How can we explain to middle school students that 0^0=1?

Please let me know.

Thank you.

Peter

 

Hi Peter,

I'm not sure we can answer your question but we can give you some information. We have received what is essentially this question before but then it was "I am a middle school teacher who is looking for a precise explanation of why zero raised to the zero power is undefined." The answer to "the question", regardless of which way you ask it depends on what you mean by mn in situations where not both m and n are zero.

One meaning of mn is in the response I received to your question from Claude.

The "advanced combinatorial way'' is that mn counts the number of functions from a set with n elements to a set with m elements. The empty function is the only function from the empty set to itself, hence 00 = 1.

This is somewhat abstract and esoteric but it is a useful way to interpret mn, especially for mathematicians whose interest is combinatorics or counting. But what if m or n is not an integer? (I am going to insist that m can't be negative for otherwise I am faced with, for example (-2)1/2 which is either undefined or I have to move to the world of complex numbers.)

One way to attempt to answer your question is to take a dynamitic approach. That is to evaluate mn where m and n are close to zero, let m and n get closer and closer to zero and watch the value of mn. The hope is that as m and n approach zero, mn will approach some value and I can say "this is the value I should take for 00 ". This is a calculus approach to the question and is the approach that Penny took when she responded to "I am a middle school teacher who is looking for a precise explanation of why zero raised to the zero power is undefined."

Another of my consultants, Sue, pointed me to more information on this "debate" on a web page at the University of Waterloo.

So we don't have an answer to your question or even a definite answer as to whether 00 should be 1 or undefined. Contrary to what many people think there is not always one definite right answer to a mathematical question.

Harley