HIP, HIP, HIP, Hooray
YOU are better than YOU think. Show yourself  how:
  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Complex Numbers
a visual, geometric approach

Modified December 1 2005:  

Welcome. 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  leads to a short and perhaps shortest possible explanation of core ideas in complex numbers and  trigonometry.  

This  webpage on 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.  This webpage replaces an earlier version in which the distributive law followed from geometric consideration instead of a geometric invariance - the  relativity of the choice of coordinate system and unit or scale for length measurment. . Moreover, in this rewrite,  trig functions are defined for all angles. Then right triangle similarity is used to express trig function of acute angles as ratios of sides of right triangles.  The rewrite may accelerate mathematics instruction in high school and college. It may further a context or application for the high school discussion of reflections and rotations.

New: The complex number applet below shows how to do arithmetic with complex numbers - add, multiply, subtract, divide, take reciprocal and powers. Visit it as you read or open the applet in its original location in a new page.   (Page improvement, March 4, 2007).

Details follow in four parts.

• 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.   The complex number applet below shows how to do arithmetic with complex numbers - add, multiply, subtract, divide, take reciprocal and powers.
• Step III  describe how axioms for complex numbers (field properties) follow from those for real numbers and from the geometric covariance assumptions that vector sums are independent of the choice of coordinate system. 
• 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. Here trig functions sine and cosine are defined for all angles first before the use of right triangle similarity to say how to compute sine and cosine for acute angles and the ratios of sides of right triangles. 

Remark: A forthcoming site   pre-calculus area will include a variant of steps I to IV and ideas from analytic geometry and number theory to provide in a single spot an alternative approach to senior high school mathematics. . 

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.  Knowledge of how to multiplication with negative numbers is optional. 

Assumption: There is a plane in which each point after a choice of unit length, and a pair of perpendicular,  coordinate axes can be represented by and also determined by a pair of rectangular coordinates [a, b] and pair of polar coordinates (r,q).  Given a coordinate system, we assume or make the extrapolation that the polar coordinates of a point (modulo 360 degrees in the angle) are unique determined by the rectangular coordinates of a points and vice-versa. 

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.

correction: 22.62  + 46.97 = 69.59 not 69.69s

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 below. 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^°
  

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.

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.

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.

Step III. Distributive Law 

Statement 

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.

Relativity of Coordinates Systems and Measurement Unit

For same approach in slightly  words, see the discussion of the distributive law for real and complex numbers in the site Number Theory area.

Rectangular and polar coordinates for a plane and displacements or vectors in the plane and are determined by a unit length (or choice of scale) and the direction and orientation of the coordinate axes.  

Assumption:  a rectangular and polar coordinate system for the plane may be defined by  the selection of a origin and  the selection of any two orthogonal, equi-length vectors - the latter define the direction of the coordinates axes. Axis Orientation: Here the vector selected to give the direction of the "y"-axis is assume to be a 90 degree (counterclockwise) of the vector selected to give the direction of the "x"-axis.

In our discussion, there is duality or correspondence between

1. the physical or coordinate free geometric approach in which we visualize arrows or vectors as displacements in the plane The vectors can applied one at a time and one after another in head to tail manner to generate more displacements. A single vector can also represent the location or displacement of a point with respect to another, say the origin. 
2. the representation and addition of the displacements in a rectangular and polar coordinates systems. 

Given a coordinate system, we can draw or define vectors in standard and non-standard position, and represent a vector as a sum of components with respect to unit lengths and vectors  in the coordinate system. The coordinate representation because of the arithmetic properties of the addition and multiplication of coordinates implies many properties of the vectors.  The geometry or physical applications of coordinate systems and vectors assumes the results of calculations with vectors, their sums in particular, are independent of the choice of coordinate system, the direction of coordinate axes, and the choice of measurement system or unit for length. 

Here the unit length can be a meter, a centimeter or any other length selected. A change in measurement systems (what is the unit length) should not affect the result albeit it may affect the coordinates.  That is, a sequence of displacements (vectors) in the plane or in a line yield an other displacement independent of the choice of unit length and associate unit vector and independent of the choice of direction and orientation of the coordinate axes. 

