Originally posted by MathiasAndrew
I was trying to convey the fact that positive and negative numbers can cancel each other out also ...
100 x 0 = 100 / 0
100 + 0 = 100 - 0
This is actually not the case. Multiplication is short-hand for addition, and the characteristics of the addition operator don't carry over.
In the case of addition:
100 + n = 100 - (-n)
0 is neither positive nor negative, so we can ignore its sign:
100 + 0 = 100 - (-0)
100 + 0 = 100 - 0
In other words, the additive reciprocal (such that the additive reciprocal of a is -a) of 0 is itself - that is, a = -a, and a = 0. Therefore:
100 + 0 = 100 - 0
This doesn't work, however, with multiplication. In this case, we have to look at the multiplicative reciprocal (such that the multiplicative
reciprocal of a is 1/a):
100 x n = 100/(1/n)
In this case, a = 1/a, and the only number for which this is true is 1. Therefore, 1 is its own multiplicative reciprocal.
There is only one number that has itself as an additive reciprocal - 0.
There is only one number that has itself as a multiplicative reciprocal - 1.
100 x 1 = 100/(1/1)
100 x 1 = 100/1
This doesn't follow for 0, because 0 is not its own multiplicative reciprocal.
We can actually prove that 0 is not its own multiplicative reciprocal.
For 0 to be its own multiplicative reciprocal, 0 would have to equal 1/0. That is,
0 = 1/0
And, I know that's exactly what this thread is suggesting. So, bear with me.
Let's look at the graph of 1/n:
The horizontal asymptote of the graph is 0 - that is, the limit of 1/n as n goes to both positive and negative infinity is 0.
The vertical asymptote of the graph is undefined - that is, the limit of 1/n as n approaches 0 from either side is a little iffy. From the left, 1/n
goes to negative infinity; from the right, 1/n goes to positive infinity.
And this is where the subject blends with what the OP was saying. What this seems to suggest is that, at 0, the graph is at both negative and positive
infinity - and, presumably, everywhere in between.
So, this is actually our definition of 1/0: it is the set of all Real numbers. Not any particular number, but all of them, simultaneously.
On the other hand, 0 is the exact opposite. 0 is defined as a null set.
1/0 is the set of all Real number.
0 is an empty set.
These two are mutually exclusive. Therefore, 1/0 cannot equal 0, and, therefore, 0 is not its own multiplicative reciprocal.
edit on 3-9-2011
by CLPrime because: (no reason given)