:
8685:
8681:
8663:
8659:
8650:
8646:
8631:
8627:
8618:
8614:
8596:
8592:
8583:
8579:
8564:
8560:
8551:
8547:
8535:
8531:
8522:
8518:
8516:
8513:
8512:
8496:
8493:
8492:
8471:
8469:
8466:
8465:
8443:
8439:
8437:
8434:
8433:
8416:
8412:
8410:
8407:
8406:
8404:imaginary parts
8386:
8382:
8373:
8369:
8367:
8364:
8363:
8343:
8339:
8330:
8326:
8317:
8313:
8304:
8300:
8291:
8287:
8278:
8274:
8265:
8261:
8252:
8248:
8243:
8240:
8239:
8222:
8218:
8209:
8205:
8203:
8200:
8199:
8179:
8175:
8166:
8162:
8157:
8154:
8153:
8133:
8129:
8120:
8116:
8111:
8108:
8107:
8090:
8086:
8084:
8081:
8080:
8063:
8059:
8057:
8054:
8053:
8048:Complex numbers
8008:
8006:
8003:
8002:
7981:
7979:
7976:
7975:
7945:
7928:
7926:
7913:
7900:
7898:
7895:
7894:
7873:
7860:
7858:
7855:
7854:
7790:
7787:
7786:
7711:
7708:
7707:
7659:
7656:
7655:
7626:
7584:
7579:
7578:
7574:
7572:
7569:
7568:
7545:
7542:
7541:
7520:identity matrix
7494:
7470:
7332:
7329:
7328:
7299:
7295:
7286:
7282:
7273:
7269:
7260:
7256:
7246:
7242:
7233:
7229:
7220:
7216:
7207:
7203:
7188:
7184:
7174:
7170:
7155:
7151:
7141:
7137:
7132:
7129:
7128:
7107:
7094:The axioms for
7004:
7001:
7000:
6911:
6908:
6907:
6878:
6875:
6874:
6858:
6852:
6832:complex numbers
6816:
6805:
6798:
6780:
6769:
6762:
6713:
6709:
6701:
6698:
6697:
6676:
6674:
6671:
6670:
6651:Inverse element
6599:
6596:
6595:
6550:
6547:
6546:
6500:
6497:
6496:
6449:
6446:
6445:
6397:
6394:
6393:
6315:
6312:
6311:
6237:
6234:
6233:
6179:
6176:
6175:
6153:natural numbers
6133:singular matrix
6130:
6128:
6113:
6095:indicates that
6073:
6062:
6049:
6021:
6019:
6009:
6005:
6003:
6000:
5999:
5975:
5969:
5936:
5932:
5926:
5915:
5899:
5894:
5890:
5876:
5872:
5866:
5855:
5836:
5831:
5827:
5818:
5814:
5808:
5794:
5788:
5785:
5784:
5754:
5750:
5744:
5733:
5717:
5704:
5700:
5694:
5683:
5677:
5674:
5673:
5655:infinity symbol
5643:
5637:
5628:
5611:
5607:
5605:
5602:
5601:
5579:
5575:
5569:
5558:
5553:
5549:
5538:
5534:
5533:
5529:
5523:
5512:
5506:
5503:
5502:
5478:
5474:
5472:
5469:
5468:
5464:
5444:
5439:
5429:
5418:
5405:
5394:
5390:
5384:
5373:
5368:
5364:
5363:
5361:
5358:
5357:
5333:
5329:
5323:
5312:
5307:
5303:
5292:
5288:
5282:
5271:
5266:
5262:
5252:
5248:
5242:
5238:
5237:
5231:
5220:
5214:
5211:
5210:
5177:
5173:
5137:
5126:
5120:
5117:
5116:
5105:
5082:
5078:
5063:
5059:
5050:
5046:
5037:
5033:
5027:
5016:
5010:
5007:
5006:
5000:
4981:
4979:
4962:
4931:
4927:
4912:
4908:
4883:
4879:
4864:
4860:
4851:
4847:
4838:
4834:
4828:
4817:
4811:
4808:
4807:
4796:
4792:
4784:
4740:
4729:
4723:
4720:
4719:
4612:
4601:
4595:
4592:
4591:
4570:
4567:
4566:
4541:
4538:
4537:
4534:
4528:
4523:
4495:. For example:
4465:
4459:
4450:
4412:
4409:
4408:
4400:
4381:
4363:
4357:
4354:
4348:
4285:
4233:
4225:
4198:Zhoubi Suanjing
4189:
4183:
4177:
4117:
4099:
4085:
4081:
4077:
4070:
4064:
4043:civilizations.
4036:
4029:
4001:
3991:
3961:
3955:
3928:
3925:
3924:
3902:
3899:
3898:
3887:
3844:
3840:
3763:
3760:
3759:
3730:
3726:
3651:
3648:
3647:
3621:
3617:
3542:
3539:
3538:
3506:
3505:
3435:
3434:
3428:
3424:
3361:
3316:
3314:
3311:
3310:
3281:
3277:
3275:
3272:
3271:
3268:
3200:
3182:
3179:
3178:
3149:
3137:
3134:
3133:
3107:
3095:
3092:
3091:
3087:
3083:
3025:
3022:
3021:
3005:
3002:
3001:
3000:; for example,
2975:
2944:
2936:
2933:
2932:
2905:
2898:
2889:
2882:
2880:
2867:
2859:
2857:
2844:
2842:
2839:
2838:
2832:
2807:additional rule
2788:
2787:
2782:
2777:
2771:
2770:
2765:
2760:
2754:
2753:
2748:
2743:
2736:
2734:
2731:
2730:
2720:
2695:
2691:
2663:
2661:
2648:
2637:
2623:
2619:
2591:
2589:
2576:
2565:
2547:
2544:
2543:
2540:is defined as:
2525:
2511:
2508:
2507:
2496:
2483:
2477:
2474:
2468:
2451:
2447:
2440:
2407:
2404:
2403:
2387:
2384:
2383:
2361:
2358:
2357:
2323:
2319:
2311:
2308:
2307:
2284:
2280:
2264:
2193:
2190:
2189:
2164:
2161:
2160:
2135:
2132:
2131:
2122:), also called
2107:
2104:
2103:
2102:for five times
2084:
2081:
2080:
2064:
2061:
2060:
2044:
2041:
2040:
2021:
2018:
2017:
1966:
1963:
1958:
1957:
1926:
1923:
1922:
1914:
1857:
1854:
1853:
1803:
1800:
1799:
1767:
1764:
1763:
1726:
1723:
1722:
1698:
1695:
1694:
1693:
1689:
1678:
1672:
1658:
1655:
1650:
1649:
1648:
1643:
1640:
1635:
1634:
1618:
1614:
1611:
1606:
1605:
1604:
1600:
1596:
1593:
1588:
1587:
1576:
1564:
1547:complex numbers
1435:
1432:
1431:
1352:
1349:
1348:
1323:
1320:
1319:
1295:
1291:
1269:
1267:
1252:
1249:
1248:
1181:
1164:
1155:
1142:
1094:
1091:
1088:
1087:
1061:
1049:
1045:
1042:
1039:
1038:
1008:
1005:
1002:
1001:
976:
970:
967:
964:
961:
960:
927:
924:
921:
920:
895:
894:
887:
882:
881:
877:
876:
869:
864:
863:
858:
855:
852:
849:
848:
816:
815:
809:
805:
804:
798:
794:
793:
787:
782:
778:
775:
772:
771:
745:
744:
735:
728:
725:
721:
720:
711:
704:
701:
696:
693:
690:
687:
686:
655:
652:
649:
648:
622:
621:
615:
605:
601:
600:
594:
584:
579:
576:
573:
570:
569:
539:
536:
533:
532:
507:
506:
500:
490:
486:
485:
479:
469:
464:
461:
458:
455:
454:
424:
421:
418:
417:
392:
391:
385:
375:
371:
370:
364:
354:
350:
349:
343:
333:
329:
328:
322:
312:
307:
304:
301:
298:
297:
278:
249:
236:
233:
230:
229:
227:
220:
217:
214:
213:
211:
204:
201:
198:
197:
195:
193:
189:
140:
129:
123:
120:
77:
75:
65:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
10952:
10942:
10941:
10936:
10931:
10929:Multiplication
10914:
10913:
10911:
10910:
10905:
10900:
10895:
10890:
10884:
10882:
10878:
10877:
10875:
10874:
10869:
10864:
10859:
10854:
10848:
10846:
10842:
10841:
10839:
10838:
10836:Super-root (4)
10833:
10828:
10823:
10818:
10812:
10810:
10803:
10802:
10800:
10799:
10794:
10789:
10784:
10779:
10774:
10769:
10763:
10761:
10757:
10756:
10749:
10748:
10741:
10734:
10726:
10717:
10716:
10712:
10711:
10688:
10686:
10672:Multiplication
10663:
10661:
10642:
10640:
10621:
10619:
10613:
10610:
10609:
10602:
10601:
10594:
10587:
10579:
10573:
10572:
10567:
10557:Multiplication
10552:
10551:External links
10549:
10548:
10547:
10527:
10507:Boyer, Carl B.
10501:
10498:
10495:
10494:
10487:
10469:
10444:
10419:
10394:
10387:
10362:
10321:
10296:
10272:
10247:
10217:
10192:
10173:
10120:
10095:
10081:Fine, Henry B.
10072:
10032:(2014-01-07).
10021:
9997:
9976:
9952:
9941:
9921:
9894:
9887:
9859:
9797:
9771:
9726:
9703:
9663:
9636:
9609:
9567:
9566:
9564:
9561:
9559:
9558:
9553:
9547:
9542:
9540:Napier's bones
9537:
9532:
9527:
9522:
9516:
9515:
9514:
9509:
9508:
9507:
9497:
9492:
9481:
9476:
9475:
9474:
9468:
9462:
9451:
9445:
9443:
9440:
9439:
9438:
9426:
9422:
9417:
9414:
9409:
9387:
9382:
9379:
9374:
9370:
9348:
9345:
9317:
9314:
9290:
9287:
9257:
9252:
9249:
9244:
9240:
9218:
9215:
9202:
9198:
9197:
9178:
9174:
9173:
9162:
9159:
9156:
9151:
9147:
9143:
9138:
9134:
9130:
9127:
9124:
9121:
9118:
9115:
9110:
9106:
9102:
9097:
9093:
9089:
9086:
9083:
9080:
9075:
9071:
9065:
9061:
9057:
9052:
9048:
9042:
9038:
9017:
9012:
9008:
9004:
9001:
8998:
8995:
8990:
8986:
8982:
8979:
8976:
8971:
8967:
8963:
8958:
8954:
8950:
8947:
8942:
8938:
8934:
8931:
8928:
8925:
8920:
8916:
8912:
8909:
8906:
8901:
8897:
8893:
8888:
8884:
8872:
8861:
8858:
8855:
8850:
8846:
8840:
8836:
8832:
8827:
8823:
8817:
8813:
8809:
8806:
8803:
8798:
8794:
8788:
8784:
8780:
8775:
8771:
8765:
8761:
8757:
8754:
8751:
8746:
8742:
8736:
8732:
8728:
8723:
8719:
8715:
8712:
8709:
8706:
8701:
8697:
8693:
8688:
8684:
8680:
8677:
8674:
8671:
8666:
8662:
8658:
8653:
8649:
8645:
8642:
8639:
8634:
8630:
8626:
8621:
8617:
8613:
8610:
8607:
8604:
8599:
8595:
8591:
8586:
8582:
8578:
8575:
8572:
8567:
8563:
8559:
8554:
8550:
8546:
8543:
8538:
8534:
8530:
8525:
8521:
8500:
8478:
8475:
8461:
8460:
8446:
8442:
8419:
8415:
8389:
8385:
8381:
8376:
8372:
8351:
8346:
8342:
8338:
8333:
8329:
8325:
8320:
8316:
8312:
8307:
8303:
8299:
8294:
8290:
8286:
8281:
8277:
8273:
8268:
8264:
8260:
8255:
8251:
8247:
8225:
8221:
8217:
8212:
8208:
8198:, the product
8187:
8182:
8178:
8174:
8169:
8165:
8161:
8141:
8136:
8132:
8128:
8123:
8119:
8115:
8093:
8089:
8066:
8062:
8050:
8044:
8043:
8036:
8030:
8029:
8015:
8012:
7988:
7985:
7960:
7957:
7954:
7951:
7948:
7943:
7940:
7937:
7934:
7931:
7925:
7920:
7917:
7912:
7907:
7904:
7880:
7877:
7872:
7867:
7864:
7851:
7845:
7844:
7841:
7830:
7827:
7824:
7821:
7818:
7815:
7812:
7809:
7806:
7803:
7800:
7797:
7794:
7784:
7772:
7769:
7766:
7763:
7760:
7757:
7754:
7751:
7748:
7745:
7742:
7739:
7736:
7733:
7730:
7727:
7724:
7721:
7718:
7715:
7705:
7680:is the sum of
7669:
7666:
7663:
7653:
7625:
7622:
7608:
7604:
7600:
7597:
7594:
7591:
7587:
7582:
7577:
7549:
7493:
7490:
7469:
7466:
7454:
7453:
7442:
7439:
7436:
7433:
7430:
7427:
7424:
7421:
7418:
7415:
7412:
7409:
7406:
7403:
7400:
7396:
7393:
7390:
7387:
7384:
7381:
7378:
7375:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7345:
7342:
7339:
7336:
7322:
7321:
7310:
7307:
7302:
7298:
7294:
7289:
7285:
7281:
7276:
7272:
7268:
7263:
7259:
7254:
7249:
7245:
7241:
7236:
7232:
7228:
7223:
7219:
7215:
7210:
7206:
7202:
7199:
7196:
7191:
7187:
7182:
7177:
7173:
7169:
7166:
7163:
7158:
7154:
7149:
7144:
7140:
7136:
7092:
7091:
7080:
7077:
7074:
7071:
7068:
7065:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
6966:
6965:
6954:
6951:
6948:
6945:
6942:
6939:
6936:
6933:
6930:
6927:
6924:
6921:
6918:
6915:
6905:
6894:
6891:
6888:
6885:
6882:
6868:Giuseppe Peano
6854:Main article:
6851:
6848:
6836:
6835:
6828:
6827:
6826:
6792:
6791:
6790:
6752:
6745:
6744:
6732:
6729:
6725:
6720:
6717:
6712:
6708:
6705:
6683:
6680:
6653:
6647:
6646:
6645:
6644:
6633:
6630:
6627:
6624:
6621:
6618:
6615:
6612:
6609:
6606:
6603:
6589:
6588:
6587:
6586:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6554:
6534:
6531:
6528:
6525:
6522:
6519:
6516:
6513:
6510:
6507:
6504:
6485:
6479:
6478:
6477:
6476:
6465:
6462:
6459:
6456:
6453:
6436:
6430:
6429:
6428:
6427:
6416:
6413:
6410:
6407:
6404:
6401:
6384:
6378:
6377:
6376:
6375:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6325:
6322:
6319:
6306:
6300:
6299:
6298:
6297:
6286:
6283:
6280:
6277:
6274:
6271:
6268:
6265:
6262:
6259:
6256:
6253:
6250:
6247:
6244:
6241:
6224:
6218:
6217:
6216:
6215:
6204:
6201:
6198:
6195:
6192:
6189:
6186:
6183:
6170:
6112:
6109:
6093:
6092:
6081:
6076:
6071:
6068:
6065:
6061:
6057:
6052:
6046:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6017:
6012:
6008:
5980:exponentiation
5973:Exponentiation
5971:Main article:
5968:
5967:Exponentiation
5965:
5961:
5960:
5949:
5945:
5939:
5935:
5929:
5924:
5921:
5918:
5914:
5908:
5905:
5902:
5898:
5893:
5889:
5885:
5879:
5875:
5869:
5864:
5861:
5858:
5854:
5848:
5845:
5842:
5839:
5835:
5830:
5826:
5821:
5817:
5811:
5806:
5803:
5800:
5797:
5793:
5774:
5773:
5762:
5757:
5753:
5747:
5742:
5739:
5736:
5732:
5726:
5723:
5720:
5716:
5712:
5707:
5703:
5697:
5692:
5689:
5686:
5682:
5639:Main article:
5636:
5633:
5614:
5610:
5598:
5597:
5582:
5578:
5572:
5567:
5564:
5561:
5557:
5552:
5548:
5541:
5537:
5532:
5526:
5521:
5518:
5515:
5511:
5481:
5477:
5461:
5460:
5447:
5442:
5438:
5432:
5427:
5424:
5421:
5417:
5413:
5408:
5403:
5397:
5393:
5387:
5382:
5379:
5376:
5372:
5367:
5355:
5342:
5336:
5332:
5326:
5321:
5318:
5315:
5311:
5306:
5301:
5295:
5291:
5285:
5280:
5277:
5274:
5270:
5265:
5261:
5255:
5251:
5245:
5241:
5234:
5229:
5226:
5223:
5219:
5197:
5196:
5185:
5180:
5176:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5140:
5135:
5132:
5129:
5125:
5110:exponentiation
5102:
5101:
5090:
5085:
5081:
5077:
5074:
5071:
5066:
5062:
5058:
5053:
5049:
5045:
5040:
5036:
5030:
5025:
5022:
5019:
5015:
4999:
4996:
4975:
4951:
4950:
4939:
4934:
4930:
4926:
4921:
4918:
4915:
4911:
4907:
4902:
4897:
4892:
4889:
4886:
4882:
4878:
4873:
4870:
4867:
4863:
4859:
4854:
4850:
4846:
4841:
4837:
4831:
4826:
4823:
4820:
4816:
4778:
4777:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4743:
4738:
4735:
4732:
4728:
4713:
4712:
4701:
4698:
4695:
4692:
4689:
4686:
4682:
4679:
4676:
4673:
4670:
4666:
4663:
4660:
4657:
4654:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4615:
4610:
4607:
4604:
4600:
4575:
4560:Greek alphabet
4546:
4527:
4524:
4522:
4519:
4518:
4517:
4514:
4511:
4501:
4500:
4474:
4473:
4461:Main article:
4458:
4455:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4350:Main article:
4347:
4344:
4340:
4339:
4336:
4335:
4330:
4325:
4321:
4320:
4315:
4310:
4306:
4305:
4302:
4299:
4284:
4281:
4269:
4268:
4224:
4223:Modern methods
4221:
4217:Warring States
4190:38 Γ 76 = 2888
4176:
4173:
4114:decimal system
4098:
4095:
4094:
4093:
4066:Main article:
4063:
4060:
4056:Central Africa
4037:Greek, Indian,
4028:
4025:
4015:, such as the
3999:
3988:
3957:Main article:
3954:
3951:
3938:
3935:
3932:
3912:
3909:
3906:
3886:
3883:
3879:
3878:
3867:
3862:
3859:
3856:
3853:
3850:
3847:
3843:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3753:
3752:
3741:
3736:
3733:
3729:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3661:
3658:
3655:
3641:
3640:
3627:
3624:
3620:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3552:
3549:
3546:
3520:
3519:
3504:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3438:
3436:
3431:
3427:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3398:
3395:
3392:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3362:
3360:
3357:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3331:
3328:
3325:
3322:
3319:
3318:
3306:, as follows:
3295:
3292:
3289:
3284:
3280:
3267:
3264:
3235:
3232:
3229:
3226:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3199:
3195:
3192:
3189:
3186:
3166:
3163:
3158:
3155:
3152:
3148:
3144:
3141:
3121:
3116:
3113:
3110:
3106:
3102:
3099:
3065:
3062:
3059:
3056:
3053:
3049:
3046:
3042:
3039:
3035:
3032:
3029:
3009:
2974:
2971:
2970:
2969:
2957:
2954:
2950:
2947:
2943:
2940:
2929:
2918:
2911:
2908:
2904:
2901:
2895:
2892:
2888:
2885:
2879:
2873:
2870:
2865:
2862:
2856:
2851:
2848:
2831:
2828:
2827:
2826:
2823:
2820:
2817:
2803:distributivity
2786:
2783:
2781:
2778:
2776:
2773:
2772:
2769:
2766:
2764:
2761:
2759:
2756:
2755:
2752:
2749:
2747:
2744:
2742:
2739:
2738:
2719:
2716:
2704:
2694:
2688:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2659:
2656:
2651:
2646:
2643:
2640:
2636:
2632:
2622:
2616:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2587:
2584:
2579:
2574:
2571:
2568:
2564:
2560:
2557:
2554:
2551:
2528:
2524:
2521:
2518:
2515:
2495:
2492:
2485:
2484:
2478:September 2023
2454:
2452:
2445:
2439:
2436:
2423:
2420:
2417:
2414:
2411:
2391:
2371:
2368:
2365:
2339:) is called a
2326:
2322:
2318:
2315:
2302:, such as the
2269:character sets
2253:
2252:
2229:
2212:
2209:
2206:
2203:
2200:
2197:
2177:
2174:
2171:
2168:
2148:
2145:
2142:
2139:
2111:
2091:
2088:
2068:
2048:
2028:
2025:
2001:
2000:
1982:
1949:
1948:
1936:
1933:
1930:
1903:
1902:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1851:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1797:
1786:
1783:
1780:
1777:
1774:
1771:
1761:
1745:
1742:
1739:
1736:
1733:
1730:
1716:infix notation
1702:
1668:Main article:
1664:
1663:
1632:
1631:Different from
1628:
1627:
1626:Different from
1623:
1622:
1585:
1578:
1577:
1574:
1571:
1570:
1563:
1560:
1487:
1486:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1398:
1397:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1333:
1330:
1327:
1316:
1315:
1304:
1294:
1288:
1284:
1281:
1278:
1275:
1272:
1265:
1262:
1259:
1256:
1149:Multiplication
1144:
1143:
1141:
1140:
1133:
1126:
1118:
1115:
1114:
1111:
1110:
1085:
1072:
1068:
1063:anti-logarithm
1060:
1057:
1048:
1036:
1033:
1032:
1025:
1024:
999:
986:
958:
955:
954:
944:
943:
918:
905:
900:
879:
878:
861:
860:
857:
846:
843:
842:
839:Exponentiation
835:
834:
820:
807:
806:
796:
795:
785:
784:
781:
768:
755:
750:
723:
722:
699:
698:
695:
684:
681:
680:
673:
672:
645:
632:
627:
613:
603:
602:
592:
582:
581:
578:
567:
564:
563:
560:Multiplication
556:
555:
530:
517:
512:
498:
488:
487:
477:
467:
466:
463:
452:
449:
448:
441:
440:
415:
402:
397:
383:
373:
372:
362:
352:
351:
341:
331:
330:
320:
310:
309:
306:
295:
292:
291:
280:
279:
277:
276:
269:
262:
254:
142:
141:
56:
54:
47:
26:
9:
6:
4:
3:
2:
10951:
10940:
10937:
10935:
10932:
10930:
10927:
10926:
10924:
10909:
10906:
10904:
10901:
10899:
10896:
10894:
10891:
10889:
10886:
10885:
10883:
10879:
10873:
10870:
10868:
10867:Logarithm (3)
10865:
10863:
10860:
10858:
10855:
10853:
10850:
10849:
10847:
10843:
10837:
10834:
10832:
10829:
10827:
10824:
10822:
10819:
10817:
10814:
10813:
10811:
10808:
10804:
10798:
10795:
10793:
10792:Pentation (5)
10790:
10788:
10787:Tetration (4)
10785:
10783:
10780:
10778:
10775:
10773:
10770:
10768:
10767:Successor (0)
10765:
10764:
10762:
10758:
10754:
10747:
10742:
10740:
10735:
10733:
10728:
10727:
10724:
10709:
10707:
10703:
10698:
10693:
10687:
10684:
10682:
10678:
10673:
10668:
10662:
10659:
10657:
10652:
10647:
10641:
10638:
10636:
10631:
10626:
10620:
10617:
10616:
10611:
10607:
10600:
10595:
10593:
10588:
10586:
10581:
10580:
10577:
10571:
10568:
10566:
10562:
10558:
10555:
10554:
10544:
10538:
10530:
10524:
10519:
10518:
10512:
10508:
10504:
10503:
10490:
10488:9780121457501
10484:
10480:
10473:
10458:
10454:
10448:
10430:
10423:
10409:
10405:
10398:
10390:
10384:
10380:
10373:
10371:
10369:
10367:
10352:
10348:
10342:
10340:
10338:
10336:
10334:
10332:
10330:
10328:
10326:
10311:
10307:
10300:
10286:
10282:
10276:
10262:
10258:
10251:
10236:
10232:
10228:
10221:
10207:
10203:
10196:
10190:
10186:
10183:
10177:
10169:
10165:
10161:
10157:
10153:
10149:
10144:
10139:
10135:
10131:
10124:
10110:
10106:
10099:
10088:
10087:
10082:
10076:
10061:
10057:
10053:
10048:
10043:
10039:
10035:
10031:
10025:
10011:
10007:
10001:
9992:
9987:
9980:
9966:
9962:
9956:
9948:
9944:
9938:
9934:
9933:
9925:
9909:
9905:
9898:
9890:
9884:
9880:
9876:
9872:
9871:
9863:
9848:
9844:
9830:
9826:
9812:
9808:
9801:
9786:
9782:
9775:
9767:
9765:
9749:
9745:
9738:
9737:
9730:
9713:
9707:
9698:
9693:
9689:
9685:
9681:
9677:
9673:
9667:
9653:
9649:
9648:
9640:
9626:
9622:
9621:
9613:
9606:
9595:
9591:
9587:
9583:
9582:Devlin, Keith
9577:
9575:
9573:
9568:
9557:
9554:
9551:
9548:
9546:
9543:
9541:
9538:
9536:
9533:
9531:
9528:
9526:
9523:
9520:
9517:
9513:
9510:
9506:
9503:
9502:
9501:
9498:
9496:
9493:
9491:
9488:
9487:
9485:
9482:
9480:
9477:
9472:
9469:
9466:
9463:
9460:
9457:
9456:
9455:
9452:
9450:
9447:
9446:
9424:
9420:
9415:
9412:
9407:
9385:
9380:
9377:
9372:
9368:
9346:
9343:
9333:
9332:division ring
9315:
9312:
9288:
9285:
9275:
9272:
9255:
9250:
9247:
9242:
9238:
9216:
9213:
9203:
9200:
9199:
9195:
9191:
9187:
9184:, above, and
9183:
9179:
9176:
9175:
9160:
9149:
9145:
9141:
9136:
9132:
9125:
9122:
9119:
9116:
9108:
9104:
9100:
9095:
9091:
9084:
9081:
9073:
9069:
9063:
9059:
9055:
9050:
9046:
9040:
9036:
9010:
9006:
9002:
8999:
8996:
8993:
8988:
8984:
8980:
8977:
8969:
8965:
8961:
8956:
8952:
8948:
8940:
8936:
8932:
8929:
8926:
8923:
8918:
8914:
8910:
8907:
8899:
8895:
8891:
8886:
8882:
8873:
8859:
8856:
8848:
8844:
8838:
8834:
8830:
8825:
8821:
8815:
8811:
8804:
8796:
8792:
8786:
8782:
8778:
8773:
8769:
8763:
8759:
8752:
8744:
8740:
8734:
8730:
8726:
8721:
8717:
8710:
8704:
8699:
8695:
8691:
8686:
8682:
8675:
8669:
8664:
8660:
8656:
8651:
8647:
8640:
8632:
8628:
8624:
8619:
8615:
8608:
8602:
8597:
8593:
8589:
8584:
8580:
8570:
8565:
8561:
8557:
8552:
8548:
8541:
8536:
8532:
8528:
8523:
8519:
8498:
8476:
8473:
8463:
8462:
8444:
8440:
8417:
8413:
8405:
8387:
8383:
8379:
8374:
8370:
8344:
8340:
8336:
8331:
8327:
8323:
8318:
8314:
8310:
8305:
8301:
8297:
8292:
8288:
8284:
8279:
8275:
8271:
8266:
8262:
8258:
8253:
8249:
8223:
8219:
8215:
8210:
8206:
8180:
8176:
8172:
8167:
8163:
8134:
8130:
8126:
8121:
8117:
8091:
8087:
8064:
8060:
8051:
8049:
8046:
8045:
8041:
8037:
8035:
8032:
8031:
8013:
8010:
7986:
7983:
7955:
7952:
7949:
7938:
7935:
7932:
7923:
7918:
7915:
7910:
7905:
7902:
7878:
7875:
7870:
7865:
7862:
7852:
7850:
7847:
7846:
7842:
7828:
7825:
7822:
7819:
7813:
7810:
7804:
7798:
7795:
7785:
7767:
7764:
7761:
7755:
7752:
7749:
7746:
7740:
7737:
7731:
7725:
7722:
7716:
7713:
7706:
7703:
7699:
7695:
7691:
7687:
7683:
7667:
7664:
7661:
7654:
7651:
7650:
7649:
7647:
7643:
7639:
7635:
7631:
7621:
7606:
7602:
7598:
7592:
7585:
7575:
7566:
7562:
7547:
7540:
7536:
7532:
7527:
7523:
7521:
7517:
7512:
7510:
7509:abelian group
7506:
7501:
7499:
7489:
7487:
7483:
7479:
7475:
7465:
7463:
7459:
7440:
7434:
7431:
7428:
7422:
7416:
7413:
7410:
7407:
7404:
7401:
7398:
7394:
7391:
7388:
7385:
7382:
7379:
7376:
7373:
7367:
7361:
7358:
7355:
7349:
7343:
7340:
7337:
7327:
7326:
7325:
7308:
7300:
7296:
7292:
7287:
7283:
7279:
7274:
7270:
7266:
7261:
7257:
7252:
7247:
7243:
7239:
7234:
7230:
7226:
7221:
7217:
7213:
7208:
7204:
7197:
7189:
7185:
7180:
7175:
7171:
7164:
7156:
7152:
7147:
7142:
7138:
7127:
7126:
7125:
7123:
7119:
7114:
7110:
7105:
7101:
7097:
7078:
7075:
7072:
7069:
7066:
7063:
7060:
7057:
7054:
7048:
7045:
7042:
7036:
7030:
7024:
7021:
7018:
7015:
7012:
7009:
7006:
6999:
6998:
6997:
6995:
6991:
6987:
6983:
6979:
6975:
6971:
6952:
6949:
6943:
6940:
6937:
6931:
6925:
6919:
6916:
6913:
6906:
6892:
6889:
6886:
6883:
6880:
6873:
6872:
6871:
6869:
6865:
6864:
6857:
6847:
6845:
6841:
6833:
6829:
6823:
6819:
6812:
6808:
6801:
6796:
6795:
6793:
6787:
6783:
6776:
6772:
6765:
6760:
6759:
6757:
6753:
6750:
6747:
6746:
6730:
6727:
6723:
6718:
6715:
6710:
6706:
6703:
6681:
6678:
6668:
6667:
6662:
6658:
6655:Every number
6654:
6652:
6649:
6648:
6631:
6628:
6622:
6619:
6613:
6607:
6604:
6594:
6593:
6591:
6590:
6573:
6570:
6567:
6564:
6558:
6555:
6529:
6526:
6520:
6517:
6514:
6508:
6505:
6495:
6494:
6492:
6491:
6486:
6484:
6481:
6480:
6463:
6460:
6457:
6454:
6451:
6444:
6443:
6441:
6440:zero property
6437:
6435:
6434:Property of 0
6432:
6431:
6414:
6411:
6408:
6405:
6402:
6399:
6392:
6391:
6389:
6385:
6383:
6380:
6379:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6332:
6329:
6326:
6320:
6317:
6310:
6309:
6307:
6305:
6302:
6301:
6284:
6278:
6275:
6272:
6266:
6263:
6260:
6257:
6254:
6248:
6245:
6242:
6232:
6231:
6229:
6225:
6223:
6220:
6219:
6202:
6199:
6196:
6193:
6190:
6187:
6184:
6181:
6174:
6173:
6171:
6169:
6166:
6165:
6164:
6162:
6158:
6154:
6150:
6146:
6138:
6134:
6126:
6122:
6117:
6108:
6106:
6102:
6098:
6079:
6074:
6069:
6066:
6063:
6059:
6055:
6050:
6044:
6040:
