PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
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Chapter P.6, Problem 54E
Solution

To prove the product of a complex number and its complex conjugate is a complex number whose imaginary part is zero.

Concept used:

  • Product of powers property, an⋅am=an+m
  • FOIL to multiply binomials, a+bc+d=a⋅c+a⋅d+b⋅c+b⋅d
  • Properties of complex number: −1=i, i2=−1, i3=−i
  • The complex conjugate of a+bi is a−bi .
  • Difference of squares formula, a+ba−b=a2−b2

Calculation:

In order to prove the product of a complex number and its complex conjugate is a complex number, let the complex number is a+bi and its conjugate is a−bi solving the expression as shown below:

  a+bia+bia−bia2−abi+abi−b2i2a2−b2i2a2−b2−1a2+b2

Hence, proved that the product of a complex number and its conjugate is a complex number, whose imaginary part is zero.

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Chapter P.6, Problem 53E
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Chapter P.6, Problem 55E

Chapter P Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

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