There's nothing imaginary about complex numbers
Lynn C. Kurtz, Ph.D.
Arizona State University
Department of Mathematics and Statistics, Retired

Abstract

This article gives a pedagogical approach to introducing complex numbers to students who haven't seen them before. Arithmetic on the number line is generalized in a natural way to arithmetic on the plane. The terms "imaginary" and "complex" are not used during the development, and the "number" whose square is -1 arises naturally. At the end of the development, the standard complex number notation is given.

1  Introduction

Typically, a High School algebra class first encounters the subject of complex numbers when they encounter an equation with the square root of a negative number. Since the squares of both positive and negative numbers are always positive, no number x satisfies the equation x2 = -1. At this point the "imaginary" unit i with the property that i2 = -1 is introduced. This is usually followed quickly by the a + bi notation and students learn to manipulate these mysterious new "complex" numbers using the usual rules in addition to the special property of i that i2 = -1. Unfortunately, the terms "imaginary" and "complex" seem very appropriate, given that, from the students' point of view, this i thing is a figment of their imagination, pulled out of nowhere, that has a property no number actually has.
In this paper, we present a pedagogical approach wherein we seek merely to answer the question of whether or not we can do 2-D arithmetic in the plane. Can we add, subtract, multiply, and divide points (a,b) in the plane just like ordinary numbers? The answer, of course, is yes, and when you have worked through this development with your students, there won't be anything magical or mysterious about the number i. It will be staring them in the face.
There is no new mathematics in this paper. After you have read it, you will see how to make a presentation for your classes that will introduce complex numbers in a natural way. Depending on the level of your class, you might choose to either elaborate on or omit some of the details given here.

2  Plane Numbers

Your students are already familiar with the Cartesian xy plane and the fact that the x axis is just a copy of the number line with 0's for the y components. For our purposes, we will call ordered pair (a,b) which represents a point in the plane a plane number. Just as we might represent an ordinary number by a letter, say a = 5 we might represent a plane number by a bold letter w = (3,4). To reduce confusion, we will usually use letters near the end of the alphabet to label plane numbers. We will define equality for plane numbers as follows. If v = (a,b) and w = (c,d), then v = w if and only if a = c and b = d.
If your students are already familiar with the idea of representing a point (a,b) as a position vector, that's all the better. Familiarity with vectors will be very helpful in motivating the necessary definitions for the arithmetic operations for plane numbers.
We will use R1 to denote the number line and R2 to denote the set of all pairs (a,b) with a and b in R1, that is, the plane numbers.
Before developing the arithmetic for plane numbers, it would be well to review the properties that number line arithmetic has. These are the properties of a field:
  1. For all a, b in R1, both a + b and a·b belong to R1. This is called closure of R1 under addition and multiplication. It is another way of saying that addition and multiplication are binary operations on R1.
  2. For all a, b, c in R1, a + (b + c) = (a + b) + c and a·(b·c) = (a·b)·c. These are the associative laws for addition and multiplication.
  3. For all a, b in R1, a + b = b + a and a·b = b·a. These are the commutative laws for addition and multiplication.
  4. For all a in R1, 0 + a = a and 1·a = a. 0 is called the additive identity, and 1 the multiplicative identity. Also 1 ≠ 0 in any field.
  5. For each a in R1, there is a number -a in R1 such that a + (-a) = 0, and for each nonzero a in R1, there is a number a-1 in R1 such that a·a-1 = 1. This says that additive and multiplicative inverses exist.
  6. For all a, b, c in R1, a·(b + c) = (a·b) + (a·c). This is called the distributive law.
These properties hold for other objects than the number line. For example, if you replace R1 by the rational numbers Q, all the statements hold. If we are to generalize our arithmetic operation to plane numbers, we will want three important things to happen:
If we can succeed with this plan, we will have extended the arithmetic from the line to the plane so that the arithmetic on the line is just part of the "bigger picture" of arithmetic in the plane.

