Multiplication: Difference between revisions

Revision as of 07:51, 7 January 2020

Revision as of 08:09, 9 January 2020

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==Products of sequences==<!--linked from below-->

===Capital pi notation===<!--This section is linked from [[Pi (letter)]], [[Capital Pi notation]], [[Capital pi notation]]-->

The product of a sequence of termsfactors can be written with the product symbol, which derives from the capital letter Π (pi) in the [[Greek alphabet]]. 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

:<math>\prod_{i=1}^4 i = 24.</math>

The subscript gives the symbol for a [[free variables and bound variables|dummybound 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. So, 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 case {{nowrap|1=''m'' = ''n''}}, the value of the product is the same as that of the single factor ''x''_''m''. If {{nowrap|''m'' > ''n''}}, the product is thean [[empty product]], with thewhose value is&nbsp;1, regardless of the expression for the factors.

===Infinite products===