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What are the characteristics of a complex number? What is the relationship between a complex number and its conjugate? Describe the usefulness of the conjugate and its effect on other complex numbers.

Mathematics
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wow so many questions
i feel like im in a job interview all over again :/
A complex number is a number which has both a real part a and imaginary part b i, where a and b are elements of the Reals, and i = sqrt(-1).

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Now im guessing for the second question that you're having trouble because you don't know what a conjugate is. Is that the case?
The relationship is that the product of a complex number and it's respective complex conjugate is a real number.
For number two. ^
Yes. But do you know how to find one?
..I still don't understand the last one.
Well it let's you turn complex numbers into reals. This comes in handy in many respects. For instance, in mathematics we often need to make the denominator real. Here's where it comes in: \[\frac{1}{3+i} = \frac{1}{3+i} \times 1 = \frac{1}{3+i} \times \frac{3-i}{3-i} = \frac{3-i}{9+1}\]
But that's a very easy way of answering your question. Aside from "aesthetics" like realizing (?) the denominator, it comes in hand when factoring polynomials. If you find one root for a polynomial that's imaginary, and the polynomial had real coefficients to begin with, then I guarantee you that its conjugate is also a root, since somewhere along the line those two roots had to cancel using the conjugate property you described to make real coefficients for the polynomial.

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