Exponent zero = 1; pi

Date: Sat, 3 Dec 1994 01:13:24 -0800 
From: Roger Gillies
Subject: Exponent Zero and pi

Dr. Math,

I am a gr. 9 math teacher in the Yukon. I have a class that wants to know
why any number with exponent zero is equal to one. Is there an explanation?

Also, how many decimal places has pi been calculated to and are there any
patterns emerging?

Thanks.

Roger Gillies, Teacher
Porter Creek Junior Secondary School, Whitehorse, Yukon Territory

Date: Mon, 5 Dec 1994 14:37:55 -0500 (EST)
From: Dr. Sydney
Subject: Re: Exponent Zero and pi

Hello there!  Thanks for writing Dr. Math!

        There are several ways to think about why a number with exponent
zero is equal to one.  Say you are wondering why 8^0 is one.  We know 8^1 is
8, 8^2 is 64, etc, right?  Also, 8^(-1) is 1/8, 8^(-2) is 1/64, etc.  We
also know that 8^0 = 8^(2 + (-2)) = 8^2 * 8^(-2) = 64(1/64) = 1.  You can do
this with any number and any exponent.  Does that make sense to you?

        One of my math buddies and fellow math doctors, Ethan, has just told
me an even better way to explain why any number to the zero power is one.
Say you are wondering why 2^0 is one.  2^4 = 16, 2^3 = 8, 2^2 = 4, 2^1 = 2,
right? (1/2) 2^4 = (2^3), (1/2) 2^3 = 2^2, (1/2) 2^2 = 2^1 right?  In
general,(1/2)2^n = 2^(n-1).  So to get 2^0, knowing that 2^1 = 2, we can say
2^0 = (1/2)2^1 = 1.

        I hope one of these explanations makes sense to you.  If not, write
back because another one of the math doctors (and another math buddy of
mine), Ken, has a different way to think about taking a number to the zero
power.  

        As for your question about pi, Ken tells me it has been calculated
to about 2 billion digits.  And no, there are no patterns in the digits --
that's why it is considered to be a transcendental number.  

        We were glad to hear from you, and we hope you'll write again soon.

--Sydney, Dr."Math Rocks"