Tuesday, February 26, 2013
What are Complex Numbers?
Fortunately, mathematicians devised complex numbers long before the practical need for them arose. In the case of the latter, complex numbers are relevant to everything from the quantum wave function to oscillating-alternating circuits and wave forms. For example, the representation of monochromatic plane waves in free space:
E = E(o) exp[i(wt – kz)]
Or the voltage at the end of a transmission line:
V(x=0) = A exp iwt[l + K]
where w = 2πf and k = 2π/ l
where f is the frequency, and K is the voltage reflection coefficient.
It was inevitable that after the invention of counting numbers, i.e. integers, a new breed would be invented which made use of the imaginary number i. This is found when one solves the quadratic:
x2 + 1 = 0
To get in the first step:
x 2 = -1
Then solving for x: x = (-1)^½
Or the square root of minus 1. This is defined and referred to as i. Thus: i×i= -1
Placement of numbers follows a similar analogy to the placement of real numbers on the Cartesian x-y axes. In this case, one uses an Argand diagram with axes:
Thus, the basis is laid to place complex numbers, say of the form: a +bi, on the above graph, where a is located by using the ordinate (x-axis) and b by using the iy (or imaginary ) axes. (Note: in most cases the i is dispensed with in the diagram)
For example, to find: 2 + 3i, you would mark off 2 on the x axis, and 3 on the iy axis.
Addition and subtraction of complex numbers follows simple rules and is straightforward. The main rule is to keep imaginary and real parts separate when performing the operations. For example:
1 + 2i + 3 – 3i = (1 + 3) + (2i – 3i) = 4 - i
7 + 8i - (3 – 3i) = (7 – 3) + (8i – (-3i)) = 4 + 11i
An Argand diagram in “action” is shown in Fig.1. Here, several vectors are represented. We see that the vector A = -2 + 2i, B = -2 – 3i, and C = 4 + 3i.
How would we combine the vectors A + B? We can use the simple addition process:
A + B = (-2 + 2i) + (-2 – 3i) = -4 –i
Using Fig. 1 to complete the parallelogram formed by the vector should lead the reader to see the resultant terminates at the coordinate (-4, -1) (Actually, -4, -i)
What about adding the vectors A + C?
A + C = (-2 + 2i) + (4 + 3i) = 2 + 5i
Completing the parallelogram, the reader should be able to satisfy himself that the resultant terminates at
(2, 5) or just beyond the upper limit of the graph.
1) Find the sum of (1 –i) and (6 – 6i)
2) Find the sum of (0- i) and (15 – 0i)
3) Subtract: (3 + 3i) from 10 – 4i
4) Find: (1 –i) + (11 – 2i) – (4 – 4i)
5) Represent the two parts of (1) as vectors on an Argand diagram, and show their resultant.
6) Using the Argand diagram of Fig. 1 and complex addition, find the resultant of the vector sum B + C.