imaginary numbers real world

Complex numbers

Overview: This article covers the definition of complex numbers of the form $$ a+ bi $$ and how to graph complex numbers.

What are complex numbers?

A complex number can be written in the form a + bi where a and b are real numbers (including 0) and i is an imaginary number

Therefore a complex number contains two 'parts':

  • one that is real
  • and another part that is imaginary
note: Even though complex have an imaginary part, there are actually many real life applications of these "imaginary" numbers including oscillating springs and electronics.

Examples of a complex number

$$ \begin{array}{c|c} \blue 3 + \red 5 i & \\\hline \blue{12} + \red{\sqrt{-3}} & \red{\sqrt{-3}} \text{ is } \blue{imaginary} \text{ part} \\\hline \blue 9 - \red i & \\\hline \blue{12} - \red{\sqrt{-25}} & \red{\sqrt{-25}} \text{ is } \blue{imaginary} \text{ part} \\\hline \end{array} $$

How do you graph complex numbers?

Complex numbers are often represented on a complex number plane (which looks very similar to a Cartesian plane).

picture of complex plan and cartesian plane
Diagram of Graph of Complex Numbers

Practice Problems of complex number

Problem 1

Identify the coordinates of all complex numbers represented in the graph below.

Identify Complex Numbers on Graph
Problem 2

In what quadrant, is the complex number $$ 2- i $$ ?

This complex number is in the fourth quadrant.

picture of graph of two minus i
Problem 3

In what quadrant, is the complex number 2i -1?

This complex number is in the 2nd quadrant.

picture of graph of two i minus 1
Problem 4

In what quadrant, is the complex number $$-i -1 $$ ?

This complex number is in the 3rd quadrant.

picture of graph of  -i minus 1
to Complex Number Home Page to Complex Numbers in Real World

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