This is Math Newsletter number 6; Wednesday, August 31, 2010.
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Division by Zero.

In some computers, provision is made for dealing with
division by zero errors. Special number codes represents 1/0
and 0/0.

1/0 + 1/0 = 1/0
1/0 + N = 1/0, where N is any valid number.

1/0 * 1/0 = 1/0
1/0 * N = 1/0

0/0 + 1/0 = 0/0
0/0 + N = 0/0, where N is any valid number.
1/0 - 1/0 = 0/0
(1/0) / (1/0) = 0/0

Suppose we want to declare 0/0 and 1/0 to be new numbers.
How shall we do this.

Lets stipulate that they need to follow the regular rules for
addition of fractions and multiplication of fractions.

a/b + c/d = (a * d + b * c)/(b d)

and

(a/b) * (c/d) = (a * c) / (b * d).

1/0 + 1 = 1/0 + 1/1 = (1 * 1 + 0 * 1) / (0 * 1) = 1/0

We can not get beyond 1/0 by adding a number to it.

This makes sense intuitively if we consider that 1/0 is
greater than any possible other previously considered number,
finite or infinite.

However,

1/0 + 1/0 = (1 * 0 + 0 * 1) / (0 * 0 ) = 0/0

This certainly violates our intuition of 1/0 as being the
largest possible number.

Also,
2 * (1/0) = (2 * 1)/0 = 2/0.

This violates the rule that for every number g,
g + g = 2 * g

The problem is that the rule

a/b + c/d = (a d + b c)/( b d)

reduces, when b = d,
to

a/b + c/b = (a + c) / b

when ever b is not = 0.

But when b = 0, it does not reduce this way,

because 0/0 is not equal to 1.

Why isn't 0/0 equal to 1?

It is because (0/0) * 2 = (0 * 2)/0 = 0/0.

1 * 2 is not equal to 1.

We could make a consistent arithmetic with the numbers
1/0 and 0/0 by defining addition and multiplication tables
as follows.

Let N represent any positive or negative number.

Then

the addition table for 0, 1, N, 0/0, and 1/0 is as follows.
0 N0/01/0
N 2*N0/01/0
0/00/00/01/0
1/01/01/02/0

We make the multiplication table for 0, N, 0/0, and 1/0 as
follows.

10N0/01/0
0000/00/0
N0N*N0/0N/0
0/00/00/00/00/0
1/00/0N/00/01/0

Note the unusual property that 1/0 * 1/0 < 2 * (1/0).

This type of relationship is characteristic of numbers
between 0 and 2.

In other ways, 1/0 behaves like a number greater than
infinity. 1/0 cannot be compared to any other number.

1/0 + 10 = 1/0 + 10/1 = (1 * 1 + 10 * 0)/(0 * 1) = 1/0.

Similarly, 0/0 is not like any other number. 0/0 behaves
like, x, a name of a number that has not yet been specified.

When we say that 0/0 and 1/0 are not numbers, what we really
mean is that 0/0 and 1/0 are newly created numbers, different
than any numbers we have considered before. They might have
slightly different rules of operation.

When we routinely create new numbers we make a mistake if we
think the word "number" has a fixed meaning.

What do we mean when we say we create new numbers. In my
opinion, we are creating these new numbers only in our
imagination. In order for us to create these new numbers,