Empty product

The empty product is the result of multiplying no numbers. Just as the empty sum --- i.e., the sum of no numbers --- is 0, so also the empty product --- the product of no numbers --- is 1. This fact is useful in combinatorics, algebra (in particular, in connection with the binomial theorem), set theory, and in the study of power series. Two often-seen instances are "a⁰ = 1", i.e., any number raised to the power 0 is 1, and "0! = 1", i.e., the factorial of 0 is 1.

Imagine a calculator that can only multiply. It has an "ENTER" key and a "CLEAR" key. If "21" is displayed, and "4" is entered, then the display will show "84", since $21\times 4=84$. If one wants to know the value of

$3\times 7\times 4,$, one presses "CLEAR"; then "3"; then "ENTER"; then "7"; then "ENTER"; then 4; then "ENTER". One hopes then to see "84" displayed. What number must then appear when "CLEAR" is pressed? It is tempting to say "0", by analogy with conventional calculators. But if "0" is displayed, then after "3" is entered, the display will show the product of 0 and 3, i.e., it will show "0", and then when "7" is entered, it will show the product of that and 7, which again is 0, and when "4" is entered it will likewise show "0", rather than "84". Only if "1" is displayed after "CLEAR" has been pressed, will the calculator perform as advertised. Therefore, when no numbers have been multiplied, the product is 1.