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al 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 x 1 x s 0 x 0 x 0 x x displaystyle x times 1 x times s 0 x times 0 x 0 x x 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 x p x m y p y m x p y p x m y m x p y m x m y p 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 the rule that 1 1 1 can then be deduced from 0 1 0 1 0 0 1 1 0 1 1 0 1 0 displaystyle 0 1 times 0 1 0 times 0 1 times 1 0 times 1 1 times 0 1 0 multiplication is extended in a similar way to rational numbers and then to real numbers citation needed multiplication with set theory edit 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 39 multiplication in group theory edit 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 the identity element is 1 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 the result is 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 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 zero is excluded this is easily seen by the nonexistence of an inverse for all elements other than 1 and 1 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 a by element b could be notated as a displaystyle cdot b or ab 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 q 0 displaystyle left mathbb q 0 cdot right 40 multiplication of different kinds of numbers edit numbers can count 3 apples order the 3rd apple or measure 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 matrices or do not look much like numbers such as quaternions integers n m displaystyle n times m is the sum of n copies of m when n and m are positive whole numbers this gives the number of things in an array n wide and m high generalization to negative numbers can be done by n m n m n m displaystyle n times m n times m n times m and n m n m displaystyle n times m n times m the same sign rules apply to rational and real numbers rational numbers generalization to fractions a b c d displaystyle frac a b times frac c d is by multiplying the numerators and denominators respectively a b c d a c b d displaystyle frac a b times frac c d frac a times c b times d this gives the area of a rectangle a b displaystyle frac a b high and c d displaystyle frac c d wide and is the same as the number of things in an array when the rational numbers happen to be whole numbers 34 real numbers real numbers and their products can be defined in terms of sequences of rational numbers complex numbers considering complex numbers z 1 displaystyle z_ 1 and z 2 displaystyle z_ 2 as ordered pairs of real numbers a 1 b 1 displaystyle a_ 1 b_ 1 and a 2 b 2 displaystyle a_ 2 b_ 2 the product z 1 z 2 displaystyle z_ 1 times z_ 2 is a 1 a 2 b 1 b 2 a 1 b 2 a 2 b 1 displaystyle a_ 1 times a_ 2 b_ 1 times b_ 2 a_ 1 times b_ 2 a_ 2 times b_ 1 this is the same as for reals a 1 a 2 displaystyle a_ 1 times a_ 2 when the imaginary parts b 1 displaystyle b_ 1 and b 2 displaystyle b_ 2 are zero equivalently denoting 1 displaystyle sqrt 1 as i displaystyle i z 1 z 2 a 1 b 1 i a 2 b 2 i a 1 a 2 a 1 b 2 i b 1 a 2 i b 1 b 2 i 2 a 1 a 2 b 1 b 2 a 1 b 2 b 1 a 2 i 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 34 alternatively in trigonometric form if z 1 r 1 cos ϕ 1 i sin ϕ 1 z 2 r 2 cos ϕ 2 i sin ϕ 2 displaystyle z_ 1 r_ 1 cos phi _ 1 i sin phi _ 1 z_ 2 r_ 2 cos phi _ 2 i sin phi _ 2 then z 1 z 2 r 1 r 2 cos ϕ 1 ϕ 2 i sin ϕ 1 ϕ 2 textstyle z_ 1 z_ 2 r_ 1 r_ 2 cos phi _ 1 phi _ 2 i sin phi _ 1 phi _ 2 34 further generalizations see multiplication in group theory above and multiplicative group which for example includes matrix multiplication a very general and abstract concept of multiplication is as the multiplicatively denoted second binary operation in a ring an example of a ring that is not any of the number systems above is a polynomial ring polynomials can be added and multiplied but polynomials are not numbers in any usual sense division often division x y displaystyle frac x y is the same as multiplication by an inverse x 1 y displaystyle x left frac 1 y right multiplication for some types of numbers may have corresponding division without inverses in an integral domain x may have no inverse 1 x displaystyle frac 1 x but x y displaystyle frac x y may be defined in a division ring there are inverses but x y displaystyle frac x y may be ambiguous in non commutative rings since x 1 y displaystyle x left frac 1 y right need not be the same as 1 y x displaystyle left frac 1 y right x 41 citation needed see also edit dimensional analysis factorial genaille lucas rulers lunar arithmetic multiplication table multiplicative inverse reciprocal napier s bones peasant multiplication product mathematics for generalizations slide rule multiplication algorithm karatsuba algorithm for large numbers toom cook multiplication for very large numbers schönhage strassen algorithm for huge numbers binary multiplier an electronic circuit booth s multiplication algorithm floating point arithmetic multiply accumulate operation a k a fused multiply add wallace tree references edit https files eric ed gov fulltext ej1246116 pdf basics of multiplication nicomachus of gerasa 1926 introduction to arithmetic translated by d ooge martin luther new york macmillan foster colin 2022 03 01 getting multiplication the right way round mathematics in school 51 2 the mathematical association uk 16 17 hdl 2134 19299752 v1 retrieved 2025 05 15 1 2 devlin keith january 2011 what exactly is multiplication mathematical association of america archived from the original on 2017 05 27 retrieved 2017 05 14 with