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.
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 trigonometry. Details follow in three 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.
- 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.
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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.
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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
For example [2,5]+ [6,2] = [8,7].
[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
Square brackets are used to indicate polar coordinates while round brackets indicate rectangular coordinates.(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:
- A = (1.5)·(2).
- B = (1.5)·(-2).
- C = (-1.5)·(-2).
- D = (1.5)·(-2).
- E = (10,45°) ·(1/20,15°).
Stop For A Summary. The polar coordinate definition
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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
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Exercises.
- Find all the square roots of 4 and -4 and plot them.
- 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.
For each point or complex number z = a+b i = (a,b) = [r,q] in this plane, we say that a is the real part of z; that b is the imaginary part of z; that r = |z| = Ö[(a2+b2)] is the magnitude, modulus or absolute value of z (different texts prefer different terms); and that q is the angle or argument of z.Three Problems.
- 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.
- L
ocate 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
Below Z, W and V stand for points in the plane or complex numbers. The following properties consequences of the rectangular and polar coordinate representation of points in the plane, alias complex numbers
- 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.
A geometric proof of the distributive law(s)
follows in
step III.
The geometric proof of the distributive law(s) in step III is quicker and shorter than other ways previously online here, if not elsewhere. |
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
- 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
- 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)
- 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.
- 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 |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically