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Chapter 22
Complex Numbers - Basic Ideas

Here is a geometric story which describes the complex numbers, or what mathematicians since Gauss in the 1840's have regarded as the complex numbers. This geometric story provides a confirmation of the law of signs for real numbers. This first part of the story could be explain to someone familiar with arithmetic and the measurement of coordinates, rectangular and polar in the plane. The second part of this story is an exercise in trig. The second part appears in the next chapter.

This applet (online only) illustrates the ideas in this chapter.

This first half 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.

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. See the foreword.

2. In Morris Kline's three-volume work Mathematical Thought from Ancient to Modern Times, see 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.

Points in the Plane

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

(a,b)+(b,d) = (a+c,c+d)

For example (2,5)+(6,2) = (8,7).

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: (1,14)+(2,8) = (1+2,14+8) = (3,22). The chapter Arrow Addition 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

[r1,q1]·[r2,q2] = [r1r2,q1+q2]

(As in the previous chapter, square brackets are used to indicate polar coordinates while round brackets indicate rectangular coordinates.)

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 69.69^° = 22.62^°+46.97^°; and that is it. See the following diagram.

Another Example. The product of the two points [3,80^°] and [4, 60^°] is [(3)(4), 80^°+ 60^°] = [12,140^°]

Stop For a Summary

The addition of points in the plane is given by means of their rectangular coordinates while multiplication is given in terms of polar coordinates. It is an exercise in trig to obtain expressions for the rectangular coordinates of a product in terms of the rectangular coordinates of the factors. See the next chapter. (Note that while your reading of this chapter requires a mastery of the algebraic way of writing and thinking, you could explain with diagrams and examples only, the ideas in this chapter to someone without this mastery. It is the on-paper communication which requires the mastery.)

Introducing the 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

FOOTNOTE: Two quantities x and y are equal modulo a third quantity c, if and only if their difference x-y = kc for some whole number or integer k.

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

• 0^° = 0^°+0^°
• 180^° = 0^°+180^° = 180^°+0^°
• 360^° = 180^°+180^° = 0^° (modulo 360^°)

• (+1) = (+1)(+1) as 0^° = 0^°+0^° 
• (-1) = (+1)(-1) = (-1)(+1) as 180^° = 0^°+180^° = 180^°+0^°
• (-1)(-1) = (+1) as 360^° = 180^°+180^°

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 pairwise products of 3=3+0i, 4=4+0i, -3=-3+i0 and -4=-4+0i using the add the angles, multiply the lengths rule.

Remark. The add the angles, multiply the length rule could be used to define the product of real numbers to people/students who know (a) about the addition of real numbers or coordinates and (b) about the multiplication of non-negative numbers. They would not need to have any previous knowledge of the law of signs.

FOOTNOTE: The add the angles, multiple the lengths rule for the multiplication of complex numbers thus yields a rule for the multiplication of real numbers once the multiplication of positive numbers with themselves or zero is understood/defined.

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

 

1. A = (1.5)·(2). 
2. B = (1.5)·(-2). 
3. C = (-1.5)·(-2).
4. D = (1.5)·(-2).
5. E = [10,45^°] ·[[1/20],15^°].

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

[r½, 1 2 q]·[r½, 1 2 q] = [r ,q]

Therefore the arrow [r^½,[1/2]q] 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,[1/2]q+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.

Exercises.

1. Find all the square roots of 4 and -4 and plot them.
2. Find the cube roots of 27 and -27 and plot them in the plane.
3. Find the square roots of \cis(45^°) = cos(45^°)+isin(45^°) = [1,45^°].

Complex Conjugates

The complex conjugate of a complex number z = a+b i with polar coordinates r = Ö[(a²+b²)] and q is the complex number [`(z)] = a-b i with polar coordinates r and -q.

*   Exercise. Show multiplying a complex number a+b i by its conjugate a-b i gives the nonnegative number r² = a²+b².

Conjugates and 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?

Two Algebraic Properties

Real Multiples of Arrows

Case 1: c ³ 0 Observe for c > 0 that c = c+i0 = [c,0] has angle 0 degrees and length c = |c|. Thus the add the angles, multiply the lengths multiplication rule yields

c·(a,b) = [c,0]·[r,q] = [cr,0+q] = [cr,q] = (ca,cb)

as before.

Case 2: c < 0 Now c = -d < 0. But d > 0 implies

(d,0)·(a,b) = [d,0]·[r,q] = [dr,q] = (da,db)

Therefore

(c,0)·(a,b) = (-1)·[d,0]·(a,b) = (-1)·[dr,q] = [dr,q+180°] = (-da,-db) = (ca,cb)

again.

Conclusion. Multiplication of a point (a,b) by a real number c = c+i0 with and without the add the angles, multiple the lengths rule gives (ca,cb).

Some Vocabulary.

Remark. The use of round brackets () in the notation for rectangular coordinates (a,b) stems from the convention in many algebra texts written before this one. The use of square bracket [] in the notation for polar coordinates [r,q] here was chosen simply because the square brackets were available. In retrospect, cosmetic appearance alone would suggest the employment of round-brackets for polar coordinates and square brackets for rectangular coordinates. The development of notation is not always cosmetically optimal.

Three Problems.

1. Locate in the plane the complex conjugate and reciprocals of the complex three numbers s = 3+4i, t = 12+(-5)i, and z = cos(120^°) +isin(120^°).
2.  Locate the three complex cube roots of 1 (unity) .Hint: divide the unit circle into three arcs each spanning an angle of 360/3 =120 degrees. The required roots are at the ends of each arc (if two arcs share the endpoint 1 = 1+i0.
3. Locate the fourth, fifth and sixth roots of unity. What is the general pattern for n-th roots of unity (where n = 2, 3, 4, ¼).?

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