Multiplication Algorithms - Revision history

Khanh at 20:44, 8 January 2022

2022-01-08T20:44:05Z

Khanh at 20:42, 8 January 2022

2022-01-08T20:42:23Z

Khanh: /* Multiplication with set theory */

2022-01-08T20:40:34Z

Multiplication with set theory

Khanh: /* Axioms */

2022-01-08T20:39:17Z

Axioms

Khanh: /* Properties */

2022-01-08T20:37:14Z

Properties

Khanh: /* Infinite products */

2022-01-08T20:20:51Z

Infinite products

Khanh: /* Product of a sequence */

2022-01-08T20:18:03Z

Product of a sequence

Khanh: /* Product of a sequence{{anchor|Product of sequences|Products of sequences}} */

2022-01-08T20:15:26Z

Product of a sequence{{anchor|Product of sequences|Products of sequences}}

Khanh: /* Products of measurements */

2022-01-08T20:12:10Z

Products of measurements

Khanh: /* Computation */

2022-01-08T20:10:33Z

Computation

@@ Line 244: / Line 244: @@
 :The same sign rules apply to rational and real numbers.
-;[[Rational number]]s
+;Rational numbers
 :Generalization to fractions <math>\frac{A}{B}\times \frac{C}{D}</math> is by multiplying the numerators and denominators respectively: <math>\frac{A}{B}\times \frac{C}{D} = \frac{(A\times C)}{(B\times D)}</math>. This gives the area of a rectangle <math>\frac{A}{B}</math> high and <math>\frac{C}{D}</math> wide, and is the same as the number of things in an array when the rational numbers happen to be whole numbers.
-;[[Real number]]s
+;Real numbers
 :Real numbers and their products can be defined in terms of sequences of rational numbers.
-;[[Complex number]]s
+;Complex numbers
 :Considering complex numbers <math>z_1</math> and <math>z_2</math> as ordered pairs of real numbers <math>(a_1, b_1)</math> and <math>(a_2, b_2)</math>, the product <math>z_1\times z_2</math> is <math>(a_1\times a_2 - b_1\times b_2, a_1\times b_2 + a_2\times b_1)</math>. This is the same as for reals <math>a_1\times a_2</math> when the ''imaginary parts'' <math>b_1</math> and <math>b_2</math> are zero.
 :Equivalently, denoting <math>\sqrt{-1}</math> as <math>i</math>, we have <math>z_1 \times z_2 = (a_1+b_1i)(a_2+b_2i)=(a_1 \times a_2)+(a_1\times b_2i)+(b_1\times a_2i)+(b_1\times b_2i^2)=(a_1a_2-b_1b_2)+(a_1b_2+b_1a_2)i.</math>
-:Alternatively, in trigonometric form, if <math>z_1 = r_1(\cos\phi_1+i\sin\phi_1), z_2 = r_2(\cos\phi_2+i\sin\phi_2)</math>, then<math display="inline">z_1z_2 = r_1r_2(\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)).</math><ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref>
+:Alternatively, in trigonometric form, if <math>z_1 = r_1(\cos\phi_1+i\sin\phi_1), z_2 = r_2(\cos\phi_2+i\sin\phi_2)</math>, then<math display="inline">z_1z_2 = r_1r_2(\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)).</math>
 ;Further generalizations

