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Chapter 1: Problem 62

Explain how to divide complex numbers. Provide an example with your explanation.

Short Answer

Expert verified
To divide complex numbers, you must multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary parts from the denominator. Simplify the result to obtain the answer. For example, the division of the complex numbers \(5 + 3i\) and \(2 - i\) results in \(2.6 + 1.4i\).

Step by step solution

01

Understand the Complex Number

A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the undefined imaginary unit, given by \(i^2 = -1\). When dividing complex numbers, say \(A/B\), where \(A = (a + bi)\) and \(B = (c + di)\), we have to eliminate the imaginary part from the denominator.
02

Find the Conjugate

Find the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign in between the real and the imaginary part. So for \(B = c + di\), the conjugate, denoted as \(\overline{B}\), would be \(c - di\).
03

Multiply with Conjugate

Multiply both numerator and denominator of the division with the conjugate of the denominator. This can be denoted as \((a + bi) * (c - di) / (c + di) * (c - di)\). This is a crucial step as multiplying with the conjugate eliminates the imaginary part in the denominator.
04

Simplify the Result

Expand and simplify the numerator and denominator separately. The denominator will now be a real number (as the imaginary part gets eliminated), and the numerator will be another complex number. Simplify the expression to obtain the result in the form of another complex number.
05

Example Illustration

Example: Consider dividing two complex numbers, \(A = 5 + 3i\) and \(B = 2 - i\). The process is described in the previous steps: \((5 + 3i) / (2 - i) = (5 + 3i) * (2 + i) / (2 - i) * (2 + i) = (13 + 7i) / 5 = 2.6 + 1.4i\)

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