Teaching Exponents

Someone recently asked what a zero exponent would mean in the real world. Here is my basic response. The important method of research is to ask and answer questions, and the important questions to ask are “how did we get the concept?” and “what does it mean?” We should look for things in the world where we take exponents. Where did we start? As far as I know, people started in history with geometry, with squares and cubes. What did we do from there? Include natural numbers as exponents. What did we do from there? Include zero as an exponent. What did we do from there? Include negatives, fractions and irrational numbers as exponents. But let’s consider squares and cubes. We would multiply a length by a length to get an area; this is why we “square,” and from where the name comes. We would multiply a length by a length by a length to get a volume; this is why we “cube,” and from where the name comes. So a cube is a volume, a square is an area, a first power is a line, and so we could define a zero power to be a what? A point. (Considering 5^0 meter to not be the same as (5 meter)^0, just as they are different for the square or cube.) The person suggested thinking about the number of apples to the zero power. Apples are not a good example, as I’m thinking. Why would we square or cube or otherwise exponentiate the number of apples? Better examples would be things we can square or cube, or situations where we do so. One good example is compound interest (and continuous interest). We find that we can multiply a dollar amount by a factor, e.g., (1 + 0.08/12), which would be 8% interest compounded monthly, to get a dollar amount for next year. We can multiply that by the same factor to get a dollar amount for two years out. When we look, we find we could’ve multiplied the original amount by the factor squared, (1 + 0.08/12)^2, to get a dollar amount two years out. And so on for three or four years: (1 + 0.08/12)^3 and (1 + 0.08/12)^4, respectively. We see we then get a formula with an exponent representing time in years. If we trace numbers down 20, 10, 5, 3, 2, 1, we would come to zero. We need to do so to integrate with our number sequence. But then we see zero here represents time. So what does it mean? Clearly, initial time of investment; the amount at that time is just how much was invested, the “principal.” And clearly, we would naturally think about the meaning of numbers like 1/2, 1 1/2, 3 3/4, 7 1/8 – and then we would think about the meaning of numbers like -1, -5, -3.741. These would mean 1/2 year, 1 1/2 year, etc., and the negatives, a year before current, five years before current, etc. As Dr. Corvini has pointed out, the negatives and irrationals allow us to achieve unit-economy in our thinking. Without them, we would need multiple formulas to solve equations (e.g., we would have no quadratic formula) or perform calculations. In this light, we see we could get future values by multiplying, and past values by dividing, using positive exponents in each case, but negative exponents and zero allow us to use a single formula and one method. They make our thinking more efficient and they save space in our minds and memories. What’s more, we can abstract away from the nature of exponents, and treat them as numbers. Then we can apply arithmetic to them. But to do so we need zero — and we need zero to bridge between negative and positive for things like past time and future time. We could also think about how exponents come up in radioactive decay, doubling times or amounts, electricity, growth, decay. How does a zero exponent come up in those situations and what does it mean? Zero exponents allow us to extend our exponents from squares and cubes to all numbers; they allow us to do arithmetic to exponents; and they allow us to extend the use of a factor to a current or initial time or state.