@@ Line 224: / Line 224: @@
 The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below 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 construction of the real numbers.
-==Multiplication in group theory==<!--linked from below-->
+==Multiplication in group theory==
-There are many sets that, under the operation of multiplication, satisfy the axioms that define [[group (mathematics)|group]] structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
+There are many sets that, under the operation of multiplication, satisfy the axioms that define group structure. These axioms are closure, associativity, and the inclusion of an identity element and inverses.
-A simple example is the set of non-zero [[rational numbers]]. Here we have identity 1, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, we must exclude zero 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, we have an [[abelian group]], but that is not always the case.
+A simple example is the set of non-zero rational numbers. Here we have identity 1, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, we must exclude zero 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, we have an abelian group, but that is not always the case.
-To see this, consider the set of invertible square matrices of a given dimension over a given [[field (mathematics)|field]]. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the [[identity matrix]]) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.
+To see this, consider the set of invertible square matrices of a given dimension over a given field. Here, it is straightforward to verify closure, associativity, and inclusion of identity (the identity matrix) and inverses. However, matrix multiplication is not commutative, which shows that this group is non-abelian.
 Another fact worth noticing is that the integers under multiplication do not form a group—even if we exclude zero. This is easily seen by the nonexistence of an inverse for all elements other than 1 and&nbsp;−1.

@@ Line 222: / Line 222: @@
 ==Multiplication with set theory==
-The product of non-negative integers can be defined with set theory using [[Cardinal number#Cardinal multiplication|cardinal numbers]] or the [[Peano axioms#Arithmetic|Peano axioms]]. See [[#Multiplication of different kinds of numbers|below]] 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 [[construction of the real numbers]].{{Citation needed|date=December 2021}}
+The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms. See below 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 construction of the real numbers.
 ==Multiplication in group theory==<!--linked from below-->

@@ Line 206: / Line 206: @@
 ==Axioms==
-In the book ''[[Arithmetices principia, nova methodo exposita]]'', [[Giuseppe Peano]] proposed axioms for arithmetic based on his axioms for natural numbers.<ref>{{cite web |url=http://planetmath.org/encyclopedia/PeanoArithmetic.html |title=Peano arithmetic |publisher=[[PlanetMath]] |access-date=2007-06-03 |archive-url=https://web.archive.org/web/20070819031025/http://planetmath.org/encyclopedia/PeanoArithmetic.html |archive-date=2007-08-19 |url-status=live }}</ref> Peano arithmetic has two axioms for multiplication:
+In the book ''Arithmetices principia, nova methodo exposita'', Giuseppe Peano proposed axioms for arithmetic based on his axioms for natural numbers. Peano arithmetic has two axioms for multiplication:
 :<math>x \times 0 = 0</math>
 :<math>x \times S(y) = (x \times y) + x</math>
-Here ''S''(''y'') represents the [[Successor ordinal|successor]] of ''y''; i.e., the natural number that follows ''y''. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including [[Mathematical induction|induction]]. For instance, ''S''(0), denoted by 1, is a multiplicative identity because
+Here ''S''(''y'') represents the successor of ''y''; i.e., the natural number that follows ''y''. The various properties like associativity can be proved from these and the other axioms of Peano arithmetic, including induction. For instance, ''S''(0), denoted by 1, is a multiplicative identity because
 :<math>x \times 1 = x \times S(0) = (x \times 0) + x = 0 + x = x.</math>
-The axioms for [[integer]]s typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (''x'',''y'') as equivalent to {{nowrap|''x'' − ''y''}} when ''x'' and ''y'' are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
+The axioms for integers typically define them as equivalence classes of ordered pairs of natural numbers. The model is based on treating (''x'',''y'') as equivalent to ''x'' − ''y'' when ''x'' and ''y'' are treated as integers. Thus both (0,1) and (1,2) are equivalent to −1. The multiplication axiom for integers defined this way is
 :<math>(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).</math>
@@ Line 219: / Line 219: @@
 :<math>(0, 1) \times (0, 1) = (0 \times 0 + 1 \times 1,\, 0 \times 1 + 1 \times 0) = (1,0).</math>
-Multiplication is extended in a similar way to [[rational number]]s and then to [[real number]]s.{{Citation needed|date=December 2021}}
+Multiplication is extended in a similar way to rational numbers and then to real numbers.
 ==Multiplication with set theory==

