E-StoreLEARN ABOUT COMPLEX NUMBERS
2 Jul 2007 19:23:49 IST
LEARN ABOUT COMPLEX NUMBERS
Basic Definitions
It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called `` The Complex Numbers." In this amazing number field every algebraic equation in z with complex coefficients
has a solution. To prove this fact we need Liouville's Theorem, but to get started using complex numbers all we need are the following basic rules.
Rules of Complex Arithmetic
-
- Every complex number has the ``Standard Form''
for some real a and b. - For real a and b,
-
-
Notice that rules 4 and 5 state that we can't get out of the complex numbers by adding (or subtracting) or multiplying two complex numbers together.
OK, so we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.
Notice that
We say that c+di and c-di are complex conjugates. To simplify a complex fraction, multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts
If z= a+bi is a complex number and a and b are real, we say that a is the real part of z and that b is the imaginary part of z and we write
Complex Conjugates
If z=a +bi is a complex number with real part a and imaginary part b, then we denote the complex conjugate of z by 
Modulus of a Complex Number
The magnitude or modulus of a complex number z is denoted |z| and defined as
The Complex Plane
Complex numbers are points in the plane
In the same way that we think of real numbers as being points on a line, it is natural to identify a complex number z=a+ib with the point (a,b) in the cartesian plane. Expressions such as ``the complex number z'', and ``the point z'' are now interchangeable.
We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of the complex numbers. The reals are just the x-axis in the complex plane.
The modulus of the complex number z= a + ib now can be interpreted as the distance from z to the origin in the complex plane.
Since the hypotenuse of a right triangle is longer than the other sides, we have
for every complex number z.
We can also think of the point z= a+ ib as the vector (a,b). From this point of view, the addition of complex numbers is equivalent to vector addition in two dimensions and we can visualize it as laying arrows tail to end. (Picture)
We see in this way that the distance between two points z and w in the complex plane is |z-w|.
the ``Parallellogram law''
The ``Triangle'' inequality
The unit circle
The fundamental trigonometric identity (i.e the Pythagorean theorem) is
From this we can see that the complex numbers
are points on the circle of radius one centered at the origin.
Think of the point 
moving counterclockwise around the circle as the real number 
moves from left to right. Similarly, the point moves clockwise if decreases. And whether increases or decreases, the point returns to the same position on the circle whenever changes by 
or by 
or by 
where k is any integer.
moving counterclockwise around the circle as the real number moves from left to right. Similarly, the point moves clockwise if decreases. And whether increases or decreases, the point returns to the same position on the circle whenever changes by or by or by where k is any integer.Exercise: Verify that
Exercise: Prove de Moivre's formula
Now picture a fixed complex number on the unit circle
Consider multiples of z by a real, positive number r.
As r grows from 1, our point moves out along the ray whose tail is at the origin and which passes through the point z. As r shrinks from 1 toward zero, our point moves inward along the same ray toward the origin. The modulus of the point is r. We call the angle 
which this ray makes with the x-axis, the argument of the number z. All the numbers rz have the same argument. We write
which this ray makes with the x-axis, the argument of the number z. All the numbers rz have the same argument. We writeJust as a point in the plane is completely determined by its polar coordinates 
, a complex number is completely determined by its modulus and its argument.
, a complex number is completely determined by its modulus and its argument.Notice that the argument is not defined when r=0 and in any case is only determined up to an integer multiple of 
.
.
Free Sign Up!
Preparing for IIT-JEE ?
Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor
@ INR 1,995/-
For Quick Info
|
|
|
|
|
| 35,229,755 Engineering Questions Attempted |
6,69,530 Visitors Monthly |
79 Engg. Exam Covered |
1156 Students Currently Online |
Connect with Us  |
 |
Vriti Communities
Privacy Policy © Vriti 2011