Problem 62 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The division of complex numbers \(a + ib\) and \(c + id\) is \((ac + bd) / (c^2 + d^2) + i((bc - ad) / (c^2 + d^2))\). The process includes multiplying by the conjugate and simplifying.

Step 1: Understanding Complex Numbers

A Complex number is a number which can be expressed in the form of \(a + ib\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i^2 = -1\). Example of a complex number could be \(2 + 3i\).

Step 2: Understanding the Conjugate

The Conjugate of a Complex Number is the identical number, but with the opposite sign in the imaginary part. So the conjugate of \(a + ib\) is \(a - ib\). If \(z = 2 + 3i\), the conjugate of \(z\) (\(z'\)) will be \(2 - 3i\)

Step 3: Division of Complex Numbers

For dividing complex numbers we make use of the conjugate. Assume that there are two complex numbers \(z1 = a + ib\) and \(z2 = c + id\). The division of \(z1\) by \(z2\) is given by \((a + ib)/(c + id)\); Multiply numerator and denominator by the conjugate of the denominator, ((a + ib)*(c - id))/((c + id)*(c - id)); expanding and simplifying gives ((ac + bd) / (c^2 + d^2)) + i((bc - ad) / (c^2 + d^2))