@@ Line 163: / Line 163: @@
 ==Properties==
-[[Image:Multiplication chart.svg|thumb|right|Multiplication of numbers 0–10. Line labels = multiplicand. X-axis = multiplier. Y-axis = product.<br>Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.<br>Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a [[singular matrix]] where the [[determinant]] is 0. In this process, information is lost and cannot be regained.]]
+[[Image:Multiplication chart.svg|thumb|right|Multiplication of numbers 0–10. Line labels = multiplicand. X-axis = multiplier. Y-axis = product.<br>Extension of this pattern into other quadrants gives the reason why a negative number times a negative number yields a positive number.<br>Note also how multiplication by zero causes a reduction in dimensionality, as does multiplication by a singular matrix where the determinant is 0. In this process, information is lost and cannot be regained.]]
-For [[real number|real]] and [[complex number|complex]] numbers, which includes, for example, [[natural number]]s, [[integer]]s, and [[rational number|fractions]], multiplication has certain properties:
+For real and complex numbers, which includes, for example, natural numbers, integers, and fractions, multiplication has certain properties:
-;[[Commutative property]]
+;Commutative property
 :The order in which two numbers are multiplied does not matter:
-::<math>x\cdot y = y\cdot x.</math><ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref><ref>{{Cite web|title=multiplication|url=https://planetmath.org/Multiplication|access-date=2021-12-29|website=planetmath.org}}</ref><ref name=":1">{{Cite book|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Oxford University Press|year=2002|isbn=978-0-19-871369-2|pages=25|language=en}}</ref>
+::<math>x\cdot y = y\cdot x.</math>
-;[[Associative property]]
+;Associative property
-:Expressions solely involving multiplication or addition are invariant with respect to the [[order of operations]]:
+:Expressions solely involving multiplication or addition are invariant with respect to the order of operations:
-::<math>(x\cdot y)\cdot z = x\cdot(y\cdot z)</math><ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref><ref name=":1">{{Cite book|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Oxford University Press|year=2002|isbn=978-0-19-871369-2|pages=25|language=en}}</ref>
+::<math>(x\cdot y)\cdot z = x\cdot(y\cdot z)</math>
-;[[Distributive property]]
+;Distributive property
 :Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:
-::<math>x\cdot(y + z) = x\cdot y + x\cdot z </math><ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref><ref name=":1">{{Cite book|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Oxford University Press|year=2002|isbn=978-0-19-871369-2|pages=25|language=en}}</ref>
+::<math>x\cdot(y + z) = x\cdot y + x\cdot z </math>
-;[[Identity element]]
+;Identity element
 :The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the '''identity property''':
