Signed Coordinates
This lesson and the previous one offers motivation for the introduction of signs. In elementary school, people learn about whole numbers n and fractions p/q before the use of signs. Ordered pairs of unsigned numbers may be introduced as coordinates in the first quadrant. Introducing signs + and - gives ordered pairs of numbers with signs as prefixes to provide coordinates for four quadrants.
If our first map extends to the left and/or below the origin, the horizontal and vertical coordinate axis's may be extended. These extensions divide the map into four regions call quadrants. To get coordinates for all four regions or quadrants we may place signs in front of numbers. See the diagram below.
In the above map, identify the points with coordinates [+2,+1], with coordinates [+2,-4], with coordinates [-2.5, -3] and lastly with coordinates [-4, +3]. By convention, + signs in front of numbers are optional. So +2 = 2 and +1 = 1.
This applet illustrates the use
of rectangular coordinates. Play with it. Move the points A and B on it and see
how their coordinates change. The coordinates are displayed in the top left
corner.
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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