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1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3.
_Why_Slopes_&_More_Math_1995

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9.  Complex Numbers More 2001.  10  Exponents & Radicals Exactly 2008
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Similarity of Right Triangles

Two right triangles are similar if they have an angle in common BESIDES a right angle. More generally, two triangles are similar if and only the angles in one equal the angles of the other in some order. 

For similar triangles, the ratio of their matching or corresponding sides are equal.  The diagram below illustrates this in a special case.

 

RgtTriSimilarity.gif (10314 bytes)

 

The observation or assumption that

the ratios of side lengths, here adjacent/hypotenuse or opposite/hypotenuse are independent of the scale factor and depend only the angle the hypotenuse makes with the adjacent side

justifies the definition and calculation of trig functions using the ratio of sides for similar right triangles.

Slopes for Straight Lines

The assumption that the ratio m = rise/run is a constant for a straight line provides the basis for what was the discussion of slopes in algebra or pre-calculus courses.   For each straight line, its slope m is the proportionality constant linking the rises to runs for any two points on the line.


The rest of this webpage is a digression


Linear Algebra and Slopes of straight lines.

If solving simultaneous equations is easy for you, see the chapter on this in the section Three Skills for Algebra, then a large part of high school equatiion solving becomes very simple -- too simple, as then solutions to problems become the task, hard or not, of extracting linear to solve from the information in a problem.    

For any two points [p,q] and [u,v] on a straight line with slope m, the rise = v - q and the run = u - p.  The assumption m = rise/run  gives the equation

                            m = (v-q)/(u-p)

Here is formula for calculating m given two points on the line. Text books prefer to use x's and y's with subscript to denote the two points.  I leave the change of notation to you.

Next  multiplication of both sides by (u-p) gives   m(u-p) = v -q   or  v = m(u-p) + q.

Suppose the slope m of the line and a  point  (p,q) on the line are given. Further  suppose [u,v] =[x,y] is unknown. Then we assume the point with coordinates [x,y] is on the  line if and only if 

                     y = m(x -p) + q

This represents the point slope form of the equation of the line.   This if and only if assumption  is needed to link coordinates with geometry. Write when and only when instead, if you like :)

 


If  (p,q) happens to be on the x-axis (the horizontal axis) then q = 0 and the foregoing equation becomes

y = m(x -p)

The previous equation might be called the x-intercept equation for a line.  For coordinates [x,y] satisfying an equation of this form, taking x to p forces y = 0.


If (p,q) happens to be on the y-axis (the vertical axis) then  p = 0, the equation becomes

y = mx + q

The latter is called the y-intercept form of the equation of a line.  For coordinates [x,y] satisfying an equation of this form, taking x to be 0 forces y = q.


Algebraic Games with Equations for Straight Lines

When a point [x,y] is on two lines, it must satisfy the equations of both.  This observation leads to two "linear" equations in two unknowns

                         ax + by = e

                         cx + dy = f

These simultaneous equations must have a unique solution if the lines are different and not parallel.  The equations of lines y = something can be rewritten in the form ax+by = e with b = 1.

The solution of these equations can be complicated by  requiring you the learner to obtain them from data such as two points on a line, or slope and point, or slope-intercept information.  So before you solve two equations to find the intersection, if any, of two  lines, you have to find a chain of reason which gives the coefficients of the equations. Exercises with this are met or avoided  in the high school or college "theoretical" development of chemistry and physics, and other subjects too.

 

 

Complex Numbers
with easy consequences of two ways to multiply complex numbers in and between vectors & trig, etc

Back ] Area Intro ] Next ]

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
 

Hint: See the (newest) Complex Number. Starter Lesson for a simple geometric introduction, then continue with easy consequence below. The clearest geometric proof of the distributive law appears in the Euclidean-Geometry/Complex No.s
folder.

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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