Purpose of Zero

Date: 17 Nov 94 18:20:48 EST
From: Mike  Romanoff
Subject: Question

Dr. Math,

        What is the actual purpose of the number zero?

                                Thank you,

                                        Regan Romanoff

Date: Sat, 19 Nov 1994 13:52:39 -0500 (EST)
From: Dr. Ken
Subject: Re: Question

Hello there!

What an excellent question!  I'm glad to see you're really thinking about
the fundamental concepts behind mathematics.  

The invention of Zero was one of the most important breakthroughs in the
history of civilization.  More important, in my opinion, than the invention
of the wheel.  I think that it's a fairly deep concept.

One crucial purpose that Zero holds is as a placeholder in our system of
notation.  When we write the number 408, we're really using a shorthand
notation.  What we really mean by 408 is "4 times 100, plus 0 times 10, 
plus 8 times 1."  Without the number zero, we wouldn't be able to have a
placeholder in the tens place, and we wouldn't be able to tell the numbers
408, 48, 480, 408000, and 4800 apart.  So yes, zero is important.

Another crucial role that zero plays in mathematics is that of an "additive
identity element."  What this means is that when you add zero to any number,
you get the number that you started with.  For instance, 5 + 0 = 5.  That
may seem obvious and trivial, but it's actually quite important to have such
a number.  For instance, when you're manipulating some numerical quantity
and you want to change its form but not its value, you might add some fancy
version of zero to it, like this:

x^2 + y^2 = x^2 + y^2 + 2xy - 2xy 
          = x^2 + 2xy + y^2 - 2xy
          = (x + y)^2 - 2xy

Now if we wanted to, we could use this as a proof that (x + y)^2 is always
greater than 2xy; the expression we started with was positive, so the one we
ended up with must be positive, too.  Therefore, subtracting 2xy from
(x + y)^2 must leave us with a positive number.  Neat stuff.

Anyway, if you're interested in finding out more about the number zero, the
field of study you should look into is called "Group Theory", a substudy of
Modern Abstract Algebra.

Thanks for the question!

-Ken "Dr." Math