Why are Operations of Zero so Strange?

Date: 03/17/97 at 21:01:38
From: Jonah Knobler
Subject: Why are Operations of Zero so Strange?

I am a student in an Algebra II class, and I'm still somewhat naive 
about stuff like division by zero.

1) Why do we say things like 1/0 are undefined?  Can't you call 1/0 
infinity and -1/0 negative infinity? Why not?

2) What is 0 * (1/0)?  Would it be zero, since whatever (1/0) is, 
we're taking it NO times?  Or would it be 1, since a/b * b = a?  Is 
this why we say it's undefined?

3) What is the quantity 0^0 (zero to the zeroth power)?  Would it be 
0, since 0 times anything is 0, or would it be 1, since anything to 
the 0 power is 1?  Or is this undefined, too?

4) If it is possible to raise something to the zeroth power, can you 
find the zeroth root of something?  Is it one?  Is it ever done?

It just seems so odd to me that zero, the very center of our entire 
concept of mathematics (something long held to be the purest, most 
perfect thing in existence) should have such gaping flaws.

- Jonah Knobler

Date: 03/18/97 at 13:06:33
From: Doctor Rob
Subject: Re: Why are Operations of Zero so Strange?

>1) Why do we say things like 1/0 are undefined?  Can't you call 1/0 
>infinity and -1/0 negative infinity? Why not?

1/0 is said to be undefined because division is defined in terms of
multiplication.  a/b = x is defined to mean that b*x = a.  There is
no x such that 0*x = 1, since 0*x = 0 for all x.  Thus 1/0 does not
exist, or is not defined, or is undefined.

You wish to introduce a new element (or maybe two elements) infinity
which you wish to append to the real number system.  That is not
prohibited.  After all, that is how we got from natural numbers to
integers (appending negative integers and zero), and from integers
to rationals (appending ratios of integers), and from rationals to
reals (appending limits of convergent sequences), and from reals to
complexes (appending the square root of -1).  What you end up with
is not the real number system, however.  Furthermore, if you wish to
define the four operations + - * and / for this new system, you
probably want them to be the same on real numbers, and just add on
the definitions of things like infinity + r and r/infinity, for any
real number r.  Some of these work fine.  It makes sense to define:

        infinity + r = r + infinity = infinity
  (-infinity) + r = r + (-infinity) = -infinity
                infinity + infinity = infinity
          (-infinity) + (-infinity) = -infinity
                       infinity - r = infinity
                    (-infinity) - r = -infinity
                       r - infinity = -infinity
                    r - (-infinity) = infinity
             infinity - (-infinity) = infinity
             (-infinity) - infinity = -infinity
        infinity * r = r * infinity = infinity  for r > 0
  (-infinity) * r = r * (-infinity) = -infinity  for r > 0
        infinity * r = r * infinity = -infinity  for r < 0
  (-infinity) * r = r * (-infinity) = infinity  for r < 0
 infinity * infinity = (-infinity) * (-infinity) = infinity
 infinity * (-infinity) = (-infinity) * infinity = -infinity
                       infinity / r = infinity  for r > 0
                    (-infinity) / r = -infinity  for r > 0
                       infinity / r = -infinity  for r < 0
                    (-infinity) / r = infinity  for r < 0
                       r / infinity = 0
                    r / (-infinity) = 0

Where we get into trouble is with defining the following:

        infinity + (-infinity)
     (-infinity) + infinity
        infinity - infinity
     (-infinity) - (-infinity)
               0 * infinity
        infinity * 0
               0 * (-infinity)
     (-infinity) * 0
        infinity / infinity
        infinity / (-infinity)
     (-infinity) / infinity
     (-infinity) / (-infinity)
        infinity / 0 = infinity
    (-infinity) / 0 = -infinity


These expressions are called "indeterminate forms."  These can all
have a large range of different values, depending on exactly where
the "infinity" parts came from.

As a result, the system you construct is not closed under addition,
subtraction, multiplication, or division.

Other indeterminate forms are 0^0, 1^infinity.  You will encounter
them again when you take calculus.

>2) What is 0 * (1/0)?  Would it be zero, since whatever (1/0) is, 
>we're taking it NO times?  Or would it be 1, since a/b * b = a?  Is 
>this why we say it's undefined?

This is one of the indeterminate forms, 0 * infinity, mentioned above.

>3) What is the quantity 0^0 (zero to the zeroth power)?  Would it be 
>0, since 0 times anything is 0, or would it be 1, since anything to 
>the 0 power is 1?  Or is this undefined, too?

This, too, is an indeterminate form.  Its logarithm is 0 * infinity.

>4) If it is possible to raise something to the zeroth power, can you 
>find the zeroth root of something?  Is it one?  Is it ever done?

Since x^0 = 1 for any nonzero x, only 1 could possibly have a zeroth
root, but, again, which x are we going to use?  This means that the
zeroth root of 1 is again indeterminate.  It is never done.

>It just seems so odd to me that zero, the very center of our entire 
>concept of mathematics (something long held to be the purest, most 
>perfect thing in existence) should have such gaping flaws.

These are not considered flaws.  Zero just has the property that you
can't divide by it.  It's a feature, not a bug!  :-)

When you learn higher abstract algebra, you will find that in some
rings there are objects called zero-divisors which, like zero here,
have no multiplicative inverse.  Stuff happens!  We have to learn to
live with this situation.

As an example of the situation in the last paragraph, consider the
arithmetic of the clock.  The set of elements are the hours {1, 2, 
...,  12}.  Addition is performed by adding the integers, and if the 
result is bigger than 12, subtract 12.  Subtraction is performed by 
subtracting the integers, and if the result is less than 1, add 12.  
Multiplication is performed by multiplying the integers, and if the 
result is bigger than 12, subtract 12 repeatedly until it is not.  

These operations have sensible interpretations with respect to real 
clocks and time measures. The additive identity (or zero) is 12.  The 
additive inverse of 12 is 12, and that of any other hour h is 12-h.  
The multiplicative identity is 1.  The hours 1, 5, 7, and 11 have 
multiplicative inverses:  themselves!  The other hours are zero-
divisors, and have no multiplicative inverses.  You can verify that 
2*h can never give the answer 1 by trying all h's.  They are called 
zero-divisors because of equations like 3*8 = 12, where 12 acts as the 
zero element, so 3 and 8 divide zero.

This example may seem strange, but it is a perfectly good example of
what is called a ring in higher abstract algebra.

Keep up the good questions!

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/