(C) Complex No. Intro

Welcome. Most of this site, except for the treatment here of complex number and trig, posts online appendices and chapters of books on understanding and explaining mathematics and pattern based reason. Writing began to explore and report a few ideas for easing or avoiding difficulties in mathematics education.

1. The immediate motivation for this approach (in this chapter) stems from three successive 1976 McGill University public lectures of the late Richard Feynman. He simply described physics as the addition and multiplication of arrows in the plane. He defined their multiplication as follows: add their angles and multiply their lengths. In terms of the polar coordinate (r₁,q₁) and (r₂,q₂) for the factors, the polar coordinates of the product is

(r,q) = (r₁r₂,q₁+q₂ )

All this was effectively presented to a general audience with no mention of vectors nor the Gauss-Argand representation of complex numbers.

2. In Morris Kline's three-volume work Mathematical Thought from Ancient to Modern Times, in volume 2, Chapter 27, the third section called The Geometrical Representation of Complex Numbers. This section briefly describes the approach of Caspar Wessel (1745-1818). Part of Wessel's work (translated into English) is reproduced in David Eugene Smith's 1929 work A Source Book in Mathematics, Dover 1959 Reprint.

After or besides the local material, visit chapters 1, 2 and 3 of Functional Trigonometry by Hillman and Alexanderson in the 1970's and posted online by Mervin E. Newton of Thiel College. It has slightly different starting point and more material including exercises!

Step I. How to Add and Multiply Points, Arrows or Complex Numbers in the Plane

This first part assumes you have some familiarity with the measurement of distances and angles, with the addition of real numbers and points in the plane, and finally with multiplication of nonnegative (that is zero and positive) real numbers

Addition of points in the plane

Coordinate Definition (Coordinate Method)

The sum of two points with the rectangular coordinates [a,b] and [c,d] is given by [a+c,b+d]. We therefore write

In words, the addition rule is simple add the rectangular coordinates of the summands to get the rectangular coordinates of the sum. With this in mind, the following question is easy: What are the rectangular coordinates of the sum of [1,14] and [2,8]? Answer:

The chapter Arrow Addition in Volume 3 discusses the addition of points or arrows in the plane further.

Multiplication

Next we define using polar coordinates the product of two points in the plane. Each point or factor is located by means of angular displacement or rotation from the positive real axis, and also a nonnegative distance from the origin. The product of two points is given by a third point. Its angular displacement is the sum of the angular displacement of the factors. Its distance to the origin is the product of the distances of the factors. This is the add the angles and multiply the lengths rule. In polar coordinate notation, the multiplication rule and definition is indicated by

Example. Two arrows are to be multiplied. One has length 1.3 and angle 22.62^°; the other factor has length 1.026 and angle 46.97^°; and so their product has length 1.3338 = 1.3·1.026 and angle 69.59^° = 22.62^°+46.97^°; and that is it. See the following diagram.

A Summary - Recapitulation

The addition of points in the plane is given by means of their rectangular coordinates while multiplication is given in terms of polar coordinates. A second way to multiply follows from the distributive law for multiplication over addition of points in the plane. See step III. The equality of two different ways to multiply has several immediate consequences given. See Step IV.

Step II. What Are Complex Numbers

Points in the plane with the operations of addition and multiplication just given are called the complex numbers. The plane with these two operations on its points is called the complex numbers plane, or more briefly the complex numbers.

We will now change to a more standard notation for them. We may and often will write the rectangular coordinates z = (a,b) as z = a+ib, We will further call the abscissa a, the real part of the complex number z = a+ib. We will also call the ordinate b, the imaginary part of the complex number z = a+ib.

We will say that the complex number z = a+ib is purely imaginary when its real part a = 0. The angle of a purely imaginary complex number z = a+ib = 0+ib = (0,b) is 90 degrees or 270 degrees (modulo 360 degrees), depending on the sign of the imaginary part b. When b > 0, the angle is 90 degrees (modulo 360 degrees). When b < 0, the angle is 270 degrees (modulo 360 degrees).

We will also say that z = a+ib is (purely) real when its imaginary part b is zero. The angle of a (purely) real complex number z = a+ib = a+i0 = (a,0) is 0 degrees or 180 degrees (modulo 360 degrees), depending on the sign of the real part a. If a > 0, this angle is 0 degrees (modulo 360 degrees) while if a > 0, this angle is 180 degrees (modulo 360 degrees).

Real Numbers as Complex Numbers

Each complex number z = a+i0 with imaginary part zero gives and is given by a real number a. We will write z = a in this situation, and say that the complex number z is also a real number.

With this practice, the real numbers can be regarded as a subset of the complex numbers; and the real number line can be identified with the horizontal axis of the plane.

Confirmation of The Law of Signs

We identify the real number line with the horizontal axis of the plane. With this identification, observe that positive numbers have angular displacement zero, modulo 360 degrees. Also observe that negative numbers have angular displacement 180 degrees, modulo 360 degrees. The magnitude of a real number is its distance to the origin.