-::<math>x\cdot 1 = x</math><ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref><ref name=":1">{{Cite book|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Oxford University Press|year=2002|isbn=978-0-19-871369-2|pages=25|language=en}}</ref>
+::<math>x\cdot 1 = x</math>
-;[[Absorbing element|Property of 0]]
+;Property of 0
 :Any number multiplied by 0 is 0. This is known as the '''zero property''' of multiplication:
-::<math>x\cdot 0 = 0</math><ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref>
+::<math>x\cdot 0 = 0</math>
-;[[Additive inverse|Negation]]
+;Negation
-:−1 times any number is equal to the '''[[additive inverse]]''' of that number.
+:−1 times any number is equal to the '''additive inverse''' of that number.
 ::<math>(-1)\cdot x = (-x)</math> where <math>(-x)+x=0</math>
 :–1 times –1 is 1.
-::<math>(-1)\cdot (-1) = 1</math>{{Citation needed|date=December 2021}}
+::<math>(-1)\cdot (-1) = 1</math>
-;[[Inverse element]]
+;Inverse element
-:Every number ''x'', [[division by zero|except 0]], has a '''[[multiplicative inverse]]''', <math>\frac{1}{x}</math>, such that <math>x\cdot\left(\frac{1}{x}\right) = 1</math>.{{Citation needed|date=December 2021}}
+:Every number ''x'', except 0, has a '''multiplicative inverse''', <math>\frac{1}{x}</math>, such that <math>x\cdot\left(\frac{1}{x}\right) = 1</math>.
-;[[Order theory|Order]] preservation
+;Order preservation
-:Multiplication by a positive number preserves the [[Order theory|order]]:
+:Multiplication by a positive number preserves the order:
-::For {{nowrap|''a'' > 0}}, if {{nowrap|''b'' > ''c''}} then {{nowrap|''ab'' > ''ac''}}.
+::For ''a'' > 0, if ''b'' > ''c'' then ''ab'' > ''ac''.
 :Multiplication by a negative number reverses the order:
-::For {{nowrap|''a'' < 0}}, if {{nowrap|''b'' > ''c''}} then {{nowrap|''ab'' < ''ac''}}.{{Citation needed|date=December 2021}}
+::For ''a'' < 0, if ''b'' > ''c'' then ''ab'' < ''ac''.
-:The [[complex number]]s do not have an ordering that is compatible with both addition and multiplication.<ref>{{Cite web|last=Angell|first=David|title=ORDERING COMPLEX NUMBERS... NOT*|url=https://web.maths.unsw.edu.au/~angell/articles/complexorder.pdf|url-status=live|access-date=29 December 2021|website=web.maths.unsw.edu.au}}</ref><ref>{{Cite web|title=Total ordering on complex numbers|url=https://math.stackexchange.com/questions/487997/total-ordering-on-complex-numbers|access-date=2021-12-29|website=Mathematics Stack Exchange}}</ref>
+:The complex numbers do not have an ordering that is compatible with both addition and multiplication.
-Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for [[Matrix (mathematics)|matrices]] and [[quaternion]]s.<ref name=":0">{{Cite web|title=Multiplication - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Multiplication|access-date=2021-12-29|website=encyclopediaofmath.org}}</ref>
+Other mathematical systems that include a multiplication operation may not have all these properties. For example, multiplication is not, in general, commutative for matrices and quaternions.
 ==Axioms==

