Based on the first teacher, 0^0 = 1.

Based on the second teacher 0^0 = 0/0 = indeterminent.

Let me encourage us to take the negative times a negative conversation over here.

How do you explain that multiplying two negatives gives a positive? I do it by showing the pattern that you get by multiplying a negative by 3, the 2, then 1, then 0, then -1, -2 etc. I’ve also told students that -7*-5 is asking you for the opposite of 7*-5. I’d love to hear additional explanations.

Some places he expands that point. I think it’s useful to do that in class. What happens if you choose not to accept that x/0 is undefined for all real numbers x (not equal to 0)? What do we do with 0/0? How about x^0 for non-zero x if you don’t want to accept that it equals 1?

I think I first raise these sorts of things in my class when we discuss order of operations and why it was important to have universally accepted, yet ultimately arbitrary rules about such things. What happens if the kids across the hall have a different set of rules for order of operations? Big deal or not?

I’ve been inclined over the last several years to define mathematics as “The Science of What If?” What if we change this definition? What if we restrict ourselves to a smaller domain? What if we look at an expanded domain? What if not all infinities are the same size?

What we’re looking at here are examples of perfect places for such discussions. The thought of allowing students to watch and accept the take in VirtualNerd video makes me ill. Such a nice young person, so friendly, and so NOT talking about mathematics.

I’ve generally argued that if the multiplication and division laws for exponents make sense, we’re forced to decide that for any real number m not 0, m^0 has to equal 1. And that follows from looking at cases like (m^5) / (m^2) = (m*m*m*m*m) / (m*m) = ((m*m*m)*(m*m)) /(m*m) = (m*m*m) = m^3, which agrees with the division law for exponents: (m^5) / (m^2) = m^(5-2) = m^3.

After students are satisfied that this makes sense, we look at a case (m^3) / (m^3) for non-zero m. Everyone agrees that this should be 1. Then we look at it from the point of view of exponent laws and we get m^0. So the notion that m^0 should be 1 seems sensible.

Finally, we discuss what happens if m itself is 0. And this gets us to our favorite “dividing a real number by 0 is undefined” situation, something we’ve already endeavored to make sense of from a couple of perspectives (none of which is “that’s the rule”).