THE TROUBLE WITH RULES, or "WHAT DOES ZERO TO THE ZERO POWER EQUAL?"

So, what is your answer?  A number of you, I am sure, said "one", and I am sure some of you said "zero".  Maybe even a few of you said "undefined".  The truth is, there is no consensus to the answer on this.  The best answer one can give is, "well, it depends on your point of view", but more on that later.

First off, I wanted to do an informal, non-scientific poll to see what people think zero to the zero equals, so naturally I asked all my facebook friends to sound off.  The results went like this:

ONE                                                             ZERO                                                  UNDEFINED
   7                                                                    2                                                                1

These were the only sure answers.  There were some less certain responses.  What I found most telling, though, was the explanations for why people chose as they did.  Here are some of the choicest quotes......

• "It has to be 1 or you break math (italics are mine).  If it were not one you couldn't solve binomials or other exponential equations".
• "1-ish".
• "A very tough question!  Still calculating.  One maybe or maybe not.  I'll get back to you (ed.: she still hasn't gotten back to me...)".
• "Anything raised to the power of zero is one.  In order of operations, exponents come first."
• "So the general consensus with my math colleagues is that one is the answer, but you aren't wrong if you say 'undefined' (ed.: this is from a friend of mine who is a high school math teacher, and she polled her school's math department on my behalf.)".
• "I have to chime in with one".
• "Gosh, I thought it was zero".
• "I think it's zero (ed.: this was from the math teacher at Forest Hills Central High whose class I have been observing the last few months)".

Quite a bit of disagreement, even amongst those that agree, no?  Here's where I think the problem occurs.  In the study of exponents, we are given two different rules.  They are as follows:

1.     

 

   2.

So, which rule applies?  Which do we follow?  There is a third way to think of this problem:



 

 With this equation we can now take the limit as x approaches zero, which looks like this:

 



 

 

To truly understand what is going on here, let's look at a graph of all three functions:

In the graph above, the purple line at the bottom is, the green line at the top is, and the red line is the graph of .  The purple and green lines are destined never to meet as they doggedly head towards their expected values of zero and one respectively, but the red line shows us something interesting.  As x gets closer and closer to zero, the graph gets closer and closer to 1!  So, clearly it equals 1, right?

 

Not so fast.  Why does something to the zero power equal one anyway?  The standard explanation goes something like this:

 

We can also solve using another rule of exponents, the sum of powers rule, like so:

And here is where the trouble really starts; try replacing 3 with 0 and we get this:

 

 

We have long been told you cannot divide by zero, so now we are desperately stuck in a hopeless paradox brought about by a dastardly construction of contradictions! And I feel the real trouble is not the rules themselves, it is how they are presented to us.

 

THE REAL TROUBLE



All our lives, math has been presented to us in a very concrete fashion, with a seemingly overwhelming amount of rules to follow that, if implemented correctly, govern all mathematical problems that ever were, are, or will be, created by the likes of Leibnitz, Newton, Euler, Pythagoras, etc.  These great mathematicians are not to be questioned or doubted.  Our teachers teach us these rules as if they are absolute and unwavering, so when they fail us, we are left scratching our heads and (perhaps) questioning the meaning of life.  It turns out that math isn't so concrete, that sometimes the rules fail, and there are still vast stores of discoveries still to be made.  Students need to know that math is fluid, dynamic, expansive, and revealing more and more of its secrets to us even today. 

 

I truly feel the four most powerful words a teacher can say when asked a question are "what do you think".  This response opens up the possibility for personal discovery by the asker, not imposition by the asked.  So, if and when my students ever get around to asking me "what does zero to the zero equal", I am definitely going to respond with "what do you think?"  Because sometimes, as we have seen here, the lack of a definitive answer is really the worthier discovery.  Cheers!