Distance Between Points in the Plane
Suppose [x1,y1] and [x2,y2]. are points in the plane. Our aim is to compute the distance c between these points.
The line segment between [x1,y1] and [x2,y2]., if it is not horizontal, nor vertical, provides the hypotenuse of right triangle (or two) with horizontal and vertical sides. The case where the line segment has a negative slope is drawn below.
The lengths of the sides are a = |x2- x1| and b = |y2- y1|
c2 = a2 + b2 = (x2- x1)2 + (y2- y1)2
So the distance c between the points [x1,y1] and [x2,y2]. is
c2 = (x2- x1)2 + (y2- y1)2
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Verification of this formula in the two cases where the line segment between [x1,y1] and [x2,y2]. is horizontal or vertical is left for you to explore.
Exercise: Redo the above proof for the case where the slope of the line segment between [x1,y1] and [x2,y2]. is positive.
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
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
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically