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.

Note: December 1, 2005. There is a new version of this webpage with a new and simple proof of the distributive law based on the Galilean relativistic  assumption that sums of vectors or displacements in the plane are independent of the coordinate system in which can be represented and/or calculated. The other geometric proofs at this site employs alternate assumptions  about the mix of coordinate and pre-coordinate Euclidean geometry. 

Complex Numbers & Trig

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.

• Steps I & II of this story or chain of reason can be followed with a knowledge of with arithmetic and the measurement of coordinates, rectangular and polar in the plane.  Step I describes the addition and multiplication of points in the plane. Step II introduces the complex numbers, and provides a confirmation (or derivation) of the law of signs.  
• Step III  (Distributive Law for Complex Numbers) describe how axioms for complex numbers (field properties) follow from those for real numbers and from some geometric assumptions about triangles and parallel lines.  Step III, sufficient by itself for people with previous command of complex numbers, provides the essential part of this story or chain of reason. The proof  stems from the usual  assumptions  usually made in the high school or college development of trigonometry from assumptions about triangles and real numbers.
• Step IV. Easy Consequences - derives the Pythagorean theorem, trig formulas for dot- and cross-products, the cosine law and a converse to the Pythagorean Theorem.

Apart from steps I to IV, a local  applet   illustrates addition and multiplication for complex numbers or points in the plane. 

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 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).?

 Field Properties

• Commutative Law for Addition:   Z + W = W + Z  
• Commutative Law for multiplication:  Z W  = W Z  
• Additive Identity Exists: The zero vector 0 = 0 + i 0  has the property 0 + Z = Z
• Multiplicative Identity Exist: The real number  1 = 1 + i 0 has length 1 and angle 0. So it has the property that 1 Z = Z.
• Reciprocals (Multiplicative Inverses) Exist for nonzero complex numbers: If Z = (r, -q) has length r> 0 and angle q then WZ = 1 if W = (1/r, -q) with length (1/r) and angle -q
• Negatives (Additive Inverses) Exist for all complex numbers:  If Z = a + ib = [a, b] then W = (-a) + i(-b) = [-a, -b]  has the property that W+ Z = 0
• Non Zero Product Law: If Z and W have lengths r and s both greater than 0 then their product has length rs > 0 (By the methods of decimal arithmetic with, the product of two positive numbers or length is positive. Alternative this follows from assuming the same law for real numbers.)

From logic, the equivalent, contrapositive form of the nonzero product law is as follows:

Zero Product Law: If the product WZ = 0 for a pair of factors W and Z given by real or complex numbers, then at least one of the factors must be zero.

This observation, an implication rule, is employed to find complex or real solutions of equations. The solution of quadratic equations, one at a time or all at once by the quadratic formula, follows from this zero product law.

Distributive Property 
(see step III. for proof)

Below  Z, W and V again stand  for complex numbers. The left and right Distributive Laws says

    Z ( W + V ) = Z W + ZV    (left distributive law)   
  ( W + V ) Z = WZ + V Z    (right distributive law)

Because multiplication commutes (that is, AB = BA), the left and right forms of the distributive law are equivalent. Each implies the other.  So a proof of one provides a proof of the other. Because they are equivalent, we the adjectives left and right may some be omitted, and we may talk about a distributive law instead of distributive laws, a harmless  variation in language.

The Rectangular Way to Compute Products

Suppose z = a + bi and w = c + di then with the aid of the associative and commutative laws for the addition and multiplication of points in the plane, and the with the aid of the distributive law (twice)

zw = (a + bi) (c+ di) = a(c+di) + bi (c+ di)   
                 (by first use of distributive law)

     =  ac+ a(di) + (bi)c+ (bi)(di)   
         (by second use of distrributive law)

     = ac + i ad  + i bc  + (-1) bd   
        ( by associative and commutative law for products)

     = ac + (-1) bd   +  i ad  + i bc  
         (by associative and commutative laws for sums)

     = 1 (ac + (-1) bd)   +  i (ad  +  bc)  
         (by the distributive law in reverse)

  =    [ac +  (-1) bd ,  ad + bc]   

The foregoing gives a second way to multiply complex numbers together using their real and imaginary parts

(a + bi) (c+ di) = (ac - bd)   +  i (ad  +  bc) 

 or equivalently, with or rectangular coordinates notation,

[a,b] [c,d] = [ac -bd, ad+ bc]

The latter formulas often the starting point for the definition of products of complex numbers before the introduction of complex number notation in the plane.

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

Step IV. Consequences - Easy Pickings

1. Another proof of the Pythagorean Theorem (B3 in the complex number site area) is a consequence of two different ways to multiply a complex number by its conjugate. The proof of the distributive law in step III does not depend on the Pythagorean theorem. Then  the Cartesian or Rectangular Distance formula (B4) is an immediate consequence of the Pythagorean Theorem
   
2.  Here cis(q) = (1,q) have real part cos(q) and sin(q) as in the unit circle definition of Trig Functions. Then 
   
   cis(A+B) = cis(A) cis(B) = 
   [ cos(A)cos(B)-sin(A)sin(B), cos(A)sin(B)+sin(A)cos(B) ]
   
   due to the rectangular way to compute products. Thus two trig identities
   
   cos(A+B) = cos(A)cos(B)-sin(A)sin(B)
   sin(A+B) =  cos(A)sin(B)+sin(A)cos(B)
   
   follow  Read about Rt Triangle Similarity (B5) before looking at the definition of unit circle definition of Trig Functions.(B6) See too  Exponential & cis functions (B9)
   
3. Multiplying one complex number by the complex conjugate of another implies trig formulas and interpretations for Dot & Cross Products (B7)of points & vectors in the plane. The trig formula for the dot product implies the  Cosine Law (B8) and a converse to the Pythagorean Theorem.
   
4. Easy Trig Identities (B10) follow from complex number based calculations with Exponential & cis functions (B9)

Trig course today could cover the above material, show how most trig identities follow from calculations with complex numbers, and give applications of trigonometry to distance calculations based on the similarity of right triangles and the values of trigonometric functions.  A course on trig and complex numbers could explore more analytic geometry,  show how to compute powers and roots for positive real numbers using the natural logarithm (defined for positive numbers) and exponential functions (defined for real numbers), and then extend these definitions to give definitions of powers and roots for complex numbers, including negative real numbers.  Calculations of roots of unity would further tie trigonometry and complex numbers together.

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Complex Numbers

Hint: See the (newest) Complex Number. Starter Lesson.  for a simple geometric introduction, then continue with easy consequence below.

The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.  

Easy Consequences
Vec & Cmplx  No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Links: Interactive Maths  

First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below -  read for review or revision .

First (Old) Complex No Intro
Distributive Law
A1 Add Poiints
A2 Polar Coords
A3 Polar Multiply
A4 Complex No.s
A5 Real Numbers
A6 Law of Signs
A7 Key Properties
B1 2nd Mult Method
C1 Unsigned Coords
C2 Signed Coords
C3 Set Codification
C4 More On Real No.s
D1 Arrow Navigation
D2 Sum of Motions
D3 Addition Method I
D4 Addition Method II
D5 Addition Method III
D6 Coordinate Addition
D7 1st Distributive Law
D8 2nd Distributive Law
D9 3rd Distributive Law

D1 to  D6 after provide a review of vectors.

More on Complex Numbers:

Chapters in Volume 3::
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc

This further  Complex Number
  Intro assumes the field properties
of real numbers in place of
deriving them geometrically

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