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Can the product of two non real complex numbers be a real number?
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.
Which arithmetic operation requires the use of complex conjugate?
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.
How do you rationalizing the denominator?
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.
Is the sum of two conjugate complex number a real 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
DO You always get an irrational number when you multiply a rational number and an irrational number?
Yes.
