- 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 Main Idea
Complex numbers are a type of number that expand the traditional notion of numbers by including imaginary numbers.
The imaginary number [latex]i[/latex] is defined to be [latex]i=\sqrt{-1}[/latex]. Any real multiple of [latex]i[/latex], like [latex]5i[/latex], is also an imaginary number.
A complex number is composed of two parts: a real part and an imaginary part, often written in the form [latex]a + bi[/latex], where [latex]a[/latex] and [latex]b[/latex] are real numbers. This expanded number system allows for solutions to equations that cannot be solved using only real numbers.
You can view the transcript for “Introduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy” here (opens in new window).
Complex Plane
The Main Idea
In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
You can view the transcript for “Plotting complex numbers on the complex plane | Precalculus | Khan Academy” here (opens in new window).
Arithmetic on Complex Numbers
The Main Idea
To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.
When we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane.
We can also multiply complex numbers by a real number, or multiply two complex numbers.
In general, multiplication by a complex number can be thought of as a scaling, changing the distance from the origin, combined with a rotation about the origin.
You can view the transcript for “Plotting complex numbers on the complex plane | Precalculus | Khan Academy” here (opens in new window).
You can view the transcript for “Visualizing complex arithmetic” here (opens in new window).
Complex Recursive Sequences
The Main Idea
A recursive relationship is a formula which relates the next value, [latex]{{z}_{n+1}}[/latex], in a sequence to the previous value, [latex]{{z}_{n}}[/latex]. In addition to the formula, we need an initial value, [latex]{{z}_{0}}[/latex].
The sequence of values produced is the recursive sequence.
In the following video we show more worked examples of how to generate the terms of a recursive, complex sequence.
You can view the transcript for “Recursive complex sequences” here (opens in new window).
Mandelbrot Set
The Main Idea
The Mandelbrot Set is a set of numbers defined based on recursive sequences.
For any complex number [latex]c[/latex], define the sequence: [latex]{{z}_{n+1}}={{z}_{n}}^{2}+c,\quad{{z}_{0}}=0[/latex]
If this sequence always stays close to the origin (within [latex]2[/latex] units), then the number [latex]c[/latex] is part of the Mandelbrot Set. If the sequence gets far from the origin, then the number [latex]c[/latex] is not part of the set.
Watch the following video for more examples of how to determine whether a complex number is a member of the Mandelbrot Set.
You can view the transcript for “Mandelbrot sequences” here (opens in new window).
If you are impressed with the Mandelbrot Set, check out this TED talk from 2010 given by Benoit Mandelbrot on fractals and the art of roughness.
You can view the transcript for “Fractals and the art of roughness” here (opens in new window).