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 |
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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.
Parallelogram Method
When two arrows or vectors (representing motions if you wish) have a tail at the same place, they may be added together by moving the tail of one to the head of the other with the aid of a parallelogram, and then using the head to tail method for addition. This gives the parallelogram method for adding a pair of arrows or vector addition. The resultant arrow does not depend on which arrow, the first or second, is moved. Here is a repeatable and reproducible methods, arbitrarily defined, for vector or arrow addition. More generally, parallelograms can be used to displace or move arrows from one location to another without changing their lengths or directions. But that is another story, or chain of reason, not illustrated here.
The following diagram shows in four steps 1, 2, 3 and 4 how to add two vectors which start at the same place, or have the same origin.
Step 5 just gives the shorthand notation for a vector from a point A to B in a figure. Outside of a figure, I will use the notation AB to denote the same vector because it not easy to put an arrow over the two letters AB. The parallelogram law says how to compute AB + AC.
Saying how to add vectors or "straight-line movements" which have tails or starting points defines a second kind of vector addition. The first first kind, head to tail addition of vectors representing movements, was described in the previous lessons. In mathematics, you should remember saying how to do a calculation or operation defines it. Remember this principle. Vector addition represents as given here represents a geometric calculation.
From the parallelogram rule or law for addition, the calculation of the sum of vectors AB + AC and the calculation of AC + AB is identical. So the order of addition is not important. Expressions of the form AB + AC can be replaced by the expression AC + AB when we describe calculations with vectors (arrows). That is, vector addition commutes. Remember this when we talk about real and complex numbers below.
The applet illustrates the parallelogram law
for the addition of vectors with tails at the origin.
www.whyslopes.com
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