Complex Numbers AxiomsHere is a review. As a matter of convenience or taste, we will write complex numbers as z = [a,b] or z = a + ib The notation z = a + ib is standard but for precision in the following, you should replace the expression a+ ib by the ordered pair [a,b]. Each complex number can be thought of a vector from the origin to the point [a, b] in the plane, or each complex number can be thought of as the position vector of a point [a,b] in a plane. In the foregoing we assume a and b are signed coordinates. Assumption B1. There is a set C = {z= a + ib | a real, b real} of complex numbers where each element z = a + ib = [a, b] has a real part a and an imaginary part b. Following this assumption, we let real part function Re(z) = a provides the real part of z (the first coordinate) and we let the imaginary part function Im(z) = b provides the imaginary part of z (the second coordinate). We also say the complex conjugate of z = a + i b is a - i b = Re(z) - i Im(z). The addition and multiplication of complex numbers may be introduced as follows [a, b] + [c,d] = [a+c, b +d] or (a+ib) + (c+id) = (a+c) + i (b+d) [a,b] x [c,d] = [ac -bd, ac + bd] or (a+ib)(c+id) = (ac-bd) + i (ad + bc) The latter operation may lack motivation if it is not introduced with the aid of the add the angles, multiply the lengths rule given earlier. It elements z, w are assumed to have the following computation properties (described algebraically. Below Z, W and V are actors standing in for complex numbers, or the result of calculations that yield complex numbers.
These computation properties also hold for real numbers, that is numbers of the form z = a +i0 on the horizontal axis. These properties are often stated as assumptions or axioms with no mention of the decimal representation of the real numbers. That represents a gap in committee driven course design. Polar Coordinates for Complex Numbers: To compute polar coordinates for complex numbers from their real and imaginary parts, you need to master two to six inverse trig functions. That subject is not yet treated here. Or, you may measure the coordinates after locating a complex number in the plane. If you start with this definition [a,b] x [c,d] = [ac -bd, ac + bd] or (a+ib)(c+id) = (ac-bd) + i (ad + bc) of complex multiplication, you will have master trigonometry before seeing the derivation of the derivation of the add the angles, multiply the lengths rule from this definition. Discussion of the Properties -- Miscellaneous stuffHere are a few arguments to suggest or imply the properties.
Pure mathematics would obtain the above properties from set theoretic constructions via long, geometric-free and decimal-free chains of reason. However most students, future mathematicians included, first master mathematics with the aid of decimal, geometric reasoning and the algebraic way of writing and reasoning. That may provide the maturity and interest to cover more material. Factors via the Unit CircleIf z = (r, alpha) in polar coordinates then z = (r, 0 degrees) * (1, alpha). A geometric arguments starting with whole numbers and then fractions imply that z = r * (1, alpha) where multiplication by r represents the rescaling of a vector and (1, alpha) is point or vector or complex number on the unit circle. It has length one. If r =0 then z = r * (1, alpha) for all choices of the angle alpha. One cannot measure an angle for a vector of length 0. Multiplication equals a rotation followed by a rescalingNext if w = (R, beta) then zw = (rR, alpha + beta) (r,0 degrees){(1, alpha)(R,beta)} = (r, 0 degrees) (R, alpha +beta) = zw Therefore zw = r{ (1,alpha)*w) This implies that multiplication of a complex number by z is equivalent
to a rotation through angle alpha followed by a rescaling using the length r of
the "vector" z . |
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 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:
|
|