Real Numbers
as Complex Numbers
and square roots of -1
Each complex number z = a+i0 with imaginary part zero gives and is given by a real number a. We will write z = a in this situation, and say that the complex number z is also a real number.
With this practice or convention, the real numbers can be regarded as a subset of the complex numbers; and the real number line can be identified with the horizontal axis of the plane.
Two Square Roots of -1
The real number -1 = -1+0i = [1,180°] has angle 180 degrees (mod 360 degrees) and length 1. The purely imaginary number (0,1) = 0+i1 = [1,90°] has angle 90 degrees and length 1. Multiplying this point or number by itself, that is, squaring it, gives the point with length 1 ×1 = 1 and angle 90°+90° = 180°. So the product equals -1+0i = -1. We call i = the principal square root of -1.
A second square root of -1 is obtained as follows. The imaginary number (0,-1) = 0+i(-1) = [1,-90°] has angle -90 degrees and length 1. Multiplying this point or number by itself, that is squaring it, gives the point with length 1 times 1 =1 and angle (-90°)+(-90°) = -180° = 180° (mod 360°). So this product equals -1+0i = -1 as well.
This provides two square roots of -1 as both [1,+90°]2 = [1,+180°] = -1 and [1,-90°]2 = [1,-180°] = -1.
Square Roots of Other Complex Numbers
The square root of a positive number or zero are real nonnegative numbers. I assume in the following that you know how to compute these square roots. The square roots of negative numbers and of other arrows or points in the coordinate plane depend on this ability.
Observe that squaring points in the plane doubles their angular displacements and squares their magnitudes (distance to the origin). That is, the add the angles, multiple the lengths rule gives
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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