[This is the first of several math-related pieces I plan to post on this website. One of the big stumbling blocks for high school algebra students--and, it appears, some professional logicians!--is the issue of zero and negative exponents. What do those little numbers mean that we put above and to the right of numbers and letters? I hope this short piece helps to strip away some of the mystery surrounding these little beasties.]

In December of 1995, while participating in the Institute for Objectivist Studies' cyberseminar on the nature of propositions, it occurred to me that there was a (fairly) simple explanation for the nature of zero exponents. Many people seem mystified by the mathematical idea that any number to the "zero" power is equal to 1. No matter how small or large, this is true. 5-to-the-0-power, nine-million-to-the-zero-power, 3/5-to-the-zero-power...you name it, and it's equal to 1. Why should this be so?

Consider positive exponents: 5-squared (five-to-the-second-power or 5x5) is equal to 25 because (we are told by our teachers) the square (exponent of two) means that you have two factors of the number 5 (i.e. you multiply 5 times itself). Thus, 5-to-the-fourth-power would mean 5x5x5x5 (or 625).

Negative exponents are a little trickier: 5-to-the-negative-second-power is equal to 1/25 because a negative power is (supposedly) defined as the exponent of a number that is the denominator of a fraction with a numerator of 1. In this case, 5-to-the-negative-second-power is equal to 1 divided by 5-squared, which is 1 divided by 25, which is 1/25.

But what possible meaning can there be to multiplying a number by itself zero times? There is no such mathematical operation, is there? And even if there were, why should the result always be 1??

Yet, when we are taught to multiply by adding exponents, we have to do it that way so that the math will come out right. For instance: 6-squared times 6-squared equals 6-to-the-fourth by the exponent rule, and (6x6)x(6x6) = 36x36 = 1296, which is 6-to-the-fourth. No problem there. And: 6-to-the-zero times 6-squared equals 6-squared by the exponent rule, and 1x(6x6) = 1x36 = 36, which is 6-squared.

So, what is the deal with zero exponents? Why do they work? Are they just a "convenient fiction"--something to help the math "come out right"? Or is there something that our math teachers aren't telling us?

The confusion comes in the way that exponents are described. The basic function of an exponent is to relate the operations of multiplication and division not (primarily) to the number of which it is an exponent, but to the unit number, 1. (This, of course, is the basis of the much-dreaded "scientific notation," which I think is how students should be taught exponents, rather than the more colloquial "times-itself" approach.)

This relation to the unit 1 is explicit for negative exponents: x-to-the-minus-3rd-power, for example, is 1 divided by x-to-the-3rd-power. For positive exponents, it is left implicit (since 1 is the multiplicative identity), but is more fully stated, for example, x-to-the-3rd-power is 1 times x-to-the-3rd-power. (x-to-the-3rd = 1 times x-to-the-3rd) .

In light of this, a zero exponent's non-fictional relation to the unit 1 becomes clearly plausible. Here's how it works:

For any real number, r, a positive exponent, n, indicates that the unit 1 is to be considered as having been multiplied by r a total of n times. A negative exponent, -n, indicates that the unit 1 is considered as having been divided by r a total of n times.

A zero exponent, by contrast, indicates that the unit 1 is considered out of any multiplicative or divisive relationship to the base number r. We are to consider only the unit 1 and to refrain from combining it (by multiplication or division) with the exponent's base number.

Now we can see the root of the confusion inherent in the way most of us are taught about zero exponents. The unit 1 should be considered not as being multiplied (or divided) by some base number "zero times," but as itself, i.e., as not having been multiplied (or divided) by the base number any times.

I apologize to the mathematicians in the group for the inelegance of this explanation. I hope it is clear enough for everyone else.