If you’ve got an adventurous spirit and are happy not knowing where things may be going, I suggest skipping the blue paragraph and going straight to The Number Line.
We present a geometric approach to introducing complex numbers. The approach views the complex numbers as a generalization of the number line to the plane. Often, students first meet complex numbers algebraically, and then the geometric interpretation is explained. Here, we reverse the process. By doing so, we hope to help some students realize complex numbers as actual objects that are as concrete as mathematical objects can be.
The Number Line
Start with a straight line. Pick out a point on this line and call it “zero” or “0”. Pick any other point and call it “one” or “1”.
We’re going to define two geometric operations and call them “addition” and “multiplication.” Both operations will take two points on the line and produce a third point on the line. So pick two points A and B and imagine arrows pointing from zero to each of A and B.
To add A and B, move the arrow pointing at B (without changing the arrow’s direction) so that its foot is on the head of the arrow pointing at A. Its head will now point at the point that is the sum of A and B, which we will denote by “A + B“.
If either A or B is zero, let their product be the point zero. Otherwise, to multiply A and B, find the arrow (whose foot is at zero) that has the same similarity relationship to the B arrow as the A arrow does to the arrow pointing from 0 to 1. In other words, find the arrow, that, together with the B arrow, is similar to the figure comprised of the A arrow and the arrow pointing from zero to one. This arrow points at the point that is the product of A and B, which we will denote by “AB”.
Notice that we haven’t talked about numbers at all. All we did was start with a line, label two special points, and then define two geometric operations on pairs of points. Of course these definitions are informed by the number line model of the real numbers, which is why we labeled the special points after the numbers they correspond to, but we didn’t have to use those names. If you wanted, you could have called the two special points “origin” and “unit,” or whatever else you fancied.
Generalizing to the Plane
And now for the punchline: The geometric definitions given above for a line work as is for the points of a plane! All you have to do is replace the word “line” with “plane” throughout, and voila, you’ve got yourself a number plane. That is the point of this blog.
In the plane, the addition described above is also known as the “parallelogram law” and multiplication involves similarity of full triangular figures (as opposed to degenerate triangles).
So defined, addition and multiplication are commutative and associative. Zero is the additive identity, one is the multiplicative identity, and multiplication distributes over addition. (That is, these operations turn the plane into a field.)
Exercise: Verify that P + 0 = 0 + P = P and P1 = 1P = P.
The real axis
Notice that the line that passes through the two special points zero and one is closed under the operations of addition and multiplication (of course…since the operations for the plane were taken straight from those of the number line). This line is known as the “real axis” because it corresponds to the number line model of the real numbers.
The point i
Let’s orient the plane so that the real axis is horizontal and one is to the right of zero.
We’ll borrow the negative sign to denote the additive inverse. That is, given a point A, there is a unique point B such that A + B is 0, and we’ll denote this unique point by “-A”. In fact, now that we’ve defined an addition and multiplication, we can write all kinds of algebraic expressions, and when we do, we will borrow all the standard conventions (i.e. the order of operations, the use of parentheses, and exponential notation).
Notice that -1 is the reflection of 1 through 0 because the arrow pointing from 0 to -1 must point opposite to and be the same length as the arrow pointing from 0 to 1.
Draw in the line that passes through 0 and is perpendicular to the real axis. For historical reasons, we’ll call this line the “imaginary axis.” This is one of those situations where it’s a really good idea not to get carried away attaching significance to the names that things have been given.
The circle centered at 0 that passes through 1 and -1 intersects the imaginary axis in two places. Call the intersection point that sits above the real axis, i. The other intersection point is i’s additive inverse, –i.
Now we ask, what is multiplication by i?
To find iA, we have to find the arrow whose relationship to the A arrow is similar to the geometric figure formed by the arrows pointing to i and 1. The geometric figure formed by the arrows pointing to i and 1 is that of an isosceles right triangle where the arrows form the legs. Thus, iA and A must form the legs of an isosceles right triangle, and since we rotate counterclockwise about 0 to get from 1 to i, we must rotate counterclockwise about 0 to go from A to iA.
We conclude that multiplication by i is counterclockwise rotation about 0 by 90 degrees.
In particular, , because if we rotate i about the origin by 90 degrees counterclockwise, we end up at -1.
So, in the number plane, -1 has a square root. (Actually, it has two square roots, since too.)
Notice that the product of any two points on the circle that passes through 1, -1, i, and –i is also a point on the circle. For suppose A and B are both on this circle. By definition, “AB is to B as A is to 1.” The arrows pointing to A and 1 form the equal sides of an isosceles triangle. Therefore, the arrows pointing to AB and B must also form the equal sides of an isosceles triangle with the same apex angle.
Exercise: Find all seven points whose seventh power is 1.
Fix a point A and consider multiplication by A. For any point P, the “product PA is to P as A is to 1.” By similarity, the angle between the arrows pointing at PA and P must equal the angle between the arrows pointing to A and 1. Also, by similarity, the ratio of the distance of PA from 0 to that of P from 0 must be equal to the ratio of the distance of A from 0 to that of 1 from 0.
This suggests expressing multiplication in terms of polar coordinates.
Since we’re finally talking about coordinates, that is, since we are finally introducing numbers into this geometric picture, we need to normalize our unit of length. Let’s declare the unit length to be the distance between 0 and 1.
In polar coordinates, we specify the point P by where is its distance from 0 and is the measure of the angle formed by the arrows pointing at P and 1 (from 0). By convention, positive angle measurements mean counterclockwise rotations about 0 from 1.
Let and be the polar coordinates of two points in the plane. Unraveling the definition of their product, we find that the product has polar coordinates .
Let’s think of the real axis as the “x-axis” and the imaginary axis as the “y-axis.” (We’ll keep the length normalization from the last section and regard the distance between 0 and 1 to be the unit length.)
Exercise: Suppose the point P has Cartesian coordinates (x, y). Show that P = x + iy.
If you apply the same generalization from the line to the plane one step further to space, things fail. Try it to find out why. (You’ll find that the problem isn’t with addition, but with multiplication.)
In this blog post, we’ve approached complex numbers as a generalization of the number line to the “number plane.” (In mathematics, this “number plane” is known as the complex plane and points in our “number plane” are complex numbers.) It’s important to note that we have not given rigorous definitions. If you want a rigorous construction of the complex numbers, algebraic approaches are more direct (see, for instance, Walter Rudin’s Principles of Mathematical Analysis, chapter 1). Still, doesn’t this realization of the complex plane as a straightforward generalization of the number line make you wonder why it wasn’t discovered a lot earlier in history? (The complex plane was discovered in 1799 by Caspar Wessel. At that time, Beethoven was working on his first and second symphonies and Gauss had just finished work on his Disquisitiones Arithmeticae.)