@@ Line 160: / Line 160: @@
 One can similarly replace ''m'' with negative infinity, and define:
 :<math>\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),</math>
-provided both limits exist.{{Citation needed|date=December 2021}}
+provided both limits exist.
 ==Properties==

@@ Line 128: / Line 128: @@
 ==Product of a sequence==
 ===Capital pi notation===
-The product of a sequence of factors can be written with the product symbol, which derives from the capital letter <math>\textstyle \prod</math> (pi) in the Greek alphabet (much like the same way the capital letter <math>\textstyle \sum</math> (sigma) is used in the context of summation). Unicode position {{unichar|220F}} contains a glyph for denoting such a product, distinct from {{unichar|03A0}}, the letter. The meaning of this notation is given by:
+The product of a sequence of factors can be written with the product symbol, which derives from the capital letter <math>\textstyle \prod</math> (pi) in the Greek alphabet (much like the same way the capital letter <math>\textstyle \sum</math> (sigma) is used in the context of summation). Unicode position U+220F ∏ contains a glyph for denoting such a product, distinct from U+03A0 Π, the letter. The meaning of this notation is given by:
 :<math>\prod_{i=1}^4 i = 1\cdot 2\cdot 3\cdot 4,</math>
 that is

@@ Line 126: / Line 126: @@
 :4.5 residents per house × 20 houses = 90 residents
-==Product of a sequence{{anchor|Product of sequences|Products of sequences}}==
+==Product of a sequence==
 ===Capital pi notation===
-The product of a sequence of factors can be written with the product symbol, which derives from the capital letter <math>\textstyle \prod</math> (pi) in the [[Greek alphabet]] (much like the same way the capital letter <math>\textstyle \sum</math> (sigma) is used in the context of [[summation]]). Unicode position {{unichar|220F}} contains a glyph for denoting such a product, distinct from {{unichar|03A0}}, the letter. The meaning of this notation is given by:
+The product of a sequence of factors can be written with the product symbol, which derives from the capital letter <math>\textstyle \prod</math> (pi) in the Greek alphabet (much like the same way the capital letter <math>\textstyle \sum</math> (sigma) is used in the context of summation). Unicode position {{unichar|220F}} contains a glyph for denoting such a product, distinct from {{unichar|03A0}}, the letter. The meaning of this notation is given by:
 :<math>\prod_{i=1}^4 i = 1\cdot 2\cdot 3\cdot 4,</math>
 that is
 :<math>\prod_{i=1}^4 i = 24.</math>
-The subscript gives the symbol for a [[free variables and bound variables|bound variable]] (''i'' in this case), called the "index of multiplication", together with its lower bound (''1''), whereas the superscript (here ''4'') gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to (and including) the upper bound. For example:
+The subscript gives the symbol for a bound variable (''i'' in this case), called the "index of multiplication", together with its lower bound (''1''), whereas the superscript (here ''4'') gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to (and including) the upper bound. For example:
 :<math>\prod_{i=1}^6 i = 1\cdot 2\cdot 3\cdot 4\cdot 5 \cdot 6 = 720.</math>
 More generally, the notation is defined as
 :<math>\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,</math>
-where ''m'' and ''n'' are integers or expressions that evaluate to integers. In the case where {{nowrap|1=''m'' = ''n''}}, the value of the product is the same as that of the single factor ''x''<sub>''m''</sub>; if {{nowrap|''m'' > ''n''}}, the product is an [[empty product]] whose value is&nbsp;1—regardless of the expression for the factors.
+where ''m'' and ''n'' are integers or expressions that evaluate to integers. In the case where ''m'' = ''n'', the value of the product is the same as that of the single factor ''x''<sub>''m''</sub>; if ''m'' > ''n'', the product is an empty product whose value is&nbsp;1—regardless of the expression for the factors.
 ==== Properties of capital pi notation====
@@ Line 144: / Line 144: @@
 :<math>\prod_{i=1}^{n}x_i=x_1\cdot x_2\cdot\ldots\cdot x_n.</math>
-If all factors are identical, a product of {{mvar|n}} factors is equivalent to [[exponentiation]]:
+If all factors are identical, a product of {{mvar|n}} factors is equivalent to exponentiation:
 :<math>\prod_{i=1}^{n}x=x\cdot x\cdot\ldots\cdot x=x^n.</math>
-[[Associativity]] and [[commutativity]] of multiplication imply
+Associativity and commutativity of multiplication imply
 :<math>\prod_{i=1}^{n}{x_iy_i} =\left(\prod_{i=1}^{n}x_i\right)\left(\prod_{i=1}^{n}y_i\right)</math> and
 :<math>\left(\prod_{i=1}^{n}x_i\right)^a =\prod_{i=1}^{n}x_i^a</math>
-if {{mvar|a}} is a nonnegative integer, or if all <math>x_i</math> are positive [[real number]]s, and
+if {{mvar|a}} is a nonnegative integer, or if all <math>x_i</math> are positive real numbers, and
 :<math>\prod_{i=1}^{n}x^{a_i} =x^{\sum_{i=1}^{n}a_i}</math>
 if all <math>a_i</math> are nonnegative integers, or if {{mvar|x}} is a positive real number.
 ===Infinite products===
-One may also consider products of infinitely many terms; these are called [[infinite product]]s. Notationally, this consists in replacing ''n'' above by the [[Infinity symbol]] ∞. The product of such an infinite sequence is defined as the [[limit of a sequence|limit]] of the product of the first ''n'' terms, as ''n'' grows without bound. That is,
+One may also consider products of infinitely many terms; these are called infinite products. Notationally, this consists in replacing ''n'' above by the Infinity symbol ∞. The product of such an infinite sequence is defined as the limit of the product of the first ''n'' terms, as ''n'' grows without bound. That is,
 :<math>\prod_{i=m}^\infty x_i = \lim_{n\to\infty} \prod_{i=m}^n x_i.</math>