3  Arithmetic operations for Plane Numbers

In this section we will develop the arithmetic operations for plane numbers. We will use the boldface + symbol for plane number addition, and the bold * symbol for plane number multiplication, so as not to confuse them with the ordinary arithmetic operations + and ·.

3.1  Addition

It seems quite natural to define the sum of two plane numbers:
(a,b) + (c,d) def
=
 
 (a + c, b + d)
Notice that if b = 0 and d = 0 this statement simply says
(a, 0) + (c,0) = (a + c, 0)
This tells us that our plane number addition reduces to and agrees with our number line addition on the x axis. At this point it is a good idea to have the students verify the commutative and associative properties for + and that (0,0) is the additive identity for plane numbers.
Those students familiar with vectors will recognize + as vector addition. Not only that, but it agrees with adding line segments directed from the origin on the number line. Now is a good time to draw a picture showing the how plane number addition is just the familiar parallelogram law in the vector interpretation. Figure 1 illustrates addition for:
v = (a,b), w = (c,d), z = v + w = (a+c,b+d)
Figure 1: Vector Addition

3.2  Multiplication

Coming up with the right definition for plane number multiplication is a bit trickier to motivate. Since plane numbers on the X axis must obey the equation
(a,0)*(c,0) = (a·c,0)
in order to agree with ordinary number line multiplication, your class may suggest the following multiplication definition:
(a,b)*(c,d) def
=
 
 (a·c,b·d)
Your students may observe that this operation satisfies most of the field properties, and it is a nice exercise to see where it fails. Unfortunately, among other things, the multiplicative identity has to be (1,1), which fails our goal to keep (1,0). This means, ultimately, that we must look elsewhere for a multiplication definition.
This is a good place to mention polar coordinates as it will be helpful in what follows. We can express the plane number (a,b) in terms of its standard polar radius r and angle θ. In order to not mix up the two notations we will use square brackets for polar notation. So the plane number w = (a,b) is the same number as w = [r, θ] where a = r cos(θ) and b = r sin(θ). We shall call r the length or magnitude of the plane number and θ its [polar] angle. As is usual with polar coordinates, this representation is not unique.
Figure 2: Polar Coordinate representation

We can always choose an angle coterminal with an angle in (-π, π] so that r > 0. We will use the absolute value sign to indicate the magnitude of a plane number, so we have  |w| = r = (a2 + b2)1/2.
Now consider for example the plane number w = (3,4) interpreted as a vector. In terms of polar coordinates, this plane number has the representation [5, φ] where φ = arctan(4/3) = .927 radians approximately. Your students may know that, as a vector, if you multiply it by 2, you should get (6,8) which is [10, φ] in polar form and if you multiply it by -2 you should get (-6,-8), which is [10, π + φ] in polar form. Also, expressing 2 and -2 as plane numbers, we have the two representations (2,0) = [2,0] and (-2,0) = [2, π]. Figure 3. illustrates this multiplication.
Figure 3: Multiplication of a plane number by a line number

Here is a tabulation of what the picture shows:
w
2
-2
2w
-2w
Rectangular
(3,4)
(2,0)
(-2,0)
(6,8)
(-6,-8)
Polar
[5,φ]
[2,0]
[2,π]
[10,φ]
[10,φ + π]
Notice that the effect of multiplying by +2 merely doubles the length while multiplying by -2 doubles the length and changes the direction. If you look at the polar coordinate line, you can see that in each case, multiplying (3,4) by either (2,0) or (-2,0) has the effect of multiplying their magnitudes and adding their polar angles.
While this argument only works for multiplying a plane number by a number on the number line, it certainly suggests another possibility for multiplying plane numbers. Perhaps to multiply any two plane numbers we could multiply their magnitudes and add their polar angles to get the answer. This would certainly agree with what we have seen so far. So if v = (a, b) = [r, θ] and w = (c, d) = [s, α], let's try the following definition for multiplication:
z = v * w def
=
 
