Chapter 3 - Complex numbers Flashcards
Consider the two complex numbers z1 = (3a+2) + j(3b-1) and z2 = (b+1) - j(a+2-b).
If the two complex numbers are equal, what can we say about the real and imaginary parts of the complex numbers?
Describe in a few steps how to find a and b.
Since the two complex numbers are equal, z1’s and z2’s real pars must be equal, and their imaginary parts must be equal:
(3a+2) = (b+1)
and
(3b-1) = (a+2-b).
- What we have is a set of equations with the same variables, so we can manipulate the equations to get e.g. b alone on one side to describe it in terms of a, and implement the result in the other equation.
How to you add and subtract complex numbers?
E.g. z1 = 3 + j2 and z2 = 4 + j6.
You add/subtract two complex numbers by adding/subtracting their real parts together, and their imaginary parts together.
In this example, z1 + z2 = (3+4) + j(2+6) = 7 + j8.
How do you multiply two complex numbers?
E.g. z1 = 3 + j2 and z2 = 4 + j6.
When multiplying two complex numbers, the normal rules for multiplying out brackets hold.
In this example:
z1 * z2 = 3*4 + 3*j6 + j2*4 + j2*j6
= 12 + j18 +j8 + j212
= 12 + j26 + (-12)
= j26
(j2 equals -1)
What is the complex conugate of z = x + jy?
You get the complex conjugate by changing the sign of the imaginary part of z:
z = x + jy
z* = x - jy
(The complex conjugate is denoted by * (z*)).
How do you divide two complex numbers?
E.g. z1 / z2 = (x1+jy1) / (x2+jy2)
You multiply both the numerator and the denominator with the conjugate of the denominator:
Given the complex numbers z = x + jy and z* = x - jy,
complete these basic complex conjugate rules:
z + z* =
z - z* =
zz* =
(z1z2)* =
z + z* = 2x = 2 Re(z)
z - z* = 2jy = 2j Im(z)
zz* = (x + jy)(x - jy) = x2 + y2
(z1z2)* = z1*z2*
From the next to last result we can see that the product of a complex number and its complex conjugate is a real number.
What are the modulus and the argument of a complex number z?
The modulus of z is the length of OP (from (0,0) in the Argand diagram to the point of z). It’s denoted as “mod z” or |z|.
The argument of z is the angle between the positive real axis and OP. It’s denoted as “arg z”.
Consider z and r in an Argand diagram.
|z| = r can also be described as:
|z| = r = sqrt(x2+y2)
Note that the modulus of z is the square root of the sum of squares of x and y, NOT of x and jy. The j part of the number has been accounted for in the representation of the argand diagram.
The figure shows that the relationships between (x, y) and (r, Ø) are:
x = r cos Ø and y = r sin Ø
Hence the complex number z = x + jy can be expressed in the form (which is called(?)):
The polar form of the complex number:
z = r cos Ø + jr sin Ø = r(cos Ø + j sin Ø).
In engineering, it is frequently written as in the picture.
Consider the two complex numbers in polar form:
z1 = r1(cos Ø1 + j sin Ø1)
z2 = r2(cos Ø2 + j sin Ø2).
How do you multiply them?
The regular rules of multiplying complex numbers also apply when the numbers are given in polar form.
z1z2 = r1r2 (cos Ø1 + j sin Ø1)(cos Ø2 + j sin Ø2)
= r1r2 (cos(Ø1+Ø2) + j sin(Ø1+Ø2)).
Hence:
|z1z2| = r1r2 = |z1| |z1|
and
arg(z1z2) = Ø1 + Ø2 = arg z1 + arg z2
What is the representation of the imaginary number j on polar form?
How do you divide two complex numbers?
Describe Euler’s formula (picture)
eeeeh Euler’s formula (picture) links the exponential and circular functions.
How do you define the logarithm of a complex number?
ln z = ln |z| + j arg z
(ln |z| + j arg z + j2nπ, n = 0, +- 1, +- 2 etc.)
Define zn and z-n in terms of sine and cosine
zn = cos nØ + j sin nØ
and
z-n = cos nØ - j sin nØ
Define de Moivre’s theorem
A complex number z may be expressed in terms of its modulus r and argument Ø in the exponential form
z = r ej<em>Ø</em>
Using the rules of indices and the property of the exponential function, we have, for any n,
zn = rn(ejØ)n = rn ej(nØ)
so that (picture (de Moivre’s theorem)
Manipulate de Moivre’s theorem to find z1/n expressed in polar form
