0 (zero) is a number representing lack of something. I.e. when counting objects, 0 means there are no objects. The number 0 is special by being neither positive nor negative; it stands exactly between positive and negative numbers. It is a whole number, an even number, a rational number, a real number and a complex number.
0 is an additive identity, meaning that x + 0 = x for any number x. Subtracting 0 from any number results in that same number (x - 0 = x). Multiplying any number by 0 gives 0.
The earliest civilization known to have had the concept of zero were Sumerians who used a placeholder symbol (double wedges as if for an empty column) where no «normal» number could go, circa 3000 BC, however it probably only served the placeholder purpose. Similar symbols also developed independently in different cultures. The full understanding of zero, i.e. its usage as a regular number in equations, first appeared in India in about 500 AD. Ancient Greeks did not use the number zero at all.
Dividing by zero is not defined because there is no such number which multiplied by 0 would give back the originally divided number. In other words given a non-zero number a, trying to solve the equation x = a/0 for x we arrive at 0 = a: a contraction implying there are no solutions. 0/0 is also undefined because it would introduce a collision of two rules: a/a = 1 and 0/b = 0; neither 1 nor 0 could be preferred as the result.
Some people naively expect that the result of division by zero should be infinity but that doesn't solve the mentioned problems; consider e.g. 6/0 = infinity, then it should hold that infinity 0 = 6* and also any other number we could possibly be dividing, which doesn't make sense. Nevertheless it is possible to work with dividions by zero using so called limits.
Even though x^0 = 1 and 0^x = 0 for non-zero x, 0^0 is also not defined because this puts the previous two equations in conflict (similarly to the issue with 0/0).
This page was last edited on 2026-09-25 02:04:49.
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