By Mike O’Hern, Center Director of Mathnasium of West Knoxville
Remember that stuff last month about exponents? You might remember that we discovered that if you raise any number – any number at all – to the power of zero you end up with an answer of 1. We arrived at that conclusion by doing a series of divisions, and I think I mentioned that if we continued on that path we might discover something about negative exponents.
Perhaps one way to think about it is that the power of positivity unites, but the power of negativity divides! (Don’t take that one too far – it just popped into my head and I thought it had a ring to it.)
Now for a very brief review, 10^2 = 100, divide by 10, 10^1 = 10, divide by 10, 10^0 = 1. Each time we divide, on the left side we simply subtract 1 from the exponent, and on the right we divide by 10 just like we learned in fourth grade.
Now I know that after reading last month’s article, inquiring minds wondered what would come next. And perhaps some of you have already figured that out! But I can’t assume that it goes without saying (and I wouldn’t miss an opportunity to say something anyway), so we’ll take the next step here together.
Let’s just divide by 10 one more time. On the left, where you have 100, you simply subtract 1 from the exponent, just like we had done before. And what is 0 – 1? Negative one: -1. So the left side is 10^-1. And on the right side, what do we get when we divide 1 by 10? You might say 0.1, and you would be right, but for the moment this will be clearer if we use the fractional notation: 1/10. So 10^-1 = 1/10.
Perhaps one way to think about it is that the power of positivity unites, but the power of negativity divides!
One more step and this starts to get interesting. (I know you’re thinking “it’s about time!” – stop it.) Divide by 10 again. 10^-2 = 1/100. What’s so interesting about that? Notice that another way to write 1/100 is 1/10^2. Here’s the connection: 10^-2 = 1/10^2. 10^-3 = 1/10^3, and so on. And this turns out to be true in general!
STOP, MR. MIKE! Why in the world would we care about negative exponents? Glad you asked! One example comes to mind right away. In the scientific world we like to talk about distances in terms of meters, for example. So when we talk about the distance to the moon we can say it’s over 100,000,000 meters. That’s 10^8 meters.
So what if we’re talking about the distance between atoms? In space the distance between atoms can be up to a meter or so, but in a chemical compound here on earth it can be something like 1/10000000000 (or 0.0000000001 if you prefer) meters (it’s slightly more, but this is a math illustration!). That would be 10^-10 meters. If we were working with atoms regularly (that is, not just comparing them to space atoms!) we would refer to that as 1 angstrom, really, but the point is that when we want to work with very small numbers or compare things that have great differences, negative exponents can make things a lot simper for us.
That will have to suffice for this month. It’s been like 10^-6 years since I filled my coffee mug, so I’d best get back to that!
Mike O’Hern, Center Director of Mathnasium of West Knoxville, earned his Bachelor’s Degree in Metallurgical Engineering at the University of Tennessee, Knoxville in 1988. He pursued graduate studies in Materials Science & Engineering while on the Research Staff at Oak Ridge National Laboratory. Mike has had a life-long love of mathematics and teaching, and feels that math is not about learning to be ready for the next math class – it’s about learning to think.