Algebra: A Combined Approach (4th Edition)

Published by Pearson
ISBN 10: 0321726391
ISBN 13: 978-0-32172-639-1

Chapter 10 - Section 10.7 - Complex Numbers - Exercise Set: 100

Answer

The product of a complex number and its conjugate is a real number because the opposite signed imaginary parts cancel each other out (in the outer and inner steps of FOIL). In addition, when the two imaginary parts are multiplied by each other (in the last step of foil), they become $i^2$, which is equal to $-1$.

Work Step by Step

$(a+bi)\cdot(a-bi) = a^2+[abi-abi]-b^2\cdot(i^2) \\ = a^2 -b^2\cdot(-1) = a^2+b^2$ The final value does not have $i$ as any coefficient.
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