In mathematics, zero, symbolized by the numeric character 0, is both:
1. In a positional number system, a place indicator meaning "no units of this multiple." For example, in the decimal number 1,041, there is one unit in the thousands position, no units in the hundreds position, four units in the tens position, and one unit in the 1-9 position.
2. An independent value midway between +1 and -1.
In writing outside of mathematics, depending on the context, various denotative or connotative meanings for zero include "total failure," "absence," "nil," and "absolutely nothing." ("Nothing" is an even more abstract concept than "zero" and their meanings sometimes intersect.)
Notation for placeholders in positional numbers is found on stone tablets from ancient (3,000 B.C.) Sumeria. Yet, the Greeks had no concept of a number like zero. In terms of modern use, zero is sometimes traced to the Indian mathematician Aryabhata who, about 520 A.D., devised a positional decimal number system that contained a word, "kha," for the idea of a placeholder. By 876, based on an existing tablet inscription with that date, the kha had become the symbol "0". Meanwhile, somewhat after Aryabhata, another Indian, Brahmagupta, developed the concept of the zero as an actual independent number, not just a place-holder, and wrote rules for adding and subtracting zero from other numbers. The Indian writings were passed on to al-Khwarizmi (from whose name we derive the term algorithm ) and thence to Leonardo Fibonacci and others who continued to develop the concept and the number.
Various arithmetic operations that include zero have sometimes been the subject of dispute such as the result of dividing zero by zero. The answer is that it can't be done. Although early mathematicians tried to wrestle some sort of result out of this operation, later ones have decided that this problem just won't bear any fruit. This is viewed as another case where language allows us to ask a question that really doesn't make sense to ask.
Zero to the zeroeth power on the other hand has three possible answers. For some apparently useful reasons, the answer is 1. But in other contexts, the answer can be either "indeterminate" (not capable of being calculated) or "undefined/nonexistent."