6037:
6034:
6031:
6028:
6025:
6022:
6015:
6010:
6006:
5998:
5997:
5996:
5994:
5990:
5986:
5982:
5981:
5974:
5964:
5947:
5943:
5937:
5933:
5927:
5922:
5919:
5916:
5912:
5900:
5891:
5887:
5883:
5877:
5873:
5867:
5862:
5859:
5856:
5852:
5843:
5837:
5828:
5824:
5819:
5815:
5801:
5798:
5795:
5791:
5783:
5782:
5781:
5779:
5760:
5755:
5751:
5745:
5740:
5737:
5734:
5730:
5718:
5710:
5705:
5701:
5690:
5687:
5684:
5680:
5672:
5671:
5670:
5668:
5664:
5660:
5656:
5653:above by the
5652:
5648:
5642:
5632:
5612:
5608:
5580:
5576:
5570:
5565:
5562:
5559:
5555:
5550:
5546:
5539:
5535:
5530:
5524:
5519:
5516:
5513:
5509:
5501:
5500:
5499:
5497:
5494:are positive
5479:
5475:
5445:
5440:
5436:
5430:
5425:
5422:
5419:
5415:
5411:
5406:
5401:
5395:
5391:
5385:
5380:
5377:
5374:
5370:
5365:
5356:
5340:
5334:
5330:
5324:
5319:
5316:
5313:
5309:
5304:
5299:
5293:
5289:
5283:
5278:
5275:
5272:
5268:
5263:
5259:
5253:
5249:
5243:
5239:
5232:
5227:
5224:
5221:
5217:
5209:
5208:
5207:
5205:
5204:commutativity
5201:
5200:Associativity
5183:
5178:
5174:
5170:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5138:
5133:
5130:
5127:
5123:
5115:
5114:
5113:
5111:
5088:
5083:
5079:
5075:
5072:
5069:
5064:
5060:
5056:
5051:
5047:
5043:
5038:
5034:
5028:
5023:
5020:
5017:
5013:
5005:
5004:
5003:
4995:
4993:
4992:empty product
4988:
4984:
4978:
4974:
4969:
4965:
4960:
4956:
4937:
4932:
4928:
4924:
4919:
4916:
4913:
4909:
4905:
4900:
4895:
4890:
4887:
4884:
4880:
4876:
4871:
4868:
4865:
4861:
4857:
4852:
4848:
4844:
4839:
4835:
4829:
4824:
4821:
4818:
4814:
4806:
4805:
4804:
4801:
4790:
4783:
4764:
4761:
4755:
4752:
4749:
4741:
4736:
4733:
4730:
4726:
4718:
4717:
4716:
4699:
4693:
4690:
4687:
4677:
4674:
4671:
4661:
4658:
4655:
4645:
4642:
4639:
4633:
4627:
4624:
4621:
4613:
4608:
4605:
4602:
4598:
4590:
4589:
4588:
4573:
4565:
4561:
4544:
4533:
4515:
4512:
4509:
4508:
4507:
4504:
4498:
4497:
4496:
4494:
4490:
4486:
4481:
4479:
4471:
4470:
4469:
4464:
4454:
4435:
4429:
4426:
4423:
4420:
4414:
4404:
4396:
4392:
4388:
4384:
4379:
4375:
4371:
4366:
4360:
4353:
4343:
4334:
4331:
4329:
4326:
4322:
4319:
4316:
4314:
4311:
4307:
4296:
4293:
4292:
4291:
4289:
4280:
4278:
4274:
4267:
4263:
4262:
4261:
4259:
4255:
4251:
4247:
4239:
4229:
4220:
4218:
4214:
4210:
4206:
4205:
4200:
4199:
4187:
4182:
4172:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4127:
4123:
4115:
4111:
4108:
4104:
4091:
4090:
4089:
4075:
4069:
4059:
4057:
4053:
4049:
4044:
4042:
4034:
4024:
4022:
4018:
4014:
4010:
4006:
3998:
3996:
3987:
3985:
3981:
3970:
3965:
3960:
3950:
3936:
3933:
3930:
3910:
3907:
3904:
3896:
3892:
3882:
3865:
3857:
3854:
3851:
3845:
3841:
3837:
3834:
3831:
3828:
3825:
3822:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3792:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3758:
3757:
3756:
3739:
3734:
3731:
3727:
3723:
3720:
3717:
3708:
3702:
3699:
3696:
3693:
3687:
3681:
3678:
3672:
3669:
3666:
3663:
3659:
3656:
3653:
3646:
3645:
3644:
3643:Furthermore,
3625:
3622:
3618:
3614:
3611:
3608:
3599:
3593:
3590:
3587:
3584:
3578:
3572:
3569:
3563:
3560:
3557:
3554:
3550:
3547:
3544:
3537:
3536:
3535:
3533:
3524:
3502:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3471:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3441:
3439:
3429:
3425:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3396:
3393:
3390:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3363:
3355:
3351:
3348:
3345:
3339:
3333:
3329:
3326:
3323:
3309:
3308:
3307:
3293:
3290:
3287:
3282:
3278:
3263:
3261:
3257:
3251:
3249:
3233:
3230:
3227:
3224:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3193:
3190:
3187:
3184:
3164:
3161:
3156:
3153:
3150:
3142:
3139:
3119:
3114:
3111:
3108:
3100:
3097:
3081:
3076:
3063:
3057:
3054:
3051:
3047:
3044:
3040:
3037:
3033:
3030:
3007:
2999:
2995:
2991:
2987:
2982:
2980:
2955:
2952:
2948:
2945:
2941:
2938:
2930:
2916:
2909:
2906:
2902:
2899:
2893:
2890:
2886:
2883:
2877:
2871:
2868:
2863:
2860:
2854:
2849:
2846:
2837:
2836:
2835:
2824:
2821:
2818:
2815:
2814:
2813:
2810:
2808:
2804:
2784:
2779:
2774:
2767:
2762:
2757:
2750:
2745:
2740:
2728:
2726:
2715:
2702:
2692:
2686:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2657:
2654:
2649:
2644:
2641:
2638:
2634:
2630:
2620:
2614:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2585:
2582:
2577:
2572:
2569:
2566:
2562:
2558:
2555:
2552:
2549:
2541:
2522:
2519:
2516:
2513:
2502:3 by 4 is 12.
