Bouncing Ball said:
Argh!!!
I am sorry, I wanted to stop myself from being totally offtopic, but this isn't right. It is impossible to divide something by 0..
I should say, an infinite number of solutions depending on the relationship between any two given numbers in an equation. It is not impossible, it is more unacceptable or not understood.
From the Univerisy of Utah Math Department site:
The reason that the result of a division by zero is undefined is the fact that
any attempt at a definition leads to a contradiction.
To begin with, how do we define
division? The ratio
r of two numbers
a and
b:
r=a/b is that number
r that satisfies
a=r*b.Well, if
b=0, i.e., we are trying to divide by zero, we have to find a number
r such that
r*0=a. (1) But
r*0=0 for all numbers
r, and so unless
a=0 there is no solution of equation (1).
Now you could say that
r=infinity satisfies (1). That's a common way of putting things, but what's infinity? It is not a number! Why not? Because if we treated it like a number we'd run into contradictions. Ask for example what we obtain when adding a number to infinity. The common perception is that infinity plus any number is still infinity. If that's so, then
infinity = infinity+1 = infinity + 2 which would imply that 1 equals 2 if infinity was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse. What about
0/0?
I said above that we can't solve the equation (1) unless
a=0. So,
in that case, what does it mean to divide by zero?
Again, we run into contradictions if we attempt to assign any number to
0/0.
Let's call the result of
0/0,
z, if it made sense.
z would have to satisfy
z*0=0. (2) That's OK as far as it goes, any number
z satisfies that equation. But it means that the result of
0/0 could be anything. We could argue that it's 1, or 2, and again we have a contradiction since
1 does not equal
2.
But perhaps there is a number
z satisfying (2) that's somehow special and we just have not identified it? So here is a slightly more subtle approach. Division is a continuous process. Suppose
b and
c are both non-zero. Then, in a sense that can be made
precise. the ratios
a/b and
a/c will be
close if
b and
c are close. A similar statement applies to the numerator of a ratio (except that
it may be zero.)
So now assume that
0/0 has some meaningful numerical value (whatever it may be - we don't know yet), and consider a situation where both
a and
b in the ratio
a/b become smaller and smaller. As they do the ratio should become closer and closer to the unknown value of
0/0.
There are many ways in which we can choose
a and
b and let them become smaller. For example, suppose that
a=b throughout the process. For example, we might pick
a=b = 1, 1/2, 1/3, 1/4, .... Since
a=b, for all choices of
a we get the ratio 1 every time! This suggests that
0/0 should equal 1. But we could just as well pick
b = 1, 1/2, 1/3, 1/4, .... and let
a be
twice as large as
b. Then the ratio is always 2! So
0/0 should equal 2. But we just said it should equal 1! In fact, by letting
a be
r times as large as
b we could get any ratio
r we please!
So again we run into contradictions, and therefore we are compelled to let
0/0 be undefined.
