Complex Numbers 2
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COMPLEX NUMBER (2)
ARGUMENT OF A COMPLEX NUMBER
If z = x + iy denotes a complex number then its argument or amplitude is given by
|
q = tan-1 x/y |
We can see that argument of complex number 0 is not defined. The argument of a complex number is not unique, i.e. if q is the value of the argument then 2np + q are also the possible values of q. Thus, the value of q which satisfies the inequality -p < q <p is called the principal value of the argument.
- arg (z1z2z3…..) = q1+q2+q3+……..
- arg zn = n arg z {from the above property}
- arg(z1/z2) = arg z1- arg z2
- arg| z1+ z2| = arg z1+ arg z2
NOTE: the polar form of a complex number is given by
|
Z= r (cosq +isinq) |
Where r is the modulus of the complex number and q its argument. This form of a complex number is sometimes also referred as the trigonometrical form of z.
Since eiq = cosq + isinq we can write z = reiq
REPRESENTATION OF A COMPLEX NUMBER ON THE ARGAND PLANE
Complex numbers can be depicted on the complex plane, also called the Argand plane, in which the x-axis is the real axis, the y - axis is the imaginary axis.(Refer figure)
So z = x+iy is represented by the point (a, b)
Let z1 =x1+iy1
z2= x2+iy2
To find z1+z2we add the real part and the imaginary part separately.
Therefore z1+z2 = (x1+x2) + i(y1+y2)
z1+z2 is actually the diagonal of the parallelogram found by z1 and z2
from the figure, if A is the point z1, and B is z2, then co-ordinates of A are (x1, y1) and that of B are (x2, y2)
i.e OA’=x1, A’A=y, OB’=x2, B’B=y2
Since angle OBB’ and angle ADP are congruent, we have AD=A’C=x2 and PD=BB’=y2
Therefore OP=OC=OA’+A’C= x1+x2
CP=CD+DP=y1+y2
Therefore z1+z2 represented by P(x1+x2, y1+y2) is the diagonal OP of parallelogram OAPB
Algebraically, if
z1=x1+iy1
z2= x2+ iy2
- z1+z2= (x1+x2)+i(y1+y2)
- z1-z2= (x1-x2)+i(y1-y2)
- z1.z2= (x1+iy1).(x2+iy2)
= x1x2+ i2 y1y2+ ix1y2+ ix2y1
= (x1x2- y1y2) + i (x1y2+ x2y1)
The roots of the equation x3 - 1 = 0 are called the cube roots of unity. On solving this equation we get the values of x as 1, (-1 ± i31/2)/2
Thus, the three cube roots of unity can be stated as
1
(-1 + i31/2)/2
(-1 - i31/2)/2
These three values of x are denoted as 1,w,w2 , as we can see that of these three, one is real and the other two are imaginary.
While working with the cube roots of unity, we shall keep in mind the following equalities,
|
1+w+w2 = 0 |
w4 = w
w5 = w2
and so on.
This theorem states that if n be any rational number, then
|
(cosq + isinq)n = cos nq +isin nq |
(cosq + isinq)-n = cos nq - isin nq
(cosq - isinq)n = cos nq - isin nq
(cosq - isinq)-n = cos nq + isin nq
The Complex Plane
In the same way that real numbers are points on a line, we can identify a complex number z =x+iy with the point (x,y) in the cartesian plane. Expressions such as “the complex number z”, and “the point z” have the same meaning.
We consider the a real number x to be the complex number x+ iy and in this way we can say that the real numbers are a subset of the complex numbers. The real numbers 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 = x+ iy as the vector (x,y). 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.
We see in this way that the distance between two points z and w in the complex plane is |z-w|.