Complex Number

Complex Number
Definition : a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = ?1. In this expression, a is the real part and b is the imaginary part of the complex number.
Another definition of Complex numbers:
A complex number is defined as order pair (a, b) of real numbers a and b subject to operational definitions given below:-
1- (a,b) = (c,d) iff a = c and b = d
2- (a,b) + (c,d) = (a+c ,b+d)
3- (a,b).(c,d) =( ac-bd,ad+bc)
4- m.(a,b) = (ma,mb)
All letters used above represent real numbers, observe that
(a,b) = (a,0) + (a,b) =a(1,0) + b(0,1) .
Note : the set of complex number is denoted by C where
C= {a+ib?a ,b ?R ,i=?(-1})

Some fundamental law:
Write z_1,z_(2 ) ,z_(3 ) belong to the set C of complex number:
1- C is closed under addition and multiplication where z_1+z_2 and z_1.z_2 also belongs to C .
2- commutative law of addition z_1+z_2 = z_2+z_1
3- a associative law of addition ?( z?_1+z_2)+z_(3 ) ?= z?_1+(z_2+z_(3 ))
4- commutative law of multiplication z_1.z_2 = z_2.z_1
5- a associative law of multiplication ?( z?_1.z_2).z_(3 ) ?= z?_1.(z_2.z_(3 ))
6- distributive law ?z_1.( z?_2+z_3)?= z?_1 z_2+z_1 z_(3 ))

a complex number is extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a,?b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology,economics, electrical,engineering, and statistics.The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century

A combination of a real and an imaginary number in the form a+bi, where a and b are real, and i is the "unit imaginary number" ?(-1)
The values a and b can be zero.
Examples: 1 + i, 2 - 6i, -5.2i, 4

Some operations of complex number:

Every complex number has the ``Standard Form

for some real a and b.
For real a and b,

addition :
multiplication
Note that (i)(2i)(–3i).
(i)(2i)(–3i) = (2 • –3)(i • i • i) = (–6)(i2 • i)
=(–6)(–1 • i) = (–6)(–i) = 6i
Note this last problem. Within it, you can see that , because i2 = –1. Continuing, we get:

This pattern of powers, signs, 1 s, and i s is a cycle:



Equality :Two complex numbers are equal if and only if both their real and imaginary parts are equal. In symbols:

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

Exercise:
Find and . (Solution)

An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = ?1. For example, ?3.5 + 2i is a complex number.
The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part. The real part of a complex number z is denoted by Re(z) or ?(z); the imaginary part of a complex number z is denoted by Im(z) or ?(z). For example,

Hence, in terms of its real and imaginary parts, a complex number z is
equal to . This expression is sometimes known as the Cartesian form of z.
A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write a ? bi with b > 0 instead of a + (-b)i, for example 3 ? 4i instead of 3 + (?4)i.


Complex Conjugates:
Definition: 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 .
(Figure 1 )
Geometric representation of z and its conjugate in the complex plane
The complex conjugate of the complex number z = x + yi is defined to

be x - yi.
Exercise:
Write in standard form.
(Solution)
Exercise:
Prove that for any pair of complex numbers and similarly .
(Proof.)
Exercise:
Prove that for any integer n.
(Proof.)
Modulus of a Complex Number:
The magnitude or modulus of a complex number z is denoted |z| and defined as

Notice that . (proof.)
Exercise:
Prove that . (Solution)
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. What about dividing one complex number by another? Is the result another complex number? Let s ask the question in another way. If you are given four real numbers a, b,c and d, can you find two other real numbers x and y so that

Now multiply both sides of the above equation by c + di and then solve for x and y to prove that the answer to our question is yes, 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.
Examples of division:
1)
2)




Complex plane:
( Figure 2)
A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; a+bi is the rectangular expression of the point.
A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram, named after Jean-Robert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 2). These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.
A position vector may also be defined in terms of its magnitude and direction relative to the origin. These are emphasized in a complex number s polar form. Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin: (a+bi)i = ai+bi2 = -b+ai.
Polar Form of a Complex Number:
We can think of complex numbers as vectors, as in our earlier example. We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry., we saw how to add, subtract, multiply and divide complex numbers from scratch.
However, it s normally much easier to multiply and divide complex numbers if they are in polar form.
Our aim in this section is to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis.

(Figure 3)
We find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and ? (the angle made with the real axis):
r is the absolute value (or modulus) of the complex number
? is the argument of the complex number.
We have z =x+iy then f(z) = w = u(x,y)+iv(x,y)
?r ?^2=x^2+y^2=?|x+iy|?^2=?|z|?^2
sin?=y?(r and cos?=x?r) then
x=r cos? and y=r sin? hence
z=r( cos? + isin?)
Note that the argument of z where ? is the angle which the line makes with the positive where argz= ?+2k?
This form is called the polar form of z
Exercise:
Find the polar form of z=-1-i (H.W.)

Complex numbers allow for solutions to certain equations that have no solutions in real numbers. For example, the equation

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i2 = ?1, so that solutions to equations like the preceding one can be found. In this case the solutions are ?1 + 3i and ?1 ? 3i, as can be verified using the fact that i2 = ?1: