Arithmetic With Complex Numbers
Learning Outcomes
- Identify the difference between an imaginary number and a complex number
- Identify the real and imaginary parts of a complex number
- Plot a complex number on the complex plane
- Perform arithmetic operations on complex numbers
- Graph physical representations of arithmetic operations on complex numbers as scaling or rotation
- Generate several terms of a recursive relation
- Determine whether a complex number is part of the set of numbers that make up the Mandelbrot set
Complex Numbers
[footnote]Portions of this section are remixed from Precalculus: An Investigation of Functions by David Lippman and Melonie Rasmussen. CC-BY-SA[/footnote] The numbers you are most familiar with are called real numbers. These include numbers like 4, 275, -200, 10.7, ½, π, and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number 3, we plot a point: To solve certain problems like , it became necessary to introduce imaginary numbers.Imaginary Number i
The imaginary number i is defined to be . Any real multiple of i, like 5i, is also an imaginary number.Example
Simplify . We can separate as . We can take the square root of 9, and write the square root of as i.
Complex Number
A complex number is a number , where- a and b are real numbers
- a is the real part of the complex number
- b is the imaginary part of the complex number
Complex Plane
In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
Example
Plot the number on the complex plane. The real part of this number is 3, and the imaginary part is . To plot this, we draw a point 3 units to the right of the origin in the horizontal direction and 4 units down in the vertical direction.
Arithmetic on Complex Numbers
Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.Example
Add and . Adding , we add the real parts and the imaginary parts.
Try It
Subtract from .Answer:
Example
Visualize the product . Multiplying we’d get\begin{align}&2\cdot1+2\cdot2i\\&=2+4i\\\end{align}
Notice both the real and imaginary parts have been scaled by 2. Visually, this will stretch the point outwards, away from the origin.
Example
Visualize the product . Multiplying, we’d get\begin{align}&i\cdot1+i\cdot2i\\&=i+2{{i}^{2}}\\&=i+2(-1)\\&=-2+i\\\end{align}
In this case, the distance from the origin has not changed, but the point has been rotated about the origin, 90° counter-clockwise.
Try It
Multiply and .Answer: Multiply
Example
Visualize the result of multiplying by . Then show the result of multiplying by again. Multiplying by ,
\begin{align}&(1+2i)(1+i)\\&=1+i+2i+2{{i}^{2}}\\&=1+3i+2(-1)\\&=-1+3i\\\end{align}
Multiplying by again,\begin{align}&(-1+3i)(1+i)\\&=-1-i+3i+3{{i}^{2}}\\&=-1+2i+3(-1)\\&=-4+2i\\\end{align}
If we multiplied by again, we’d get . Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by 45°.