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Tutors - All Subjects
(use at your own risk)
AU:
tutorfinder.com.au
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findatutor.ca
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CDN:
Montreal Tutors
NZ: findatutor.co.nz
UK:
tutorhunt.com
USA: wiziq.com
USA: ziizoo.com
YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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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
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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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.
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Complex Numbers Axioms
Here is a review.
As a matter of convenience or taste, we will write complex numbers as
z = [a,b] or z = a + ib
The notation z = a + ib is standard but for precision in the following, you
should replace the expression a+ ib by the ordered pair [a,b]. Each
complex number can be thought of a vector from the origin to the point [a, b] in
the plane, or each complex number can be thought of as the position vector of a
point [a,b] in a plane. In the foregoing we assume a and b are signed
coordinates.
Assumption B1. There is a set
C = {z= a + ib | a real, b real}
of complex numbers where each element z = a + ib = [a, b] has a real
part a and an imaginary part b.
Following this assumption, we let real part function
Re(z) = a
provides the real part of z (the first coordinate) and we let the imaginary
part function
Im(z) = b
provides the imaginary part of z (the second coordinate). We also
say the complex conjugate of z = a + i b is
a - i b = Re(z) - i Im(z).
The addition and multiplication of complex numbers may be introduced as
follows
[a, b] + [c,d] = [a+c, b +d] or (a+ib) + (c+id)
= (a+c) + i (b+d)
[a,b] x [c,d] = [ac -bd, ac + bd] or (a+ib)(c+id) = (ac-bd) +
i (ad + bc)
The latter operation may lack motivation if it is not introduced with the aid
of the add the angles, multiply the lengths rule given earlier.
It elements z, w are assumed to have the following computation
properties (described algebraically. Below Z, W and V are actors standing
in for complex numbers, or the result of calculations that yield complex
numbers.
- Commutative Law for Addition:
Z + W = W + Z
- Commutative Law for multiplication:
Z W = W Z
- Distributive Law: Z (W +
V) = Z W + ZV (Applying in reverse factors ZW + ZV)
- 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 has length s > 0 and angle theta
then WZ = 1 if W is given the length (1/s) and angle -theta. Here we write W
= 1/Z. In coordinates, we may use the formula W =
a/(a2+b2) - ib/(a2+b2) to define
a complex number W with the property WW = 1.
- Negatives(Additive Inverses) Exist for all complex numbers:
If Z = a + ib = [a, b] then W = (-a) + i(-b) has the property that W+ Z = 0
- Zero Product Law Holds: 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. The contrapositive of this law is of most interest for finding
solutions of equations via factorization).
These computation properties also hold for real numbers, that is numbers of
the form z = a +i0 on the horizontal axis. These properties are often stated as
assumptions or axioms with no mention of the decimal representation of the real
numbers. That represents a gap in committee driven course design.
Polar Coordinates for Complex Numbers:
To compute polar coordinates for complex numbers from their real and
imaginary parts, you need to master two to six inverse trig functions. That
subject is not yet treated here. Or, you may measure the coordinates after
locating a complex number in the plane. If you start with this definition
[a,b] x [c,d] = [ac -bd, ac + bd] or (a+ib)(c+id) = (ac-bd) +
i (ad + bc)
of complex multiplication, you will have master
trigonometry before seeing the derivation of the derivation of the add the
angles, multiply the lengths rule from this definition.
Discussion of the Properties -- Miscellaneous stuff
Here are a few arguments to suggest or imply the properties.
The sums Z + W and W + Z are may be computed using the same
parallelogram rule for the addition of vectors based at the origin.. From this
viewpoint, there is no difference between Z + W and W + Z. They should be
equal. This observation applies even when Z = a + ib and W = c + i d.
Computation of Z + W using coordinates gives the expression (a+c) + i(b+d)
while the computation of W+Z gives (c+a) + i(d+d). These two expressions
should give the same result when computed. So the commutative
law for addition holds for complex numbers.
The commutative law for multiplication comes from the multiply
the lengths, add the angles rule for multiplication of vectors. Here Z =
(r, alpha) and W = (s, beta) gives ZW = (rs, alpha + beta). But for unsigned
numbers, lengths included, the order of multiplication and addition is does
not affect the result. So rs = sr and alpha + beta = beta + alpha.
Therefore WZ = (sr, beta + alpha) =ZW.
The distributive law Z(W+V) = Z(W+V) is implied or
suggested by our previous observations about multiplication by scale
factors and about rotations. We assume Z =(r, alpha) in polar
coordinates. Multiplication by Z can be performed in two steps. The
first step consist of a rotation. The second consists of rescaling, namely
multiplying by the length r of the vector - the position vector for Z in
the complex plane. both rescaling and rotation distribute over
addition. So Z(W+V) = r * (rotate by alpha) * (W+V) = .... =
r (rotate by theta) W + r (rotate by theta) V
= Z (W + V)
The ... represents a few steps that are missing.
If we assume 0 + a = a when a is real, then the coordinate rule for
addition of points or position vectors, namely
(a+ib) + (c+id) = (a+c) + i(b+d)
implies
(a+ib) + (0+i0) = (a+0) + i(b+0) = a + ib
So 0 = 0+ i0 is the additive identity: Z + 0 = Z
Reciprocals or Multiplicative inverses for complex numbers:
Pure mathematics would obtain the above properties from set theoretic
constructions via long, geometric-free and decimal-free chains of reason.
However most students, future mathematicians included, first master mathematics
with the aid of decimal, geometric reasoning and the algebraic way of
writing and reasoning. That may provide the maturity and interest to cover
more material.
Factors via the Unit Circle
If z = (r, alpha) in polar coordinates then z = (r, 0 degrees) * (1,
alpha). A geometric arguments starting with whole numbers and then fractions
imply that z = r * (1, alpha) where multiplication by r represents the rescaling
of a vector and (1, alpha) is point or vector or complex number on the unit
circle. It has length one. If r =0 then z = r * (1, alpha) for all
choices of the angle alpha. One cannot measure an angle for a vector of length
0.
Multiplication equals a rotation followed by a rescaling
Next if w = (R, beta) then
zw = (rR, alpha + beta)
(r,0 degrees){(1, alpha)(R,beta)} = (r, 0 degrees) (R, alpha +beta) = zw
Therefore
zw = r{ (1,alpha)*w)
This implies that multiplication of a complex number by z is equivalent
to a rotation through angle alpha followed by a rescaling using the length r of
the "vector" z .
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www.whyslopes.com
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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