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.

They follow the rules that

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 | N | 0/0 | 1/0 |
---|---|---|---|

N | 2*N | 0/0 | 1/0 |

0/0 | 0/0 | 0/0 | 1/0 |

1/0 | 1/0 | 1/0 | 2/0 |

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

follows.

1 | 0 | N | 0/0 | 1/0 |
---|---|---|---|---|

0 | 0 | 0 | 0/0 | 0/0 |

N | 0 | N*N | 0/0 | N/0 |

0/0 | 0/0 | 0/0 | 0/0 | 0/0 |

1/0 | 0/0 | N/0 | 0/0 | 1/0 |

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,

the possibility of their existence had to already be there.

Kermit