Zero

Zero (0) is the number which is the size of the empty set. The numeral? zero is used in [positional number systems]?, where the position of a digit signifies its value, and the digit zero is used to skip a position. This use was introduced into mathematics relatively late, during the early 800s in Indian books. The definition originally used was, "In subtraction, when nothing is left over, write a dot to keep the place." Prior mathematical systems, such as the Babylonian or Greek, did not use zero at all, but still succeded quite well for everyday purposes. The number zero was not accepted until later still.

The following are some basic rules for dealing with zero. These rules apply for any complex number x, unless otherwise stated.

• Addition: x + 0 = x and 0 + x = x. (That is, 0 is an identity element with respect to addition.)
• Subtraction: x - 0 = x and 0 - x = -x.
• Multiplication: x × 0 = 0 and 0 × x = 0.
• Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse.
• Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts. For all positive x, 0^x = 0.

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