@@ Line 112: / Line 112: @@
 ==Products of measurements==
-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:<ref name="Devlin"/>
+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:
 :[4 bags] × [3 marbles per bag] = 12 marbles.
-When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by [[dimensional analysis]]. This analysis is routinely applied in physics, but it also has applications in finance and other applied fields.
+When two measurements are multiplied together, the product is of a type depending on the types of measurements. The general theory is given by dimensional analysis. This analysis is routinely applied in physics, but it also has applications in finance and other applied fields.
-A common example in physics is the fact that multiplying [[speed]] by [[Time in physics|time]] gives [[distance]]. For example:
+A common example in physics is the fact that multiplying speed by time gives distance. For example:
 :50 kilometers per hour × 3 hours = 150 kilometers.
 In this case, the hour units cancel out, leaving the product with only kilometer units.

@@ Line 53: / Line 53: @@
 [[file:צעצוע מכני משנת 1918 לחישובי לוח הכפל The Educated Monkey.jpg|200px|right|thumb|The Educated Monkey – a tin toy dated 1918, used as a multiplication "calculator". <small>For example: set the monkey's feet to 4 and 9, and get the product – 36 – in its hands.</small>]]
-Many common methods for multiplying numbers using pencil and paper require a [[multiplication table]] of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the [[Ancient Egyptian multiplication|peasant multiplication]] algorithm, does not. The example below illustrates "long multiplication" (the  "standard algorithm", "grade-school multiplication"):
+Many common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9). However, one method, the peasant multiplication algorithm, does not. The example below illustrates "long multiplication" (the  "standard algorithm", "grade-school multiplication"):
         23958233
@@ Line 65: / Line 65: @@
     139676498390 ( = 139,676,498,390        )
-Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. [[Common logarithm]]s were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The [[slide rule]] allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical [[calculator]]s, such as the [[Marchant Calculator|Marchant]], automated multiplication of up to 10-digit numbers. Modern electronic [[computer]]s and calculators have greatly reduced the need for multiplication by hand.
+Multiplying numbers to more than a couple of decimal places by hand is tedious and error-prone. Common logarithms were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculators, such as the Marchant, automated multiplication of up to 10-digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
 ===Historical algorithms===
-Methods of multiplication were documented in the writings of [[Ancient Egypt|ancient Egyptian]], {{Citation needed span|text=Greek, Indian|date=December 2021|reason=This claim is not sourced in the subsections below.}}, and [[History of China#Ancient China|Chinese]] civilizations.
+Methods of multiplication were documented in the writings of ancient Egyptian, Greek, Indian, and Chinese civilizations.
-The [[Ishango bone]], dated to about 18,000 to 20,000&nbsp;BC, may hint at a knowledge of multiplication in the [[Upper Paleolithic]] era in [[Central Africa]], but this is speculative.
+The Ishango bone, dated to about 18,000 to 20,000&nbsp;BC, may hint at a knowledge of multiplication in the Upper Paleolithic era in Central Africa, but this is speculative.
 ====Egyptians====
-The Egyptian method of multiplication of integers and fractions, which is documented in the [[Rhind Mathematical Papyrus]], was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining {{nowrap|1=2 × 21 = 42}}, {{nowrap|1=4 × 21 = 2 × 42 = 84}}, {{nowrap|1=8 × 21 =  2 × 84 = 168}}. The full product could then be found by adding the appropriate terms found in the doubling sequence:<ref>{{Cite web|title=Peasant Multiplication|url=http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml|access-date=2021-12-29|website=www.cut-the-knot.org}}</ref>
