The term empty product is most often used in the above sense when discussing arithmetic operations.
Allowing a "product" with zero factors reduces the number of cases to be considered in many mathematical formulas.
Such a "product" is a natural starting point in induction proofs, as well as in algorithms.
For these reasons, the "empty product is one" convention is common practice in mathematics and computer programming.
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.
Under the perhaps more familiar n-tuple interpretation, that is, the singleton set containing the empty tuple.
Note that in both representations the empty product has cardinality 1 – the number of all ways to produce 0 outputs from 0 inputs is 1.
An n-fold categorical product can be defined as the limit with respect to a diagram given by the discrete category with n objects.
Classical logic defines the operation of conjunction, which is generalized to universal quantification in predicate calculus, and is widely known as logical multiplication because we intuitively identify true with 1 and false with 0 and our conjunction behaves as ordinary multiplier.
This is related to another concept in logic, vacuous truth, which tells us that empty set of objects can have any property.
It can be explained the way that the conjunction (as part of logic in general) deals with values less or equal 1.
Reducing the number of conjoined propositions increases the chance to pass the check and stay with 1.
If such a language has a function that returns the product of all the numbers in a list, it usually works like this: (Please note: prod is not available in the math module prior to version 3.8.)