Law of Signs
for Real Numbers
We identify the real number line with the horizontal axis of the plane. With this identification, observe that positive numbers have angular displacement zero, modulo 360 degrees. Also observe that negative numbers have angular displacement 180 degrees, modulo 360 degrees. The magnitude of a real number is its distance to the origin.
Suppose z = a+i0 and w = c+i0. We
want to compute the product zw with the multiply the lengths, add the
angles rule. Each factor has length |a| or |c|. Each factor has angle 0 or 180 degrees (modulo 360
degrees). The relationships
- (+1) = (+1)(+1) as 0° = 0°+0°
- (-1) = (+1)(-1) = (-1)(+1) as 180° = 0°+180° = 180°+0°
- (-1)(-1) = (+1) as 360° = 180°+180°
For the second example, the number -2 is identified with the point (-2,0) = [2,180°]. See the figure below.
Now multiplying the point [2,180°] by itself leads to the product [2,180°]2 = [22,180°+180°] = [4,360°] = [4,0°]. Thus the point on the horizontal axis identified with -2 when squared gives the point identified with +4 indicated above. The 360 degrees in the diagram for the number or point 4 = (4,0) represents the doubling of the angle 180 degrees.
For an example or exercise, compute the pairwise products of 3=3+0i, 4=4+0i, -3=-3+i0 and -4=-4+0i using the add the angles, multiply the lengths rule.
Remark. The add the angles, multiply the length rule could be
used to define the product of real numbers to people/students who know (a) about
the addition of real numbers or coordinates and (b) about the multiplication of
non-negative numbers. They would not need any previous knowledge of the law of
signs.
In summary, The add the angles, multiple the lengths rule for the multiplication of complex numbers thus yields a rule for the multiplication of real numbers once the multiplication of positive numbers with themselves or zero is understood/defined.Note each factor gives a point or arrow in the coordinate plane.More Exercises. Compute the following using the multiply the lengths, add the angles rule for multiplication of points in the plane.
- A = (1.5)·(2).
- B = (1.5)·(-2).
- C = (-1.5)·(-2).
- D = (1.5)·(-2).
- E = [10,45°] ·[[1/20],15°].
Stop For A Summary. The polar coordinate definition
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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