Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

Online Volumes (Book Orders)
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

Mathematics Course Designers: LAMP offers food for thought.
More Site Areas 
1. Help Your Child or Teen Learn 
2. Solving Linear Equations
3. Fractions Ratios Rates Proportions & Units
4. Euclidean Geometry
5. Analytic Geometry/Functions 
6. Number Theory
7. More Calculus
More Site Areas 
8. Complex Numbers 
9. Qc Maths  Education  
10. Secondary IV(?) maths
11. Real  Analysis 
12. LaTeX2HotEqn:
13. Electric Circuits Etc  
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15. Algebra, Odds & Ends, Etc
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16. Math Education Essays
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20. Statistics Useful, or Not.
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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.


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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.

Coordinate Method

In a plane, the intersection of two perpendicular lines, one horizontal and the other vertical, defines a reference point or origin for the plane. The head of each arrow in the plane has coordinates. Ordered pairs of vertical and horizontal coordinates, ordinates and abscissa, can be employed to add arrows together or find the position of the head of their sum, when the tails of the vectors in the sum are both located at the origin. This gives a fourth  method for arrow addition given by the addition of coordinates. This method is very similar to the third method for addition with components. Here is another technical observation with little motivation except for consequences that will follow.

The next figure shows each vector as a sum of components.  We assume the vector heads are located at [a,b] and [c,d].  Each vector from the origin may be defined (or drawn) by giving the location of its head.

wpe1B.gif (3319 bytes)

The next step is to arrive at a formula for vector addition in  terms of the coordinates [a,b] and [c,d] of the head locations of the vectors or summands in this vector addition.

wpe1D.gif (7585 bytes)

The foregoing suggests we represent points in the plane and vectors ending at those points by the coordinates of the head of the vector, that is the coordinates of the points.  The foregoing also suggests the vector sum of the arrows with heads at [a,b] and [c,d] respectively has its head at [a+c, b+d].  So to represent the vector addition of arrows, we put

        [a+c, b+d]  =   [a,b] + [c,d]

This gives the coordinate way to add points and their position vectors in the plane.   The position vector of a point goes from the origin (its tail) to the point (its head).

See applet   for examples. Have it display rectangular coordinates for points and vectors.

Recap and Examples

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

The addition rule is simple add the rectangular coordinates to get the rectangular coordinates of the sum. This coordinate addition follows from our previous discussion of methods for arrow addition.


Example [2,5]+ [6,2]=[8,7].                   
           | 
         7 !=======+------------//-------* [8,7] 
           |                             % 
           |       [2,5]                 % 
         5 |=======*                     | 
           |       |                     | 
           |       |                     | 
           |       |            [6,2]    | 
         2 |-------|-//----------*       | 
           |       |             %       | 
           |       |             %       | 
------------------------------------------------- 
           |       2             6      8 
           | 
           |                  

    Figure 2. Addition of Points [2,5] + [6,2] = [6+2, 5+2]

 

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Complex Numbers

HInt: See the (newest)
Complex Number. Starter Lesson Then continue with easy consequence below.

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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a 1983 McGill. Ph. D. in mathematics
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