Friday, October 20, 2017

Why Zero To The Power Of Zero Is One

While speaking with my daughter about her math homework earlier this week, we got on the topic of exponents and the proper expression of 00 (That's zero to the power of zero or 0^0). The inclination is to say that 00 is undefined because of two distinct math laws that seem to contradict each other: 

  1. Any number to the power of zero is 1.  
  2. Zero to any power is 0. 

Before I get into any explanations, let's keep in mind that I'm not even close to being a "real" mathematician. I use a lot of math in my computer programming background, but, really, I'm just a dad who has a burning hatred for this so-called "common core math" that teaches math by rote rather than by theory. 

Dividing Exponents

We know that, when you multiply exponents with the same base, you add the exponents:
  • xa * xb = xa+b
And, when you divide exponents with the same base, you subtract exponents:

  • xa ÷ xb = xa-b
So, under the law of dividing exponents, x5 ÷ x5 = x5-5 = x0 = 1
Plug in a value for x, and remember that any number divided by itself is 1: 25 ÷ 25 = 32 ÷ 32 = 1

Aha, but you can't divide by zero, right? So 05 ÷ 05 = 0 ÷ 0 = ????

Almost Zero

If we can't divide by zero, how close can we get? Using the exponent calculations below, you'll see that, as the base and the exponent get closer to zero, the product approaches 1.

It could certainly be pointed out that this doesn't directly prove that 00 = 1, but if we want the xx function to be right continuous at 0, then 00 must be 1.

How We Think About Division

We tend to think of multiplication as addition of stacks and division as subtraction of stacks. Exponents are then shorthand for the multiplication or division of the stacks of the same amount. In order to get to the bottom of this 00 question, we have to alter our way of thinking about multiplication and division. Instead of seeing them as tools of addition and subtraction, we have to try to see it as ways to map elements on a grid. With that in mind:

  • 2 = 32 ways to map a 2 element set to a 5 element set. 
  • 20 = 0 ways to map a 2 element set to a 0 element set
But there is one way to map a zero element set to a zero element set. A zero element set is an empty set, so when you map an empty set to itself: The identity function which draws a line straight through coordinates 0,0. 

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