2500:
2491:
2481:
2471:
2466:
2462:
2458:
2455:This section
2453:
2444:
2443:
2435:
2421:
2418:
2415:
2412:
2409:
2389:
2369:
2366:
2363:
2355:
2354:
2349:
2344:
2342:
2324:
2320:
2316:
2313:
2305:
2301:
2297:
2296:factorization
2292:
2290:
2278:
2274:
2270:
2262:
2258:
2250:
2246:
2242:
2238:
2237:cross product
2234:
2230:
2227:
2207:
2198:
2175:
2169:
2143:
2137:
2129:
2125:
2109:
2089:
2086:
2066:
2046:
2026:
2023:
2015:
2014:juxtaposition
2011:
2007:
2003:
2002:
1998:
1997:
1992:
1988:
1983:
1980:
1976:
1972:
1971:decimal point
1955:
1951:
1950:
1934:
1931:
1928:
1920:
1912:
1911:
1910:
1908:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1852:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1798:
1784:
1781:
1778:
1775:
1772:
1769:
1762:
1759:
1743:
1740:
1737:
1734:
1731:
1728:
1721:
1720:
1719:
1717:
1700:
1687:
1683:
1677:
1671:
1661:
1646:
1633:
1629:
1624:
1586:
1584:
1579:
1572:
1567:
1559:
1557:
1552:
1548:
1544:
1539:
1537:
1536:
1530:
1528:
1524:
1521:
1517:
1512:
1510:
1506:
1502:
1498:
1493:
1490:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1430:
1429:
1428:
1426:
1422:
1417:
1415:
1411:
1407:
1404:) and 4 (the
1403:
1400:Here, 3 (the
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1347:
1346:
1345:
1331:
1328:
1325:
1302:
1292:
1286:
1282:
1279:
1276:
1273:
1270:
1263:
1260:
1257:
1254:
1247:
1246:
1245:
1243:
1242:
1237:
1236:
1231:
1230:
1225:
1221:
1220:whole numbers
1216:
1214:
1213:
1208:
1204:
1200:
1196:
1192:
1189:
1184:
1180:
1176:
1172:
1171:juxtaposition
1167:
1163:
1158:
1154:
1150:
1139:
1134:
1132:
1127:
1125:
1120:
1119:
1117:
1116:
1086:
1070:
1055:
1046:
1037:
1034:
1030:
1026:
1000:
984:
959:
956:
952:
950:
945:
919:
903:
898:
847:
844:
840:
836:
833:
779:
769:
753:
748:
685:
682:
678:
674:
671:
646:
630:
625:
611:
590:
568:
565:
561:
557:
531:
515:
510:
496:
475:
453:
450:
446:
442:
416:
400:
395:
381:
360:
339:
318:
296:
293:
289:
285:
282:
281:
275:
270:
268:
263:
261:
256:
255:
252:
248:
247:
186:
178:
170:
163:
158:
150:
146:
138:
135:
127:
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: β
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
19:
10862:Division (2)
10826:Division (2)
10797:Hexation (6)
10776:
10772:Addition (1)
10689:
10671:
10666:
10664:
10643:
10622:
10565:cut-the-knot
10516:
10509:(revised by
10478:
10472:
10461:. Retrieved
10459:. 2018-04-11
10456:
10447:
10436:. Retrieved
10422:
10411:. Retrieved
10407:
10397:
10378:
10354:. Retrieved
10350:
10313:. Retrieved
10309:
10299:
10288:. Retrieved
10284:
10275:
10264:. Retrieved
10260:
10250:
10239:. Retrieved
10231:cacm.acm.org
10230:
10220:
10209:. Retrieved
10205:
10195:
10176:
10133:
10129:
10123:
10112:. Retrieved
10108:
10098:
10085:
10075:
10064:. Retrieved
10037:
10024:
10013:. Retrieved
10009:
10000:
9979:
9968:. Retrieved
9964:
9955:
9945:– via
9931:
9924:
9912:. Retrieved
9897:
9869:
9862:
9851:. Retrieved
9833:. Retrieved
9815:. Retrieved
9800:
9789:. Retrieved
9774:
9761:
9755:. Retrieved
9735:
9729:
9718:. Retrieved
9706:
9679:
9675:
9666:
9656:, retrieved
9646:
9639:
9629:, retrieved
9619:
9612:
9604:
9598:. Retrieved
9521:, reciprocal
9512:Wallace tree
9273:
8403:
8034:Real numbers
7701:
7697:
7693:
7689:
7685:
7681:
7637:
7633:
7629:
7628:Numbers can
7627:
7564:
7560:
7538:
7534:
7530:
7528:
7524:
7513:
7502:
7495:
7478:Peano axioms
7471:
7462:real numbers
7460:and then to
7455:
7323:
7121:
7117:
7112:
7108:
7103:
7099:
7093:
6993:
6985:
6981:
6973:
6969:
6967:
6861:
6860:In the book
6859:
6856:Peano axioms
6837:
6821:
6817:
6810:
6806:
6799:
6785:
6781:
6774:
6770:
6763:
6751:preservation
6696:, such that
6664:
6656:
6488:
6439:
6387:
6142:
6124:
6120:
6100:
6096:
6094:
5992:
5988:
5978:
5976:
5962:
5777:
5775:
5666:
5662:
5650:
5644:
5599:
5496:real numbers
5462:
5198:
5103:
5001:
4986:
4982:
4976:
4972:
4967:
4963:
4958:
4954:
4952:
4802:
4779:
4714:
4535:
4505:
4502:
4482:
4475:
4466:
4402:
4394:
4390:
4386:
4382:
4364:
4358:
4355:
4341:
4332:
4327:
4317:
4312:
4286:
4273:Al Khwarizmi
4270:
4264:
4243:
4209:Rod calculus
4202:
4196:
4194:
4168:
4164:
4160:
4156:
4152:
4148:
4144:
4140:
4136:
4132:
4128:
4125:
4100:
4071:
4048:Ishango bone
4045:
4030:
4002:
3992:
3977:
3888:
3880:
3754:
3642:
3529:
3269:
3252:
3077:
2983:
2976:
2833:
2811:
2806:
2729:
2721:
2542:
2505:
2488:
2475:
2467:for details.
2460:
2456:
2352:
2345:
2293:
2254:
2123:
1994:
1967:DOT OPERATOR
1954:dot operator
1953:
1904:
1679:
1615:DOT OPERATOR
1540:
1533:
1531:
1513:
1494:
1491:
1488:
1418:
1413:
1409:
1406:multiplicand
1405:
1401:
1399:
1317:
1240:
1239:
1234:
1233:
1229:multiplicand
1228:
1227:
1217:
1210:
1182:
1165:
1162:dot operator
1156:
1153:cross symbol
1148:
1147:
948:
617:multiplicand
559:
145:
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
10651:Subtraction
7646:quaternions
7533:by element
6844:quaternions
6137:determinant
5985:superscript
4376:reduce the
4283:Grid method
4250:Brahmagupta
4107:sexagesimal
4103:Babylonians
4097:Babylonians
4078:2 Γ 21 = 42
4013:calculators
3953:Computation
3895:quaternions
3891:quaternions
2994:truncations
2697: times
2625: times
2438:Definitions
2341:coefficient
2245:dot product
2128:parentheses
1601:×
1297: times
1203:subtraction
737:denominator
445:Subtraction
10923:Categories
10513:) (1991).
10463:2023-06-23
10438:2021-12-29
10413:2022-04-19
10356:2021-12-29
10315:2021-12-29
10290:2020-08-16
10266:2020-08-16
10241:2020-01-25
10211:2020-01-25
10114:2022-04-22
10066:2014-01-22
10015:2021-12-29
9970:2022-03-15
9914:2015-11-10
9853:2023-09-25
9835:2023-09-25
9817:2023-09-25
9791:2023-09-25
9757:2017-08-03
9720:2017-04-25
9658:2017-03-07
9631:2017-03-07
9600:2017-05-14
9563:References
9556:Slide rule
7684:copies of
6135:where the
6111:Properties
5665:terms, as
4179:See also:
4009:slide rule
2812:In words:
1996:The Lancet
1975:interpunct
1682:arithmetic
1674:See also:
1645:MIDDLE DOT
1619:⋅
1421:properties
1408:) are the
1402:multiplier
1235:multiplier
1195:arithmetic
1188:elementary
607:multiplier
541:difference
502:subtrahend
124:April 2012
94:newspapers
10537:cite book
10257:"Product"
10168:205861906
10160:0885-064X
10143:1407.3360
10056:130132289
10030:Qiu, Jane
9991:1204.1019
9525:Factorial
9146:ϕ
9133:ϕ
9126:
9105:ϕ
9092:ϕ
9085:
9007:ϕ
9003:
8985:ϕ
8981:
8937:ϕ
8933:
8915:ϕ
8911:
8779:−
8727:×
8692:×
8657:×
8625:×
8529:×
8474:−
8459:are zero.
