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Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may serve as back ground info for partial fraction decomposition. |
Coordinates in 1, 2 & 3 Dimensions
Summary: This lesson and the next 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.
For a Line Segment or Line
Unsigned and Signed Coordinates For Line Segments
finite or infinite
For a line segment, we can measure distance from one end using unsigned numbers.
Figure 1: Line segment with origin at left end.
The choice of which end to start the numbering or measuring gives an orientation.
For a line segment, we can also measure distance and direction from a point in the middle using signed coordinates.
Figure 2: Line segment with origin in middle.
Here the positive sign (+) indicates distance and direction to the right while the negative sign (-) indicates distance and direction to the left.
One Dimensional Assumption
Geometric Assumption: An infinite line can be described by signed coordinates with an origin (the zero point) anywhere on the line, and the and +1 point anywhere on the line except the at the origin. Note for later: the directed line segment from 0 to +1 defines the line orientation and the unit length for the coordinate system.)
Thus every coordinate (real number) determines a point on the line and vice-versa. For sake of argument, we assume infinite (straight) lines exist.
For Plane Geometry
Unsigned Coordinates for Rectangular Maps
origin at corner, Unsigned Coordinates
Figure 3: Rectangular Coordinates for Maps.
Ordered pairs of numbers without signs such [1,4] or [3,2] may be used to locate points on a map when the origin or reference point is at the bottom left corner.
On such maps there is no need for signs. More generally, you use coordinates such as [1.5, 3.27] or [a, b] to locate points on the map -- provide their rectangular coordinates. Here a and b stand for any pair of unsigned numbers including zero that may be used as coordinates.
In the above map, the left edge of the map region give the vertical coordinate axis while the bottom edge gives a horizontal vertical axis for coordinate use.
The word rectangular is used above as "polar coordinates" will be introduced later. Rectangular coordinates are also called Cartesian Coordinates.
Signed Coordinates in the Plane
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.
Figure 4. Plane with Signed Coordinate System
Here axes are perpendicular (orthogonal, at right angles) to each other..
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.
Remark: Descartes employed pairs of unsigned numbers to locate points in rectangular region with the origin of the coordinate system at a corner of the rectangular region (possibly a map or a plan). The use of signed coordinates came later. Non-negative coordinates are sufficient for finite regions. The current mathematical habit is to write the pair of coordinates as an ordered pair (x,y) or (a,b) where the variables a, b, x and y will be given by real numbers.
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.
Two Dimensional Assumption
Geometric Assumption: An extended or infinite flat plane can be covered by a rectangle coordinates using infinite lines parallel to a horizontal and vertical axis and real numbers as coordinates to locate points in the plane, regardless of the choice of unit length and orientation & placement of the horizontal and vertical axes.
The foregoing assumption extrapolates our comprehension or familiarity with finite rectangular regions. It gives a powerful numerical or algebraic model of \work with points, lines, circles and further geometric objects in the plane.
By describing geometry with numbers, alone, in ordered pairs or in ordered triplets, the arithmetic properties of numbers can be used to arrive at conclusions about geometry in one, two and three dimension via chains of reasons as coordinates provide a precision missing in useful but suggestive and approximately drawn diagrams on a small or large scale.
Geometric Model *(Assumption): An extended or infinite flat plane can covered by a rectangle coordinates using infinite lines parallel to a horizontal and vertical axis and real numbers as coordinates to locate points in the plane.
The foregoing assumption extrapolates our comprehension or familiarity with finite regions regions. It gives a powerful numerical or algebraic model of \work with points, lines, circles and further geometric objects in the plane.
By describing geometry with numbers, alone, in ordered pairs or in ordered triplets, the arithmetic properties of numbers can be used to arrive at conclusions about geometry in one, two and three dimension via chains of reasons as coordinates provide a precision missing in useful but suggestive and approximately drawn diagrams on a small or large scale.
C. Spatial Geometry
Figure 5. Space with Signed Coordinate System.
Image of a rectangle coordinate system with origin in middle and signed coordinates in three mutually orthogonal directions
. In this image, the point at (a, b, c) would have positive coordinates.Cases
where one or more of the coordinates a, b and c is non-positive is left to the
reader to imagine.
Three Dimensional Assumption
Geometric Model *(Assumption): Space can covered by a rectangle coordinates using infinite lines parallel to two horizontal and one vertical axis and real numbers as coordinates to locate points in the plane.
Again, the foregoing assumption extrapolates our comprehension or familiarity with finite regions in the plane and in space. . It gives a powerful numerical or algebraic model of \work with points, lines, circles and further geometric objects in the plane and in space.
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