In for a penny, in for a pound. High school trigonometry and geometry without coordinates depends on diagrams to define trig functions and to arrive at their properties.  The use of diagrams and geometric or physical assumptions implies high school mathematics is mixed and not pure.  This essay on complex numbers implies mixed mathematical development of trig and allied concepts has presented in high school has many possible ports of entry. In college calculus,  more mixed mathematics appears in calculus in   geometric area comparison proofs that the value 1  is the limit of (1/h)sin(h) as h approaches zero.  It seems that pure mathematics, the diagram and context free development of mathematics, comes after calculus.  

Proof of Distributive Law For Real Numbers
from a change of unit length or scale. 
(Proof Corrected May 3, 2006)

Let u₁ and  u₂ represent displacements (vectors) in straight line. Let k be a nonzero vector in the real line. Let k define our unit length and positive direction along the line. Then   u₁ =  a₁ k and  u₂ =  a₂ k for some real coefficients a₁  and a₂. Let  u₃ = u₁ + u₂. Here head of  u₁ to tail  u₂ addition of  u₁ and  u₂ defines  u₁ + u₂  = u₃ = a₃ k for some real coefficient  a₃.   But at the same time, we can compute  a₃ = a₁  + a₂   from the addition of coordinates with respect to the "unit vector" k.   Here we assume any nonzero vector can be used to define coordinates for vector addition. 

Now let K = (1/s) k for some positive (or nonzero)  number  s.  For p = 1, 2 and 3, by substitution and the associative law, we have  u_p =  a_pk = a_p (sK)  =  (sa_p)K. = A_pK where A_p  = sa_p 

Let  u₃ = u₁ + u₂ be calculated via head to tail addition. The same geometric assumption about vector addition with coordinates that give a₃ = a₁  + a₂  also give  A₃ = A₁  + A₂   for the coordinated based calculation of vector addition with respect to the new unit vector K.  

Via the substitutions A_p  = sa_p  for p = 1, 2 and 3, we rewrite   A₃ = A₁  + A₂ as sa₃ = sa₁  + sa₂ . Now make the further replacement or substitution   a₃ = a₁  + a₂  to obtain the distributive law 

s(a₁  + a₂) = sa₁  + sa₂  

The latter holds when s is a nonzero number.  

Remark:  The condition that the scale change s be positive can be replaced by the condition that the s be nonzero. Think about that later. 
Remark: The distributive law for real numbers is equivalent to the assumption that a change of scale will not affect the sum of displacements along a coordinate line in which the coordinates are real number determined the choice of origin and choice of scale or unit length (direction included).

Proof of Distributive Law For Complex Numbers
from a change of scale (Proof Corrected May 3, 2006)

 Let k serve be a nonzero vector for the plane. Pick a counterclockwise direction for the plane. Then rotation of k through 90 degrees counterclockwise defines a vector rot k of the same length of k. Since k and rot(k) are perpendicular, The use of rectangular coordinates imply every vector u in the plane can be written in as x k + y rot(k) for some real numbers x and y .  If we write i k = rot(k) then  u  = x k +  y ik defines the vector (x+yi)k.  Hence every vector u in the plane can be written as a complex number x+yi times k. 

Let u₁ and  u₂ represent displacements (vectors) in the plane. Then   u₁ =  a₁ k and  u₂ =  a₂ k for some complex number coefficients a₁  and a₂ . The  head of  u₁ to tail  u₂ addition of  u₁ and  u₂ defines  u₁ + u₂  = u₃ = a₃ k for some real coefficient  a₃.  

But at the same time, we can compute  a₃ = a₁  + a₂   from the rectangular addition of coordinates in a coordinate system with axes given by  k and (1,90^o) k = ik

Let s = (r,q.) be a non-zero complex number.  Now   K = (1/s) k = (1/r,-q.)k  is nonzero vector. Let  u₃ = u₁ + u₂  as before. 

