Editing Empty product (section)

=== Relevance of defining empty products ===
The notion of an empty product is useful for the same reason that the number [[0|zero]] and the [[empty set]] are useful: while they seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentation of many subjects.

For example, the empty products 0!&nbsp;=&nbsp;1 (the [[factorial]] of zero) and ''x''<sup>0</sup>&nbsp;=&nbsp;1 shorten [[Taylor series#Definition|Taylor series notation]] (see [[zero to the power of zero]] for a discussion of when ''x''&nbsp;=&nbsp;0). Likewise, if ''M'' is an ''n''&nbsp;×&nbsp;''n'' matrix, then ''M''<sup>0</sup> is the ''n''&nbsp;×&nbsp;''n'' [[identity matrix]], reflecting the fact that applying a [[linear map]] zero times has the same effect as applying the [[identity function|identity map]].

As another example, the [[fundamental theorem of arithmetic]] says that every positive [[integer]] greater than 1 can be written uniquely as a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and its proof) become longer.<ref>{{cite web |url=http://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EWD1073.html |title=How Computing Science created a new mathematical style |author=[[Edsger Wybe Dijkstra]] |date=1990-03-04 |work=EWD |access-date=2010-01-20 | quote=Hardy and Wright: 'Every positive integer, except 1, is a product of primes', Harold M. Stark: 'If ''n'' is an integer greater than 1, then either ''n'' is prime or ''n'' is a finite product of primes'. These examples — which I owe to A. J. M. van Gasteren — both reject the empty product, the last one also rejects the product with a single factor.}}</ref><ref>{{cite web |url=https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD993.html |title=The nature of my research and why I do it |author=[[Edsger Wybe Dijkstra]] |date=1986-11-14 |work=EWD |access-date=2024-03-22 |quote=But also 0 is certainly finite and by defining the product of 0 factors — how else? — to be equal to 1 we can do away with the exception: 'If ''n'' is a positive integer, then ''n'' is a finite product of primes.' }}</ref>

More examples of the use of the empty product in mathematics may be found in the [[binomial theorem]] (which assumes and implies that ''x''<sup>0</sup> = 1 for all ''x''), [[Stirling number]], [[König's theorem (set theory)|König's theorem]], [[binomial type]], [[binomial series]], [[difference operator]] and [[Pochhammer symbol]].