YOU are better than YOU think. Show yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
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. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
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.
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.
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]
ADVERTISEMENT
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