Simplifying Complex Number Division: A Comprehensive Guide

When dealing with complex numbers, it's essential to understand how to manipulate them in both rectangular and polar forms. This article will guide you through the process of dividing complex numbers using these forms and explain the steps involved in doing so. Let's explore how to solve the question 1i ÷ 1-i and provide a comprehensive understanding of complex number division.

Complex Numbers in Rectangular Form

Complex numbers are typically expressed in rectangular form, which is the standard format when dealing with addition, subtraction, multiplication, and division. In this form, a complex number is expressed as:

A Bi

where:

A is the real part B is the imaginary part i is the imaginary unit, defined as the square root of -1 (i.e., i^2 -1)

For instance, the complex number 1 i is composed of a real part of 1 and an imaginary part of 1. Similarly, the complex number 1 - i consists of a real part of 1 and an imaginary part of -1.

Complex Numbers in Polar Form

Complex numbers can also be represented in polar form. In polar form, a complex number is expressed as:

r∠θ

where:

r is the magnitude (or modulus) θ is the argument (or angle)

To convert a complex number from rectangular form to polar form, use the following formulas:

r √(a^2 b^2)

θ tan^(-1)(b/a)

And to convert back from polar to rectangular form:

a rcos(θ)

b rsin(θ)

Converting Between Rectangular and Polar Form

The choice of form often depends on the type of operation being performed. For addition and subtraction, rectangular form is the most convenient. For multiplication and division, however, polar form is more efficient. Let's dive into the process of converting between these forms and performing division.

Performing Division with Complex Numbers

To divide complex numbers, we can follow a straightforward process:

Convert the complex numbers to polar form. Divide the magnitudes. Subtract the angles.

Let's apply this to the example 1i ÷ 1-i.

Example: Dividing 1i by 1-i

First, let's convert 1i to polar form:

1i 1.4142∠45°

Next, convert 1-i to polar form:

1-i 1.4142∠-45°

Now, perform the division in polar form:

Divide the magnitudes: 1.4142 ÷ 1.4142 1.0 Subtract the angles: 45° - (-45°) 90°

Therefore, the result is:

1.0∠90° i

Using the Conjugate to Simplify Division

Another method to divide complex numbers involves using the conjugate of the denominator. The conjugate of a complex number is formed by changing the sign of the imaginary part. For example, the conjugate of 1-i is 1 i.

To divide 1i by 1-i using the conjugate:

Multiply the numerator and denominator by the conjugate of the denominator: 1i(1-i) ÷ (1-i)(1 i) The denominator becomes 1 - i^2, and since i^2 -1, this simplifies to 1 - (-1) 2 The numerator simplifies to 2i The final result is 2i/2 i

This method is particularly useful when working manually or using basic calculators that do not support complex number operations.

In conclusion, the division of complex numbers 1i ÷ 1-i can be solved by converting them to polar form and dividing the magnitudes and subtracting the angles. Alternatively, using the conjugate of the denominator provides a simpler approach. Understanding both methods and the transformations between rectangular and polar forms is crucial for efficient and accurate complex number arithmetic.