# Is zero a number?

Dear Straight Dope:

Is zero a number?

Two ways to answer this one, the simple answer and the complicated mathematical answer.

The simple answer: Yes, of course zero is a number. (I'm giving you the benefit of the doubt and not saying, "you dolt.") What, you think it's maybe an animal or vegetable?

The mathematical answer: Well, it depends on what you mean by "number." This is not sarcasm, there are different sets of numbers that build up to the Real Number system (the unique complete ordered field).

Zero is clearly an element of the set of Real Numbers, it's the "additive identity" -- the number that, when added to any other number x, doesn't change the value of x. (Similarly, 1 is the multiplicative identity -- the number that, when multiplied by any other number x, doesn't change the value of x.) Thus, zero is a number, just as any other element of the set of Real Numbers is a number.

But before you get to the Real Numbers, you probably start with the Counting Numbers or Natural Numbers, the set N = {1, 2, 3, ...} in set notation. Zero isn't a member of the set of Natural Numbers since you normally don't start counting with zero. A primitive society developing a counting system wouldn't think of "none" ... they'd start counting with "one." Thus, if by "number" you mean "the set of all Natural Numbers," then zero isn't among

them. Of course, the concept of zero makes its appearance pretty early historically (the idea of using zero as a placeholder digit comes later, but that's notational, and a different story).

Let's ignore history and get back to mathematical development. After you have developed the Counting Numbers, you get the Positive Integers, and that's when zero steps onto the stage. The Positive Integers (more technically correct, the Non-Negative Integers) are the set of Natural Numbers and zero, usually designated P = {0, 1, 2, 3...} At that stage in the development of your number system, zero becomes a number.

The next step is usually the set of all Integers, which is the set of Natural Numbers, zero, and the additive inverses of the Natural Numbers (negative numbers). Thus, I =

{..., -3, -2, -1, 0, 1, 2, 3,...}

We can stop here. If you must know, to get to the real numbers, we need to add multiplicative inverses for all numbers except zero, thus, through the group operations of addition and multiplication, building Q, the set of all Rational Numbers, which is an "ordered field" in group theory terminology. It's ordered because you can always compare two distinct numbers and one of them will be larger than the other. It's a field because it follows the field axioms for addition + and multiplication X (and every element has an inverse under these operations, except there's no inverse for zero under multiplication). And then finally, we add the irrational numbers to get the property of completeness (viz., every set which is bounded above has a least upper bound), and voila! the Real Numbers, the unique complete ordered field. Aren't you glad you asked?

All these different stages of development of the Real Number system contain zero, except for the first stage, the Natural Numbers. If you limit your definition of "number" to the Natural Numbers, then, no, zero isn't a "number." Of course, under that definition, 1/2 and -5 and pi aren't "numbers" either.

Conclusion: yes, zero is a number, unless you are defining "number" in the restricted mathematical sense of Counting Numbers.