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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 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.
Component 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. Each arrow in the plane is equal to the sum or resultant of a horizontal and vertical arrows, its so-called horizontal and vertical components. This representation or decomposition of an arrow as the sum of horizontal and vertical components leads to a third method for arrow addition given by the addition of components. The horizontal components of an arrow sum is given the arrow sum of the horizontal components. Likewise, the vertical components of an arrow sum is given the arrow sum of the vertical components. Here is a technical observation with little motivation except for consequences that will follow.
Each vector from the origin in a rectangular coordinate is the sum of vectors parallel to the coordinate axis's. The following diagram shows how these "component vectors" parallel to the axis's may be used instead of the parallelogram rule.
See applet for examples. Have it display rectangular coordinates for points and vectors. To learn more, visit the chapter vectors in Calculus and Beyond.
Transition to Coordinates
If [a,b] and [c,d] are the heads of two vectors, we the head of the sum of the vectors will be at location [e, f] = [a,b] +[c,d] where the + operation indicates arrow addition. In the case where both heads are in the first quadrant, we have e = a +c and f = b +d. In the case where one or both are not in the first quadrant, put a + c = e and b + d = f. This defines the addition of coordinates. Exercise: Show this addition is well-defined.
Observe, addition of vectors using the first method (parallelogram method) commutes. That the order of addition does not affect the result. Therefore addition commutes in all further methods that produce the same result. This implies the addition of coordinates just defined commutes.
www.whyslopes.com
Complex NumbersHint: 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