Converting Complex Numbers to Standard Form: A Comprehensive Guide

Understanding how to convert complex numbers to their standard form is crucial in various fields, including mathematics, engineering, and physics. The standard form of a complex number is expressed as:

z a bi

where:

a is the real part b is the imaginary part i is the imaginary unit, defined such that i^2 -1

Steps to Convert to Standard Form

Identify the Complex Number

Make sure you know the form of the complex number you have. It might be given in various forms such as:

Polar Form: z rcos(θ) isin(θ) Rectangular Form: z a bi

Convert from Polar to Rectangular if Applicable

If the complex number is in polar form, z rcos(θ) isin(θ), you can convert it to standard form using:

a r cos(θ) b r sin(θ)

Therefore, the standard form would be:

z r cos(θ) r sin(θ) middot; i

Combine Real and Imaginary Parts

If the complex number is already in a form like a b middot; i, simply identify a and b.

Write in Standard Form

Ensure that your final expression is in the format a bi.

Example

Given

z 5 3i

Here, a 5 and b 3.

The standard form is already:

z 5 3i

Given in Polar Form

z 2cos(π/3) isin(π/3)

Let's calculate a and b:

a  2 cos(π/3)  2 middot; 1/2  1b  2 sin(π/3)  2 middot; √3/2  √3

Therefore, the standard form is:

z 1 √3i

Additional Advice

Just treat i like you would the variable x because the rules that apply are the same for either situation. Until you get a little farther in your lessons, you'll learn a few properties of i that are different from other variables.

Example Problem: Solving and Writing in Standard Form

There is an example problem on your homework which reads:

2 - 3i middot; 65i

It wants us to solve and write it in standard form of abmiddot;i, where a is a real number and i is imaginary.

Step-by-Step Solution

Rewrite the expression by combining the real and imaginary parts separately:

Add the real parts: 2

Add the imaginary parts: -3i 65i 62i