+The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42, 4 × 21 = 2 × 42 = 84, 8 × 21 =  2 × 84 = 168. The full product could then be found by adding the appropriate terms found in the doubling sequence:
 :13 × 21 = (1 + 4 + 8) × 21 = (1 × 21) + (4 × 21) + (8 × 21) = 21 + 84 + 168 = 273.
 ====Babylonians====
-The [[Babylonians]] used a [[sexagesimal]] [[positional number system]], analogous to the modern-day [[decimal expansion|decimal system]]. Thus, Babylonian multiplication was very similar to modern decimal multiplication.  Because of the relative difficulty of remembering {{nowrap|60 × 60}} different products, Babylonian mathematicians employed [[multiplication table]]s. These tables consisted of a list of the first twenty multiples of a certain ''principal number'' ''n'': ''n'', 2''n'', ..., 20''n''; followed by the multiples of 10''n'': 30''n'' 40''n'', and 50''n''. Then to compute any sexagesimal product, say 53''n'', one only needed to add 50''n'' and 3''n'' computed from the table.{{Citation needed|date=December 2021}}
+The Babylonians used a sexagesimal positional number system, analogous to the modern-day decimal system. Thus, Babylonian multiplication was very similar to modern decimal multiplication.  Because of the relative difficulty of remembering 60 × 60 different products, Babylonian mathematicians employed multiplication tables. These tables consisted of a list of the first twenty multiples of a certain ''principal number'' ''n'': ''n'', 2''n'', ..., 20''n''; followed by the multiples of 10''n'': 30''n'' 40''n'', and 50''n''. Then to compute any sexagesimal product, say 53''n'', one only needed to add 50''n'' and 3''n'' computed from the table.
 ====Chinese====
-[[File:Multiplication algorithm.GIF|thumb|right|250px|{{nowrap|1=38 × 76 = 2888}}]]
+[[File:Multiplication algorithm.GIF|thumb|right|250px|38 × 76 = 2888]]
-In the mathematical text ''[[Zhoubi Suanjing]]'', dated prior to 300&nbsp;BC, and the ''[[Nine Chapters on the Mathematical Art]]'', multiplication calculations were written out in words, although the early Chinese mathematicians employed [[Rod calculus]] involving place value addition, subtraction, multiplication, and division. The Chinese were already using a [[Chinese multiplication table|decimal multiplication table]] by the end of the [[Warring States]] period.<ref name="Nature">{{cite journal | url =http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482 | title =Ancient times table hidden in Chinese bamboo strips | journal =Nature | author =Jane Qiu | date =7 January 2014 | access-date =22 January 2014 | doi =10.1038/nature.2014.14482 | s2cid =130132289 | archive-url =https://web.archive.org/web/20140122064930/http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482 | archive-date =22 January 2014 | url-status =live }}</ref>
+In the mathematical text ''Zhoubi Suanjing'', dated prior to 300&nbsp;BC, and the ''Nine Chapters on the Mathematical Art'', multiplication calculations were written out in words, although the early Chinese mathematicians employed Rod calculus involving place value addition, subtraction, multiplication, and division. The Chinese were already using a decimal multiplication table by the end of the Warring States period.
 ===Modern methods===
-[[Image:Gelosia multiplication 45 256.png|right|250px|thumb|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 {{nowrap|1=45 × 256 = 11520}}. This is a variant of [[Lattice multiplication]].]]
+[[Image:Gelosia multiplication 45 256.png|right|250px|thumb|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 45 × 256 = 11520. This is a variant of Lattice multiplication.]]
-The modern method of multiplication based on the [[Hindu–Arabic numeral system]] was first described by [[Brahmagupta]]. Brahmagupta gave rules for addition, subtraction, multiplication, and division. [[Henry Burchard Fine]], then a professor of mathematics at [[Princeton University]], wrote the following:
+The modern method of multiplication based on the Hindu–Arabic numeral system was first described by Brahmagupta. Brahmagupta gave rules for addition, subtraction, multiplication, and division. Henry Burchard Fine, then a professor of mathematics at Princeton University, wrote the following:
-:''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 division they did cumbrously.''<ref>{{cite book |last=Fine |first=Henry B. |author-link=Henry Burchard Fine |title=The Number System of Algebra – Treated Theoretically and Historically |edition=2nd |date=1907 |page=90 |url=https://archive.org/download/numbersystemofal00fineuoft/numbersystemofal00fineuoft.pdf}}</ref>
+:''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 division they did cumbrously.''
-These place value decimal arithmetic algorithms were introduced to Arab countries by [[Al Khwarizmi]] in the early 9th&nbsp;century and popularized in the Western world by [[Fibonacci]] in the 13th century.{{Citation needed|date=December 2021}}
+These place value decimal arithmetic algorithms were introduced to Arab countries by Al Khwarizmi in the early 9th&nbsp;century and popularized in the Western world by Fibonacci in the 13th century.
 ====Grid method====
-[[Grid method multiplication]], 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:
+Grid method multiplication, 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:
 :{| class="wikitable" style="text-align: center;"
@@ Line 109: / Line 109: @@
 ===Computer algorithms===
-The classical method of multiplying two {{math|''n''}}-digit numbers requires {{math|''n''<sup>2</sup>}} digit multiplications. [[Multiplication algorithm]]s have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the [[Discrete Fourier transform#Multiplication of large integers|discrete Fourier transform]] reduce the [[computational complexity]] to {{math|''O''(''n'' log ''n'' log log ''n'')}}. In 2016, the factor {{math|log log ''n''}} was replaced by a function that increases much slower, though still not constant.<ref>{{Cite journal|last1=Harvey|first1=David|last2=van der Hoeven|first2=Joris|last3=Lecerf|first3=Grégoire|title=Even faster integer multiplication|year=2016|journal=Journal of Complexity|volume=36|pages=1–30|doi=10.1016/j.jco.2016.03.001|issn=0885-064X|arxiv=1407.3360|s2cid=205861906}}</ref> In March 2019, David Harvey and Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of <math>O(n\log n).</math><ref>David Harvey, Joris Van Der Hoeven (2019). [https://hal.archives-ouvertes.fr/hal-02070778 Integer multiplication in time O(n log n)] {{Webarchive|url=https://web.archive.org/web/20190408180939/https://hal.archives-ouvertes.fr/hal-02070778 |date=2019-04-08 }}</ref> The algorithm, also based on the fast Fourier transform, is conjectured to be asymptotically optimal.<ref>{{Cite web|url=https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/|title=Mathematicians Discover the Perfect Way to Multiply|last=Hartnett|first=Kevin|website=Quanta Magazine|date=11 April 2019|language=en|access-date=2020-01-25}}</ref> The algorithm is not practically useful, as it only becomes faster for multiplying extremely large numbers (having more than {{math|2<sup>1729<sup>12</sup></sup>}} bits).<ref>{{Cite web|url=https://cacm.acm.org/magazines/2020/1/241707-multiplication-hits-the-speed-limit/fulltext|title=Multiplication Hits the Speed Limit|last=Klarreich|first=Erica|website=cacm.acm.org|language=en|access-date=2020-01-25|archive-url=http://archive.today/2020.10.31-123457/https://cacm.acm.org/magazines/2020/1/241707-multiplication-hits-the-speed-limit/fulltext|archive-date=31 October 2020|url-status=live}}</ref>
+The classical method of multiplying two {{math|''n''}}-digit numbers requires {{math|''n''<sup>2</sup>}} digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers. Methods based on the discrete Fourier transform reduce the computational complexity to {{math|''O''(''n'' log ''n'' log log ''n'')}}. In 2016, the factor {{math|log log ''n''}} 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 <math>O(n\log n).</math> 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 {{math|2<sup>1729<sup>12</sup></sup>}} bits).
 ==Products of measurements==