Question 896000: Are these statements always, sometimes, or never true? If never, what is a counterexample. If sometimes true, give an example AND counterexample.
1) The difference between any complex number a+bi (b doesn't equal zero) and its conjugate is a real number.(I said ALWAYS because it turned out to always be 0)
2) The product of any two imaginary numbers bi (b doesn't equal zero) and di (d doesn't equal zero) is a positive real number. (I said NEVER TRUE because i * i is -1)
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Please Answer! Thanks for the help!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! when you add a + bi and a - bi, the answer is going to always be 2a + 0i because bi - bi = 0i which is equal to 0.
since the imaginary component becomes 0, the result will always be a real number.
this statement is always true.
the second statement is:
The product of any two imaginary numbers bi (b doesn't equal zero) and di (d doesn't equal zero) is a positive real number.
bi * di is equal to -b*d because i^2 is always equal to -1.
here's where you can get tripped up.
the only requirement for b and d is that they are not equal to 0.
they can be positive or negative.
if they are both positive, then their product is positive and so the result will be negative when you multiply them by -1.
if they are both negative, then their product is positive and so the result will be negative when you multiply them by -1.
if one of them is positive and one of them is negative, then their product is negative and the result will be positive when you multiply them by -1.
so the answer to the second questions is sometimes, since there is no restriction on whether b or d is positive or negative.
the only restriction is that they do not equal 0.
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