Editing Multiplication (section)

==Multiplication of different kinds of numbers==<!--linked from above-->
Numbers can ''count'' (3&nbsp;apples), ''order'' (the 3rd&nbsp;apple), or ''measure'' (3.5&nbsp;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 [[Matrix (mathematics)|matrices]]) or do not look much like numbers (such as [[quaternion]]s).

; Integers
: <math>N\times M</math> 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
: <math>N\times (-M) = (-N)\times M = - (N\times M)</math> and
: <math>(-N)\times (-M) = N\times M</math>
: The same sign rules apply to rational and real numbers.

; [[Rational number]]s
: 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.<ref name=":0"/>

; [[Real number]]s
: Real numbers and their products [[Construction of the real numbers#Construction from Cauchy sequences|can be defined in terms of sequences of rational numbers]].

; [[Complex number]]s
: 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>, <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><ref name=":0"/>
: 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"/>

; Further generalizations
: See [[#Multiplication in group theory|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 (mathematics)|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, <math>\frac{x}{y}</math>, is the same as multiplication by an inverse, <math>x\left(\frac{1}{y}\right)</math>. Multiplication for some types of "numbers" may have corresponding division, without inverses; in an [[integral domain]] ''x'' may have no inverse "<math>\frac{1}{x}</math>" but <math>\frac{x}{y}</math> may be defined. In a [[division ring]] there are inverses, but <math>\frac{x}{y}</math> may be ambiguous in non-commutative rings since <math>x\left(\frac{1}{y}\right)</math> need not be the same as <math>\left(\frac{1}{y}\right)x</math>.{{Citation needed|date=December 2021}}