8402:when the
8380:×
8337:×
8311:×
8285:×
8272:−
8259:×
8216:×
8001:high and
7953:×
7936:×
7911:×
7871:×
7826:×
7811:−
7805:×
7796:−
7765:×
7756:−
7747:×
7738:−
7723:−
7717:×
7700:wide and
7665:×
7603:⋅
7548:⋅
7414:×
7402:×
7389:×
7377:×
7350:×
7293:×
7267:×
7240:×
7214:×
7165:×
7046:×
7022:×
7010:×
6990:induction
6978:successor
6941:×
6917:×
6884:×
6707:⋅
6620:−
6614:⋅
6605:−
6556:−
6527:−
6515:⋅
6506:−
6455:⋅
6403:⋅
6357:⋅
6345:⋅
6321:⋅
6276:⋅
6267:⋅
6255:⋅
6246:⋅
6197:⋅
6185:⋅
6161:fractions
6060:∏
6045:⏟
6038:×
6035:⋯
6032:×
6026:×
5913:∏
5907:∞
5904:→
5888:⋅
5853:∏
5847:∞
5844:−
5841:→
5810:∞
5805:∞
5802:−
5792:∏
5731:∏
5725:∞
5722:→
5696:∞
5681:∏
5556:∑
5510:∏
5416:∏
5371:∏
5310:∏
5269:∏
5218:∏
5165:⋅
5162:…
5159:⋅
5153:⋅
5124:∏
5076:⋅
5073:…
5070:⋅
5057:⋅
5014:∏
4925:⋅
4917:−
4906:⋅
4901:⋯
4896:⋅
4877:⋅
4858:⋅
4815:∏
4727:∏
4599:∏
4574:∑
4545:∏
4427:
4277:Fibonacci
4139:, ..., 20
4062:Egyptians
4021:computers
3934:⋅
3908:⋅
3858:ψ
3852:φ
3838:⋅
3832:⋅
3814:⋅
3802:⋅
3784:⋅
3778:−
3772:⋅
3735:ψ
3724:⋅
3709:ψ
3703:
3688:ψ
3682:
3673:⋅
3626:φ
3615:⋅
3600:φ
3594:
3579:φ
3573:
3564:⋅
3493:⋅
3481:⋅
3463:⋅
3457:−
3451:⋅
3422:⋅
3416:⋅
3404:⋅
3384:⋅
3372:⋅
3340:⋅
3291:−
3248:sequences
3228:⋅
3217:∈
3205:∈
3188:⋅
3154:∈
3112:∈
3058:…
3008:π
2953:≠
2903:⋅
2887:⋅
2855:⋅
2780:−
2775:−
2768:−
2751:−
2741:×
2687:⏟
2677:⋯
2635:∑
2631:≡
2615:⏟
2605:⋯
2563:∑
2559:≡
2553:⋅
2523:∈
2465:talk page
2422:π
2419:×
2413:×
2390:π
2370:π
2367:×
2271:(such as
2010:variables
1932:⋅
1881:×
1875:×
1869:×
1863:×
1827:×
1815:×
1809:×
1773:×
1732:×
1701:×
1660:FULL STOP
1501:rectangle
1441:×
1358:×
1329:×
1287:⏟
1277:⋯
1258:×
1175:computers
1173:, or, on
1096:logarithm
1056:
1029:Logarithm
730:numerator
612:×
591:×
497:−
476:−
10697:Division
10630:Addition
10235:Archived
10185:Archived
10136:: 1β30.
10083:(1907).
10060:Archived
9908:Archived
9847:Archived
9829:Archived
9811:Archived
9785:Archived
9748:Archived
9746:. 1982.
9652:archived
9625:archived
9594:Archived
9442:See also
9201:Division
7652:Integers
7642:matrices
7096:integers
6840:matrices
6663:, has a
6661:except 0
6545:, where
6483:Negation
6157:integers
5993:exponent
4782:variable
4493:distance
4401:log log
4393:log log
4219:period.
4155:, and 50
4017:Marchant
2949:′
2910:′
2894:′
2872:′
2864:′
2463:See the
2402:, as is
2353:multiple
2261:asterisk
1964:⋅
1688:(either
1612:⋅
1581:In
1562:Notation
1551:matrices
1535:division
1497:counting
1207:division
1199:addition
1179:asterisk
1177:, by an
972:radicand
871:exponent
800:quotient
789:fraction
706:dividend
677:Division
288:Addition
18:Multiply
10807:Inverse
10760:Primary
10706:∕
10656:−
10646:−
10109:bbc.com
9684:Bibcode
7638:measure
7476:or the
6149:complex
5600:if all
4789:integer
4453:bits).
4175:Chinese
4118:60 Γ 60
4105:used a
4054:era in
4041:Chinese
3995:Germany
3969:tin toy
2348:product
2289:FORTRAN
2263:(as in
2241:vectors
2239:of two
2130:(e.g.,
2016:(e.g.,
2006:algebra
1583:Unicode
1520:derived
1509:lengths
1414:product
1410:factors
1241:factors
1212:product
951:th root
713:divisor
657:product
492:minuend
345:summand
335:summand
240:
228:
224:
212:
208:
196:
162:scaling
108:scholar
10702:÷
10692:÷
10681:·
10677:×
10667:×
10525:
10485:
10385:
10166:
10158:
10054:
10038:Nature
9939:
9885:
9676:Nature
9303:" but
9028:, then
7480:. See
6850:Axioms
6802:< 0
6766:> 0
6159:, and
5498:, and
4953:where
4491:gives
2277:EBCDIC
2259:, the
2249:scalar
2059:times
1961:
1959:U+22C5
1919:period
1758:equals
1690:×
1656:.
1653:
1651:U+002E
1641:·
1638:
1636:U+00B7
1609:
1607:U+22C5
1594:×
1591:
1589:U+00D7
1549:, and
1205:, and
978:degree
596:factor
586:factor
387:addend
377:augend
366:addend
356:addend
110:
103:
96:
89:
81:
10635:+
10625:+
10432:(PDF)
10164:S2CID
10138:arXiv
10090:(PDF)
10052:S2CID
9986:arXiv
9751:(PDF)
9740:(PDF)
9715:(PDF)
7688:when
7634:order
7630:count
7516:field
7498:group
7482:below
7116:when
6968:Here
6820:<
6815:then
6809:>
6804:, if
6784:>
6779:then
6773:>
6768:, if
6756:order
6749:Order
5659:limit
4985:>
4980:; if
4485:speed
4167:and 3
3177:then
3052:3.141
2273:ASCII
1979:comma
1760:six")
1169:, by
1031:(log)
929:power
889:power
811:ratio
115:JSTOR
101:books
10559:and
10543:link
10523:ISBN
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