- Understand the difference between imaginary numbers and complex numbers
- Learn how to plot a complex number on a special graph called the complex plane
- Do math operations with complex numbers, and learn how these operations can be shown as scaling or rotation
- Create a series of numbers that are made by repeating a rule, and learn how to find specific terms in the series
- Determine whether a complex number is part of a special set of numbers called the Mandelbrot Set
Complex Numbers
The numbers you are most familiar with are called real numbers. These include numbers like [latex]4, 275, -200, 10.7, \frac{1}{2}, π[/latex], and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number [latex]3[/latex], we plot a point on the number line equidistant between [latex]2[/latex] and [latex]4[/latex]. To solve certain problems like [latex]x^{2}=–4[/latex], it became necessary to introduce imaginary numbers.
imaginary number
The imaginary number [latex]i[/latex] is defined to be [latex]i=\sqrt{-1}[/latex].
Any real multiple of [latex]i[/latex], like 5[latex]i[/latex], is also an imaginary number.
Example: [latex]\sqrt{9}=3[/latex] because [latex]3 \ast 3 = 9[/latex].
It is also true that [latex](-3)\ast (-3) = 9[/latex] although we agreed that using the radical symbol requests only the principle root, the positive one. But, there is no number that when multiplied by itself results in a negative number.
The property of integer multiplication states that both a negative number squared and a positive number squared result in a positive number. Indeed, you saw in the review section to this module that the square root of a negative number does not exist in the set of real numbers. Mathematicians realized the helpfulness of being to do calculations with such numbers though, so they assigned a value to [latex]\sqrt{-1}[/latex], calling it the imaginary unit [latex]i[/latex].
A complex number is the sum of a real number and an imaginary number.
complex number
A complex number is a number [latex]z = a + b i[/latex], where
- [latex]a[/latex] and [latex]b[/latex] are real numbers
- [latex]a[/latex] is the real part of the complex number
- [latex]b[/latex] is the imaginary part of the complex number