 [ rs, θ + α]
Expressing this in rectangular coordinates gives
z
=
(rs cos(θ + α), rs sin(θ + α))
=
(rs (cos(θ)cos(α) - sin(θ)sin(&alpha)), rs (sin(θ)cos(α) + cos(θ)sin(α))
=
(rcos(θ) scos(α) - rsin(θ) ssin(α), rsin(θ) scos(α) + rcos(θ) ssin(α))
=
(ac - bd, bc + ad)
where the last step comes from the polar coordinate conversions: v = (a, b) = (rcos(θ), rsin(θ)) and w = (c,d) = (scos(α), ssin(α)).
To summarize, our prospective definition for multiplication of plane numbers in rectangular form is:
(a,b) * (c,d) = (ac - bd, bc + ad)
and in polar form it is:
[r, θ] * [s,α] = [rs, θ + α]
and we know that geometrically the result is a plane number whose magnitude is the product of the magnitudes of the originals and whose polar angle is the sum of the polar angles of the originals.
You should have your students get familiar with these formulas. At this point I would hand out a sheet with a bunch of addition and multiplication exercises. Somewhere buried in the exercises put (0,1)*(0,1) and see if anyone notices that squaring this plane number gives (-1,0). And, of course, there are many things to check. Does it all check out with the number line? Is (1,0) the multiplicative identity? What about all the field properties? Point out that some of the field properties are easier to show using the polar form. I would do a couple of the easy ones and perhaps talk about the multiplicative inverse.

3.3  An example, the multiplicative inverse

If w = (a,b) ≠ (0,0) does w have a multiplicative inverse? In other words, can we find (x,y) such that (a,b)*(x,y) = (1,0)? This means (ax - by, bx + ay) = (1,0). This gives the linear system:
ax - by
=
1
bx + ay
=
0
with solutions
x = a
a2 + b2
 , y = -b
a2 + b2
It is easy to check that
w-1 = ( a
a2 + b2
 , -b
a2 + b2
 )
satisfies w * w-1 = (1,0) so long as w ≠ (0,0).
Notice what the same problem becomes in polar form. If r ≠ 0, find [s, α] such that
[r, θ] * [s, α] = [1,0]
This requires that rs = 1 and θ + α is coterminal with 0. So we may simply take s = 1/r and α = -θ, so the inverse of w = [r,θ] is w-1 = [1/r, -θ]. Have one of your students check that these two versions of w-1 are the same.

4  Wrapping it up

What you do from here is a matter of taste. You can define the conjugate of w = (a,b) as

and develop various formulas such as

and all the others without ever mentioning i or "complex" numbers a + bi.

Sooner or later you will want to introduce the standard notation and terminology. To that end, you can define i = (0,1) and 1 = (1,0) (note the bold face on the "1" to distinguish the plane number name of the multiplicative identity from the line number 1). (a,b) can be written as (a,0)* (1,0) +  (b,0)* (0,1), which can be expressed as (a,0) *1 + (b,0)* i. Now with only a slight abuse of the notation, we can note that the first term is just the plane number version of the line number a, and instead of writing it as (a,0)* 1, just write it as a and drop the * symbol and the 1 symbol. Similarly, in the second term we can drop the * symbol and unbold the i. This gives the representation:
(a,b) = a + bi
where we have now also dropped the + symbol. To convert to this new notation we just need to note that
(a,b) + (c,d)
=
(a + bi) + (c + di) = (a + c) + (b + d)i
(a,b) * (c,d)
=
(a + bi)(c + di) = (ac - bd) + (bc + ad)i
Of course, the advantage of this notation is that the multiplication formula can be obtained by formally expanding the product using the fact that i2 = -1 and grouping like terms. It also hides the implied use of the plane number operations.
And there is nothing imaginary about i because, using our conventions:
i2 = (0,1)*(0,1) = (-1,0) = -1
This document is "copyleft" i.e., not copyrighted 2008. It is for educational purposes and may be freely distributed either electronically or in printed form as long as it is distributed in its entirety. Comments, corrections, and suggestions are welcome. Let me know if you find it useful. My email address is kurtz@asu.edu.