Editing Multiplication (section)

==Definitions==

{{Expert needed|mathematics
| talk = Merging new section with "Multiplication of Different Kinds of Numbers"
| reason = defining multiplication is not straightforward and different proposals have been made over the centuries, with competing ideas (e.g. recursive vs. non-recursive definitions)
| section = yes
| date = September 2023|section
}}

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.

===Product of two natural numbers===
[[File:Three by Four.svg|thumb|3 by 4 is 12.]]

The product of two natural numbers <math>r,s\in\mathbb{N}</math> is defined as:

<math display="block"> 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}} . </math>

===Product of two integers===
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 [[absolute value|positive amounts]], combined with the sign derived from the following rule:

{| class="wikitable" style="margin-left:1.6em; text-align: center;"
! style="padding:0.2em 1em;" | {{math|×}}
! style="padding:0.2em 1em;" | {{math|+}}
! style="padding:0.2em 1em;" | {{math|−}}
|-
! style="padding:0.2em 1em;" | {{math|+}}
| {{math|+}} || {{math|−}}
|-
! style="padding:0.2em 1em;" | {{math|−}}
| {{math|−}} || {{math|+}}
|}

(This rule is a consequence of the [[distributivity]] of multiplication over addition, and is not an ''additional rule''.)

In words:

* A positive number multiplied by a positive number is positive (product of natural numbers),
* A positive number multiplied by a negative number is negative,
* A negative number multiplied by a positive number is negative,
* A negative number multiplied by a negative number is positive.

===Product of two fractions===
Two fractions can be multiplied by multiplying their numerators and denominators:

:<math display="block"> \frac{z}{n} \cdot \frac{z'}{n'} = \frac{z\cdot z'}{n\cdot n'} , </math>
:which is defined when <math> n,n'\neq  0 </math>.

=== Product of two real numbers ===

There are several equivalent ways to define formally the real numbers; see [[Construction of the real numbers]]. The definition of multiplication is a part of all these definitions. 

A fundamental aspect of these definitions is that every real number can be approximated to any accuracy by [[rational number]]s. A standard way for expressing this is that every real number is the [[least upper bound]] of a set of rational numbers. In particular, every positive real number is the least upper bound of the [[truncation]]s of its infinite [[decimal representation]]; for example, <math>\pi</math> is the least upper bound of <math>\{3,\; 3.1,\; 3.14,\; 3.141,\ldots\}.</math>

A fundamental property of real numbers is that rational approximations are compatible with [[arithmetic operation]]s, and, in particular, with multiplication. This means that, if {{mvar|a}} and {{mvar|b}} are positive real numbers such that <math>a=\sup_{x\in A} x</math> and <math>b=\sup_{y\in B} y,</math> then <math>a\cdot b=\sup_{x\in A, y\in B}x\cdot y.</math> In particular, the product of two positive real numbers is the least upper bound of the term-by-term products of the [[sequence]]s of their decimal representations.

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 {{slink|#Product of two integers}}. The construction of the real numbers through [[Cauchy sequence]]s is often preferred in order to avoid consideration of the four possible sign configurations.

===Product of two complex numbers===
Two complex numbers can be multiplied by the distributive law and the fact that <math> i^2=-1</math>, as follows:
:<math>\begin{align}
  (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{align}</math>

[[File:Komplexe zahlenebene.svg|thumb|upright=1.25|A complex number in polar coordinates]]

The geometric meaning of complex multiplication can be understood by rewriting complex numbers in [[polar coordinates]]:

:<math>a + b\, i = r \cdot ( \cos(\varphi) +  i \sin(\varphi) ) = r \cdot  e ^{ i \varphi} </math>

Furthermore,
:<math>c + d\, i = s \cdot ( \cos(\psi) +  i\sin(\psi) ) = s \cdot e^{i\psi},</math>

from which one obtains
:<math>(a \cdot c - b \cdot d) + (a \cdot d + b \cdot c) i = r \cdot s \cdot e^{i(\varphi + \psi)}.</math>

The geometric meaning is that the magnitudes are multiplied and the arguments are added.

===Product of two quaternions===
The product of two [[quaternion]]s can be found in the article on [[quaternions]]. Note, in this case, that <matH>a \cdot b</math> and <math>b \cdot a</matH> are in general different.