SOLUTION: Do all complex numbers in the form a-bi/bi-a result in -1 and Why?

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Question 1011591: Do all complex numbers in the form a-bi/bi-a result in -1 and Why?
Found 3 solutions by josgarithmetic, ikleyn, MathLover1:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Yes.

You wrote this: a-bi/bi-a

The steps go this way:
a-1-a
a+(-1)+(-a)
a+(-a)+(-1)
0+(-1)
-1

You can identify the Number Properties used in the steps if you want to refer to them.

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
Do you know that bi - a = -a + bi = -(a - bi) ?

If you know it, then

%28a+-+bi%29%2F%28bi-a%29 = %28a-bi%29%2F%28-%28a-bi%29%29 = -1.

This is why.


Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
answer is yes, and here is why:

%28a-bi%29%2F%28bi-a%29=+-1+...to eliminate i from denominator, multiply both numerator and denominator by %28bi%2Ba%29

%28%28a-bi%29%28bi%2Ba%29%29%2F%28%28bi-a%29%28bi%2Ba%29%29=+-1+

%28abi%2Ba%5E2-bi%2Abi-abi%29%2F%28%28bi%29%5E2-a%5E2%29=+-1+

%28cross%28abi%29%2Ba%5E2-b%5E2%2Ai%5E2-cross%28abi%29%29%2F%28b%5E2%2Ai%5E2-a%5E2%29=+-1+

%28a%5E2-b%5E2%2A%28-1%29%29%2F%28b%5E2%2A%28-1%29-a%5E2%29%29=+-1+

%28a%5E2%2Bb%5E2%29%2F%28-b%5E2-a%5E2%29=+-1+

%28a%5E2%2Bb%5E2%29%2F%28-%28b%5E2%2Ba%5E2%29%29=+-1+

cross%28%28a%5E2%2Bb%5E2%29%291%2F%28-cross%28%28a%5E2%2Bb%5E2%29%291%29=+-1+

1%2F-1=+-1+

-1=+-1+