Problem 62 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator, simplify, and separate the real and imaginary parts. For example, \((3 + 4i) / (1 + 2i) = 5/3 - 2i\).

Step 1: Understanding the Structure of Complex Numbers

Firstly, we need to understand the structure of complex numbers. A complex number is a number of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which satisfies the equation \(i^2 = -1\). In the complex number \(a + bi\), \(a\) is called the real part and \(b\) is called the imaginary part.

Step 2: Introduction to the Conjugate of a Complex Number

The conjugate of a complex number is obtained by changing the sign of its imaginary part. So the conjugate of the complex number \(a + bi\) is \(a - bi\).

Step 3: Division of Complex Numbers

The division of complex numbers is carried out by multiplying both the numerator and denominator by the conjugate of the denominator. Assume we have two complex numbers \(Z1 = a + bi\) and \(Z2 = c + di\). The division \(Z1 / Z2\) can be written as \((a + bi) / (c + di)\). To perform the division, we multiply the numerator and the denominator by the conjugate of the denominator: \((a + bi) / (c + di) = (a + bi)(c - di) / (c + di)(c - di)\).

Step 4: Solving the Division

Perform the multiplication in the numerator and denominator, simplify and separate the real and imaginary parts to get the result of the division. The final division result is in the form \(p + qi\).

Step 5: Example: Division of Complex Numbers

Let's take an example of \(Z1 = 3 + 4i\) and \(Z2 = 1 + 2i\). The division \(Z1 / Z2 = (3 + 4i) / (1 + 2i) = (3 + 4i)(1 - 2i) / (1 + 2i)(1 - 2i) = (3 - 8 + 4i + 2i) / (1 - 4) = -5 + 6i / -3 = 5/3 - 2i\). So, the division result of the given complex numbers is \(5/3 - 2i\).