Polar Coordinates
A quick review
Polar coordinates (R, a) can be used to locate points [a, b] in the plane. Polar coordinates (R, a) for points [a,b] in the plane can also be measured. So rectangular and polar coordinates are interchangeable at least through measurement. Trigonometry provides further methods.
Given a point A in a planar map with coordinates, we can measure its distance R from the origin and measure in a counterclockwise manner, the angle a it makes with the horizontal coordinate axis. Geometrically or physically we assume, the distance R and the angle a uniquely determine the point, and the vector from the origin to the point
The ordered pair (r units, a ) with round brackets provides the polar coordinates of a point A in the plane. Here r units is the length of the the associated "position" vector OA which goes from the origin O to the point A. This vector makes an angle alpha with the horizontal axis.
Polar to Rectangular Coordinates, and Back
We will write
[a,b] = (R,q)
when both sides locate, determine or correspond to the same point in the plane. I assume you know how to measure the rectangular coordinates [a,b] and polar coordinates (R, q) of points in the plane, given the location. This provides a geometric mechanism for determining rectangular coordinates from polar coordinates, and vice-versa. (Methods based on trigonometry will be available later.)
The point with polar coordinates (R,q) has length R and angle q. [Angles are determined for each point, modulo 360 degrees.]
| b | * [a,b] =(R, q) | o | o | o | o | o \ angle q | o | --------o----------+----------------- | a | |
Figure 1. Rectangular and Polar Coordinates of a Point.
Each point [a,b] in the plane can be identified with the arrow, head at it, and tail at the origin. For the following topic, recall the discussion of arrow addition using horizontal and vertical coordinates.
Points in the plane may be located using polar coordinates or rectangular coordinates.
round () versus square brackets []
In the following lessons, if I remember, I will use round brackets () with polar coordinates. Your textbooks may use round brackets for both polar and rectangular coordinates. Matching pairs of round (), square brackets [] and braces {} will also be used in computations to indicate the order in which calculations are done. This could lead to bracket and parenthesis over use.
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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