# How would you define R^0?

I know that R^2 is 2-dimensional space, R^3 is 3-dimensional space, and so on and so forth to R^n.

My question is, how would you define R^0? Does such a thing exist?

### 3 Answers

- spoon737Lv 610 years agoFavorite Answer
R^2 and R^3 are shorthand for RxR and RxRxR, where the operation "x" is a Cartesian product. We can form a Cartesian product from any set, it doesn't have to be the reals. For example, if A = {1,2,3}, then A^3 is the set of all ordered triples (a,b,c), where a, b, and c are one of the numbers 1, 2, or 3. In general, a set of the form X^n is the set of all n-tuples of elements of X.

So, R^0 would be the set of all 0-tuples of real numbers. However, a 0-tuple is essentially something containing nothing, so the entire idea of a 0-tuple containing real numbers is a contradiction. Thus, no such object exists, meaning the only reasonable interpretation of R^0 is that it is the empty set.

Edit: Steven and Melanie, try actually reading the question. He is not talking about exponents, he is talking about a Cartesian product. The notation is the same, but the concepts are completely different. R is not a number, it stands for the set of real numbers. The Cartesian product R^0 is certainly NOT 1. It isn't even a number, it's a set.

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- StevenLv 710 years ago
R^0 is 1

when you multiply exponents, you add

when you divide exponents, you subtract

so

R^5/R^5= R^(5-5)= R^0

and any number divided by itself is 1

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