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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.
Real Numbers
as Complex Numbers
and square roots of -1
Each complex number z = a+i0 with imaginary part zero gives and is given by a real number a. We will write z = a in this situation, and say that the complex number z is also a real number.
With this practice or convention, the real numbers can be regarded as a subset of the complex numbers; and the real number line can be identified with the horizontal axis of the plane.
Two Square Roots of -1
The real number -1 = -1+0i = [1,180°] has angle 180 degrees (mod 360 degrees) and length 1. The purely imaginary number (0,1) = 0+i1 = [1,90°] has angle 90 degrees and length 1. Multiplying this point or number by itself, that is, squaring it, gives the point with length 1 ×1 = 1 and angle 90°+90° = 180°. So the product equals -1+0i = -1. We call i = the principal square root of -1.
A second square root of -1 is obtained as follows. The imaginary number (0,-1) = 0+i(-1) = [1,-90°] has angle -90 degrees and length 1. Multiplying this point or number by itself, that is squaring it, gives the point with length 1 times 1 =1 and angle (-90°)+(-90°) = -180° = 180° (mod 360°). So this product equals -1+0i = -1 as well.
This provides two square roots of -1 as both [1,+90°]2 = [1,+180°] = -1 and [1,-90°]2 = [1,-180°] = -1.
Square Roots of Other Complex Numbers
The square root of a positive number or zero are real nonnegative numbers. I assume in the following that you know how to compute these square roots. The square roots of negative numbers and of other arrows or points in the coordinate plane depend on this ability.
Observe that squaring points in the plane doubles their angular displacements and squares their magnitudes (distance to the origin). That is, the add the angles, multiple the lengths rule gives
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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