Complex Numbers


Henri Picciotto


During my many years teaching high school math, I developed a graphical approach to the teaching of complex numbers, geared to Algebra 2 students. See a fun kinesthetic introduction to this, by Michael Pershan and Max Ray, here.

The basic idea is to start by defining complex numbers as (corresponding to) points in the plane, which can be thought of in rectangular form (with normal Cartesian coordinates,) or in polar form. These numbers have their own arithmetic: addition works like vector addition, and multiplication is defined as multiplying the r's and adding the θ's. We verify that arithmetic for numbers on the x-axis (the real numbers) still works as expected. We see that in this system, -1 has a square root, which we call i.

That the approach makes sense need not be proved formally at the Algebra 2 level, but the proof is accessible to students in a later elective course, such as Precalculus, or my Space course. (The proof is somewhat involved, and thus requires some mathematical maturity, but the only prerequisite is an understanding of similar triangles.) In Algebra 2 it is sufficient to confirm in several cases that the distributive rule in a + bi form yields the same result as the geometric multiplication in polar form.

The advantages of this approach are many.
- It provides some review and practice of basic trig.
- It gives a visual, concrete interpretation to complex numbers.
- It lays a foundation for the formulas for the sine and cosine of a sum, and the double angle formulas.
- It prepares students for the calculation of images of points under geometric transformations, including the rotation matrix.

In fact, I found this approach to matrices to be far more effective than introducing them as a way to solve systems of equations. Read about this on my blog.



These online games help provide a solidly intuitive introduction to complex number arithmetic.
Addition (rectangular form)
Subtraction (rectangular form)
Multiplication (polar form)
Division (polar form)
Download all four games (GeoGebra files)
(It's easy to cheat to improve your score, but that wouldn't do much to help you understand complex number arithmetic, now, would it.)


From most basic to most advanced:
Kinds of Numbers (a worthwhile conversation starter)
Complex Numbers in Algebra 2
Complex Numbers, Transformations, and Matrices
— (classroom-tested, includes the proof that this approach makes sense,
— and some activities involving the TI-89 calculator)
GeoGebra version (not tested, but possibly useful)

Kinesthetic Introduction to Complex Numbers

Geometric transformations, complex numbers, and matrices

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Related pages on this site:
Signed Number Arithmetic Games
Seeking Depth in Algebra 2
Transformational Geometry
Applets Directory
Games Directory