“Any number to the zero power is equal to 1,” my teachers would declare. “Why?” the students would ask. “Because.” the teachers would declare. And this would usually be the end of it. Sometimes a persistent student would again ask, “Why?” to which he’d get the slightly more creative answer, “That’s just the way it is.” Because of the mystery surrounding the zero power in my classrooms growing up, I always got the feeling that there was something supernatural about it and that my teachers either held the key and didn’t want to share or didn’t know how to.
It wasn’t until graduate school that I finally found out why a number to the zero power always equaled one and how to explain this once mysterious phenomenon.
Now as a math teacher, I take the time to explain the zero power. It’s not mysterious, it’s just division. Whenever a question about the zero power comes up, I stop, go to a side board, and ask “What’s 2^3? How about 2^2?” When the kids answer “8” and “4”, I then ask, “Ok, what’s “2^1?” I create a table of their answers on the board:
2^3 2^2 2^1
8 4 2
We then step back and look at the pattern.
Soon the students see that the common difference between numbers is 2. “We divide by 2 to get to the next number,” they say. I then go on to ask, “Then what’s 2^0?”
Of course most students will answer 0 at first. I’ve realized that it’s a natural reaction to answer “zero” whenever hearing “zero” and “multiply” or “divide” within the same lesson. Other students will put 2^0 into the calculator and answer “1”, but that’s what I did as a student and it wasn’t good enough. I always wanted to know why.
I direct all students back to the table we created to find the next term “2^0”.
2^3 2^2 2^1 2^0
8 4 2 ??
“If we divide by 2 each time to get to the next term, what’s 2 divided by 2?” I ask. The students will answer, with an “oh, obviously!” tone, “1.”
I make sure to explain to the students that this isn’t a proof that any number to the zero power is equal to 1; it’s just an example. Still, I challenge my students to create a table with their favorite number, starting by raising it to the 3rd power and working down to the power of zero.
[...] “Any number to the zero power is equal to 1,” my teachers would declare. “Why?” the students would ask. “Because.” the teachers would declare. And this would usually be the end of it. Sometimes a persistent student would again ask, “Why?” to which he’d get the slightly more creative answer, “That’s just the way it is.” Because of the mystery surrounding the zero power in my classrooms growing up, I always got the feeling that there was something … Read More [...]
[...] Donohue presents two zero-based posts at her appropriately-named blog The ZeroSum Ruler: To the Zero! [power] and Dividing by Zero Blows up the Universe!. The middle school students I work with seem to think [...]
The statement “a number to the zero power always equaled one” needs a caveat, viz., the number raised to the zero power can’t be zero itself.
0^0 is a completely different matter.
You can argue as before: 0^3 = 0, 0^2 = 0, 0^1 = 0. Hence, 0^0 = 0, although the argument should be upgraded somewhat because the division by zero is forbidden.
As it comes out, in different branches of mathematics 0^0 is sometimes declared to be 0 and sometimes 1.
Definitions in mathematics are given to serve a purpose, mostly to simplfy various formulations and derivations. As you showed, defining 2^0 = 1 is a convenience. It could be exploited to define negative powers.
0^0 can be rewritten as 0^(1-1) or 0^(2-2), or 0^(3-3), etc…. Let’s just use 0^(1-1).
0^(1-1) can also be written as (0^1)/(0^1) and anything divided by itself is equal to zero as it asks “how many 0^1s fit into 0^1?
Therefore, 0^0 = 1. I can’t see how 0^0 would ever equal zero.