Empty Product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive identity.

When a mathematical recipe says "multiply all the numbers in this list", and the list contains, say, 2, 3, 2 and 4, we multiply first the first number by the second, then the result by the third, and so on until the end of the list, so the product of (2,3,2,4) would be 48. If the list contains only one number, so that we cannot multiply first by second, common convention holds that the 'product of all' is that same number, and if the list has no numbers at all, the 'product of all' is 1. This value is necessary to be consistent with the recursive definition of what a product over a sequence (or set, given commutativity) means. For example,

\begin{align} \text{prod}(\{2,3,5\}) & = \text{prod}(\{2,3\}) \times 5 = \text{prod}(\{2\}) \times 3 \times 5 \\ & = \text{prod}(\{\}) \times 2 \times 3 \times 5 = 1 \times 2 \times 3 \times 5. \end{align}

In general, we define

The empty product is used in discrete mathematics, algebra, the study of power series, and computer programs.

The term "empty product" is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming; these are discussed below.

Read more about Empty Product:  0 Raised To The 0th Power, Nullary Conjunction and Intersection, Nullary Cartesian Product, Nullary Categorical Product, In Computer Programming

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