multiplication you have a multiplicand written second multiplied by a multiplier written first devlin keith january 2011 what exactly is multiplication profkeithdevlin org archived from the original on 2024 12 12 retrieved 2024 12 12 1 2 musser gary l peterson blake e burger william f 2013 mathematics for elementary teachers a contemporary approach john wiley sons p 101 isbn 978 1 118 48700 6 1 2 humez alexander humez nicholas 2008 10 02 on the dot the speck that changed the world oxford university press p 103 isbn 978 0 19 971718 7 klose orval 1966 the number systems and operations of arithmetic pergamon press p 39 isbn 978 1 4831 3709 4 cajori florian 1929 a history of mathematical notations the open court publishing company p 267 isbn 1616405716 cite book isbn date incompatibility help victory on points nature 218 5137 111 1968 bibcode 1968natur 218s 111 doi 10 1038 218111c0 the lancet formatting guidelines for electronic submission of manuscripts pdf retrieved 2017 04 25 tall david 1983 introducing algebra on the computer today and tomorrow mathematics in school 12 5 37 40 jstor 30213874 peterson dave 2019 10 14 order of operations implicit multiplication algebra pemdas the math doctors archived from the original on 2023 09 24 retrieved 2023 09 25 peterson dave 2023 08 18 implied multiplication 1 not as bad as you think algebra ambiguity pemdas the math doctors archived from the original on 2023 09 24 retrieved 2023 09 25 peterson dave 2023 08 25 implied multiplication 2 is there a standard algebra arithmetic ambiguity pemdas the math doctors archived from the original on 2023 09 24 retrieved 2023 09 25 peterson dave 2023 09 01 implied multiplication 3 you can t prove it algebra pemdas the math doctors archived from the original on 2023 09 24 retrieved 2023 09 25 gookin dan 2004 c for dummies 2nd ed wiley p 88 isbn 978 0 7645 7392 7 fuller william r 1977 fortran programming a supplement for calculus courses universitext springer p 10 doi 10 1007 978 1 4612 9938 7 isbn 978 0 387 90283 8 multiplicand mathematics britannica www britannica com encyclopædia britannica inc retrieved 2024 11 15 weisstein eric w multiplicand mathworld wolfram com wolfram research inc retrieved 2024 11 15 litvin chester 2012 advance brain stimulation by psychoconduction trafford pp 2 3 5 6 isbn 978 1 4669 0152 0 via google book search multiplication mathematische basteleien de retrieved 2022 03 15 pletser vladimir 2012 04 04 does the ishango bone 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wayback machine hartnett kevin 2019 04 11 mathematicians discover the perfect way to multiply quanta magazine retrieved 2020 01 25 klarreich erica january 2020 multiplication hits the speed limit cacm acm org retrieved 2020 01 25 cite web cs1 maint deprecated archival service link weisstein eric w product mathworld wolfram com retrieved 2020 08 16 summation and product notation math illinoisstate edu retrieved 2020 08 16 weisstein eric w exponentiation mathworld wolfram com retrieved 2021 12 29 1 2 3 4 5 6 7 8 9 multiplication encyclopedia of mathematics retrieved 2021 12 29 1 2 3 4 biggs norman l 2002 discrete mathematics oxford university press p 25 isbn 978 0 19 871369 2 weisstein eric w multiplicative inverse wolfram mathworld retrieved 2022 04 19 angell david ordering complex numbers not pdf unsw sydney school of mathematics and statistics retrieved 2021 12 29 cawagas raoul e carrascal alexander s bautista lincoln a maria john p sta urrutia jackie d nobles bernadeth 2009 the subalgebra structure of the cayley dickson algebra of dimension 32 trigintaduonion arxiv 0907 2047v3 math ra 10 2 building the real numbers mathematics libretexts 2018 04 11 retrieved 2023 06 23 burns gerald 1977 introduction to group theory with applications new york academic press isbn 9780121457501 ervin heather k winter 2017 fraction multiplication and division models a practitioner reference paper pdf international journal of research in education and science 3 1 258 279 issn 2148 9955 further reading edit boyer carl b revised by merzbach uta c 1991 history of mathematics john wiley and sons inc isbn 978 0 471 54397 8 cite book cs1 maint multiple names authors list link external links edit multiplication and arithmetic operations in various number systems at cut the knot modern chinese multiplication techniques on an abacus v t e elementary arithmetic addition subtraction multiplication or division or v t e hyperoperations primary successor addition multiplication exponentiation tetration left inverse predecessor subtraction division root extraction super root right inverse predecessor subtraction division logarithm super logarithm related articles ackermann function conway chained arrow notation grzegorczyk hierarchy knuth s up arrow notation steinhaus moser notation authority control databases international gnd national united states france bnf data japan czech republic israel other yale lux retrieved from https en wikipedia org w index php title multiplication oldid 1377208951 categories multiplication elementary arithmetic mathematical notation hidden categories articles with short description short description is different from wikidata articles needing additional references from april 2012 all articles needing additional references use dmy dates from september 2023 wikipedia articles needing clarification from may 2025 cs1 errors isbn date all articles with unsourced statements articles with unsourced statements from may 2025 articles needing expert 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