New (August 3, 2001):  Two  webpages on Complex Numbers (this one) and on Distributive Law for Complex Numbers  offer a short way to reach and explain trigonometry, the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem.  

The explanation below is a must for students of engineering, and a bonus for students of  trig or calculus.

Complex Numbers Introduction

To learn More,Read the following pages:

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.

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

[a,b] + [b,d] = [a+c,b+d]

Associative and commutative Axioms for real numbers imply addition of points in the plane is associative and commutative. 

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 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

(r₁,q₁)·(r₂,q₂) = (r₁r₂,q₁+q₂) 

Axioms for real numbers immediately imply this multiplication is commutative and associative. 

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.

Another Example. The product of the two points (3,80^°) and (4, 60^°) is 

(3 ^. 4, 80^°+ 60^°) = (12,140^°)

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.

Note: 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 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).

Exercise: Use  b = sign(b)|b| to show that  bi = b^. i where i = [0,1]

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^°
  

should use round brackets here for polar coordinates

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

should use round brackets here for polar coordinates

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.

Teachers: The add the angles, multiple the lengths rule for the multiplication of complex numbers gives a rule for the multiplication of real numbers once the multiplication of nonnegative numbers with themselves is mastered. There are now three ways to introduce the law of signs. (i) give it as as part of a rule for multiplication of real numbers after students have learnt to multiply unsigned numbers;  (ii) derive it from the axioms for real numbers;  and (iii) derive it from the add the angles, multiple the lengths rule for multiplication of complex numbers, after signed numbers have been introduced as a coordinates in or along a real line and in rectangular coordinates for the plane. Approach (ii) presumes or forces a mastery of the algebraic way of reading and writing. Thus (i) and/or (iii) could be best for novices.  Both 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.

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.

should use round brackets here
 for polar coordinates

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, ½q]·[r½,½q] = [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.

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.

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?

Some Vocabulary.

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 = (1, 120^°) in polar coordinates.
2.  Locate the three complex cube roots of 1 (unity) .Hint: divide the unit circle into three arcs each spanning an angle of  one third of 360 =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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