impossible statements

[Dan Zollman]

I've been using these proofs for a while, and after seeing a math and science forum, I decided to mention them.

Proof #1:
A + A - 2A = A + A - 2A (both members are identical)
A + (A - 2A) = (A + A) - 2A (associative property)
A + (-A) = (2A) - 2A (simplify contents of parentheses)
A - A = 2A - 2A
1(A - A) = 2A - 2A (factoring)
1(A - A) = 2(A - A) (factoring again)
1(A - A)/(A - A) = 2(A - A)/(A - A) (divide both sides by the quantity)
1 = 2

I have a supporting argument for this one. When someone finds the counterstatement, I'll continue.

Proof #2:
I found this while answering the question of 'what is 1 to the i power?'.
Since 1 = i^4, then
1^i = (i^4)^i
We also know that 1 to any power is 1, so we can add:
1 = (i^4)^i
and...
= i^4i = (i^i)^4
...take the 4th root of 1...
1^i = i^i
1=i, etc.

The simple answer to Proof #2 is that exponentiation is not defined for imaginary numbers (which has been confirmed to me by a few teachers and a math professor).
Now that leaves me with the questions: If exponentiation is not defined for imaginary numbers, then how can it be said that e^i pi = -1, and why does a TI graphing calculator say that i^i is .20something (while confirming that 1^i = 1)?
I haven't taken trig, calc, or stat yet in school, so part of my confusion may be because I haven't learned anything about that yet.

I know that these proofs are somewhat ignorant to the general purpose of the math systems involved. They aren't useful in the actualI'm just wondering what people would have to say about both of these proofs, and if they knew the answers to my questions.