Complex Plane

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.
Exercise: Prove this last statement algebraically. (Proof.)
Exercise: Prove the ``Parallellogram law''
(Proof.)
The ``Triangle'' inequality

is easily seen to hold. (Proof.)
Exercise: Prove the Triangle inequality for n complex numbers

(Proof.)

Trigo.....!

TRIGONOMETRY - MEASURE OF AN ANGLE
Any real number may be interpreted as the radian measure of an angle as follows: If , think of wrapping a length of string around the standard unit circle C in the plane, with initial point P(1,0), and proceeding counterclockwise around the circle; do the same if , but wrap the string clockwise around the circle. This process is described in Figure 1 below.

Figure 1
If Q(x,y) is the point on the circle where the string ends, we may think of as being an angle by associating to it the central angle with vertex O(0,0) and sides passing through the points P and Q. If instead of wrapping a length s of string around the unit circle, we decide to wrap it around a circle of radius R, the angle (in radians) generated in the process will satisfy the following relation:

Observe that the length s of string gives the measure of the angle only when R=1.
As a matter of common practice and convenience, it is useful to measure angles in degrees, which are defined by partitioning one whole revolution into 360 equal parts, each of which is then called one degree. In this way, one whole revolution around the unit circle measures radians and also 360 degrees (or ), that is:

Each degree may be further subdivided into 60 parts, called minutes, and in turn each minute may be subdivided into another 60 parts, called seconds:

EXAMPLE 1 Express the angle in Degree-Minute-Second (DMS) notation.
Solution: We use Equation 3 to convert a fraction of a degree into minutes and a fraction of a minute into seconds:

Therefore, .
EXAMPLE 2 Express the angle in radians.
Solution: From Equation 2 we see that

EXAMPLE 3 Find the length of an arc on a circle of radius 75 inches that spans a central angle of measure .
Solution: We use Equation 1, , with R=75 inches and , to obtain

Here are some more exercises in the use of the rules given in Equations 1,2, and 3.
EXERCISE 1 Express the angle radians in (a) decimal form and (b) DMS form.
Solution
EXERCISE 2 Express the angle in radians.
Solution
EXERCISE 3 Assume that City A and City B are located on the same meridian in the Northern hemisphere and that the earth is a sphere of radius 4000 mi. The latitudes of City A and City B are and , respectively.
(a)
Express the latitudes of City A and City B in decimal form.
(b)
Express the latitudes of City A and City B in radian form.
(c)
Find the distance between the two cities.

Topic 1

Rational and irrational number....
-Real number....