For p = 1, 2 and 3, by substitution and the associative law, we have  u_p =  a_pk = a_p (sK)  =  (sa_p)K. = A_pK where A_p  = sa_p  But the coordinate based calculation of A₃ of u₃ with respect to the new  vector K and iK gives A₃ = A₁  + A₂ 

Via the substitutions A_p  = sa_p  for p = 1, 2 and 3, we rewrite   A₃ = A₁  + A₂ as sa₃ = sa₁  + sa₂ . Now make the further replacement or substitution   a₃ = a₁  + a₂  to obtain the distributive law 

s(a₁  + a₂) = sa₁  + sa₂  

for any nonzero-complex number s. The extension to  case where where s = 0 holds by inspection.  We may conclude the following

the left distributive law s(a₁  + a₂) = sa₁  + sa₂   whenever s, a₁  and a₂ are complex numbers. 

From commutatively of complex multiplication, we also have 

the right distributive law  (a₁  + a₂)s = a₁s +   a₂ s whenever s, a₁  and a₂ are complex numbers.  

Remark: The distributive law for complex  numbers is equivalent to the assumption that a change of scale will not affect the sum of displacements along a coordinate line in which the coordinates are real number determined the choice of origin and choice of scale or unit length (direction included).
Remark:  Since multiplication of complex numbers is equivalent to a positive scale change and rotation (adding an angle),  the above argument could be divided into two steps: one based on a scale and the other on rotation. 

The Rectangular Way to Compute Products. Product computation using real and imaginary parts.

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. An Accelerated Approach to Trigonometry. Let 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  
   
   With polar coordinates and  the real and imaginary parts of the cis(q) function, a function defined for all angles, we have defined trig functions for all angles. Now  similarity assumptions for triangles to say how to compute trig functions for acute angles from ratios of sides of right triangles. The latter reverses the standard path in which the ratios of right triangles side are employed to introduce trigonometry. See the  Right Triangle Similarity (B5)  
   
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.

Appendix: Another Approach to Complex Numbers

The logic chapters Islands and Divisions of Knowledge indicates different and equivalent entry points for the logical development of a subject. The impure logical development of mathematics in high schools and colleges before any study of pure mathematics could be done in a more accessible and more rapid manner if the following was done.

1. Arrows or vectors are used to denote displacements along a line or in plane, the existence of which is assumed.
2. The sum of displacements from a point is represented by the Head to tail placement of arrows.
3. Arrows can be rotated clockwise or counterclockwise
4. Given a unit length, the length of vectors can be measured, and signed real numbers can be used as coordinates along each straight line. 
5. Given a unit length and vector of unit length,  there is a coordinate system with the vector and its rotation via 90 degrees providing direction of the x and y coordinate axes. 
6. For each coordinate system in item 6, the vector addition corresponds to  coordinate addition. 
7. Arrows can be multiplied by nonnegative  real numbers. The latter correspond in a coordinate to multiply the arrow components by the same non-negative real number. 

In the foregoing we make the addition of vectors to be the initial operation and the addition of coordinates to be the implied operation.  The Associativity of the head to tail addition of arrows implies coordinate addition is associative.  The distributive law for  multiplication  a non-negative number over addition follows from from the independent of vector addition with respect to the choice of coordinate system.  The distributive law for rotation over addition also follows from from the independent of vector addition with respect to the choice of coordinate system.  If we select an origin for the plane, a unit vector in the plane (say in standard position),  and then define rectangular and polar coordinates using  both, then we can also define complex numbers in the plane using these linked rectangular and polar coordinate. The distributive law for multiplication of complex numbers over their sum (a vector addition) is an immediate consequent of the foregoing considerations. If we assume the head to tail addition of vectors in the plane is commutative then the implied  addition of coordinates and complex numbers is commutative.  If we assume the multiplication of non-negative numbers is commutative and associative (see the site number theory area)  then multiplication of complex numbers is also commutative and associative. 

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