Editing Multiplication (section)

===Capital pi notation{{Anchor|Capital Pi notation}}===<!--This section is linked from [[Pi (letter)]], [[Capital Pi notation]], [[Capital pi notation]]-->
{{Further information|Iterated binary operation#Notation}}

The product of a sequence of factors can be written with the product symbol <math>\textstyle \prod</math>, which derives from the capital letter Π (pi) in the [[Greek alphabet]] (much like the same way the [[summation symbol]] <math>\textstyle \sum</math> is derived from the Greek letter Σ (sigma)).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Product|url=https://mathworld.wolfram.com/Product.html|access-date=2020-08-16|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|title=Summation and Product Notation|url=https://math.illinoisstate.edu/day/courses/old/305/contentsummationnotation.html|access-date=2020-08-16|website=math.illinoisstate.edu}}</ref> The meaning of this notation is given by
:<math>\prod_{i=1}^4 (i+1) = (1+1)\,(2+1)\,(3+1)\, (4+1),</math>
which results in
:<math>\prod_{i=1}^4 (i+1) = 120.</math>

In such a notation, the [[variable (mathematics)|variable]] {{mvar|i}} represents a varying [[integer]], called the multiplication index, that runs from the lower value {{math|1}} indicated in the subscript to the upper value {{math|4}} 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.

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.

==== Properties of capital pi notation====
By definition, 
:<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]]:
:<math>\prod_{i=1}^{n}x=x\cdot x\cdot\ldots\cdot x=x^n.</math>

[[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 non-negative integer, or if all <math>x_i</math> are positive [[real number]]s, and
:<math>\prod_{i=1}^{n}x^{a_i} =x^{\sum_{i=1}^{n}a_i}</math>
if all <math>a_i</math> are non-negative integers, or if {{mvar|x}} is a positive real number.