Suppose z = a+i0 and w = c+i0. We want to compute the product zw with the multiply the lengths, add the angles rule. Each factor has length |a| or |c|. Each factor has angle 0 or 180 degrees (modulo 360 degrees). The relationships

For the second example, the number -2 is identified with the point [-2,0] = (2,180^°). See the figure below.

Now multiplying the point (2,180^°) by itself leads to the product (2,180^°)² = (2²,180^°+180^°) = (4,360^°) = (4,0^°). Thus the point on the horizontal axis identified with -2 when squared gives the point identified with +4 indicated above. The 360 degrees in the diagram for the number or point 4 = [4,0] represents the doubling of the angle 180 degrees.

For an example or exercise, compute the pair-wise products of 3=3+0i, 4=4+0i, -3=-3+i0 and -4=-4+0i using the add the angles, multiply the lengths rule.

More Exercises. Compute the following using the multiply the lengths, add the angles rule:

Stop For A Summary. The polar coordinate definition

(r₁,q₁)·(r₂,q₂) = (r₁r₂,q₁+q₂)

of the product of two point in the plane, involves the multiplication of lengths (= distances to the origin) and the addition of angles. For points on the horizontal axis, the angles of the factors are zero or 180^° (modulo 360^°). Computing the angle of the product will involve one of the following expressions:

0^°+0^°

0^°

0^°+180^°

180^°

180^°+0^°

180^°

180^°+180^°

360^°

Since the angle 180 degrees is associated with -1, and the angles 0 and 360 degrees are both associated with the number +1, the polar coordinate definition of multiplication of points in the plane agrees with (or yields) the law of signs for the multiplication of positive and negative numbers.

Square Root of -1

The real number -1 = -1+0i = (1,180^°) has angle 180 degrees (mod 360 degrees) and length 1. The purely imaginary number [0,1] = 0+i1 = (1,90^°) has angle 90 degrees and length 1. Multiplying this point or number by itself, that is, squaring it, gives the point with length 1 ×1 = 1 and angle 90^°+90^° = 180^°. So the product equals -1+0i = -1. We call i, the principal square root of -1.

A second square root of -1 is obtained as follows. The imaginary number (0,-1) = 0+i(-1) = [1,-90^°] has angle -90 degrees and length 1. Multiplying this point or number by itself, that is squaring it, gives the point with length 1 times 1 =1 and angle (-90^°)+(-90^°) = -180^° = 180^° (mod 360^°). So this product equals -1+0i = -1 as well.

This provides two square roots of -1 as both (1,+90^°)² = (1,+180^°) = -1 and (1,-90^°)² = (1,-180^°) = -1.

Square Roots of Other Complex Numbers

The square root of a positive number or zero are real nonnegative numbers. I assume in the following that you know how to compute these square roots. The square roots of negative numbers and of other arrows or points in the coordinate plane depend on this ability.

Observe that squaring points in the plane doubles their angular displacements and squares their magnitudes (distance to the origin). That is, the add the angles, multiple the lengths rule gives

[r^½, ½q]·[r^½,½q] = [r ,q]

Therefore the arrow (r^½, ½q) when squared (meaning multiplied by itself) yields (r,q) . So it is called a square root of the arrow (r,q). Another square root is located by the polar coordinates (r^½, ½q+180^°) since (r,q) = (r,q+360^°) both locate the same point in the plane. You should consider the special case of positive numbers z = a+i0 = (a,0^°) where the angle q = 0 degrees.

Complex Conjugates

The complex conjugate of a complex number z = a+b i with polar coordinates (r, q) is the complex number `z = a-b i with polar coordinates (r, -q). Multiplying a complex number a+b i by its conjugate a-bi gives the nonnegative number r²> 0

Conjugates and Multiplicative Inverses (Reciprocals)

Observe that p = [(a)/(r²)]-i[(b)/(r²)] = [1/(r²)][`(z)] has angle -q and length [1/(r)]. Here p = [1/(r²)][r,-q] = [[1/(r)],-q].) Multiplying number p = [[1/(r)],-q] by z = [r,q] gives the complex number [1,0] with length 1 and angle 0, that is, the real number 1. And multiplication of any point (c,d) by 1 = [1,0^°] yields back the point (c,d)

The reciprocal (or multiplicative inverse) of the complex number z = a+b i with length r > 0 and angle q is the complex number p with length 1/r and angle -q.

Observe that if r > 1 then the length of the reciprocal [1/(r)] < 1 < r, that is, the length of the reciprocal is less than 1 and the length of the original number. In contrast, if 0 < r < 1 then [1/(r)] > 1 > r. Question: Which of these two cases is represented in the above diagram? What happens in the case r = 1?

Complex Numbers Introduction