a pure real number
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Yes. Consider as the simplest example: i * i = -1. But there are others: (a + bi)(a - bi) = a² + b². When you multiply conjugates, the result is always real. This is useful when dividing to get a pure real number in the denominator.
It can be used as a convenient shortcut to calculate the absolute value of the square of a complex number. Just multiply the number by its complex conjugate.I believe it has other uses as well.
Generally, the process involves multiplying the numerator and denominator of the fraction by the same number. This number is selected so that the original denominator becomes rational. In the process the numerator may become rational. If the original denominator is of the form √b then you multiply the numerator and denominator by √b/√b. If the original denominator is of the form a+√b then you multiply the numerator and denominator by (a-√b)/(a-√b). NOTE change of sign. There is a similar process, using complex conjugates, if the denominator is a complex number.
Not necessarily. It can be wholly imaginary.For example, 1 + i actually has two complex conjugates. Most schools will teach you that the complex conjugate is 1 - i. However, -1 + i is also a conjugate for 1 + i. (Their product is -1 times the product of the "normal" conjugate pair).The sum of 1 + i and -1 + i = 2i
Yes.