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- z
_{1}= a+ib and z_{2 }= c+id then z_{1}= z_{2 }implies that a = c and b = d. - If we have a complex number z where z = a+ib, the conjugate of the complex number is denoted by z* and is equal to a-ib. In fact, for any complex number z, its conjugate is given by z* = Re(z) – Im(z).
- Division of complex numbers: The numerator as well as denominator should first be multiplied by the conjugate of the denominator and then simplified.

**Argument of a complex number:**

1. Argument of a complex number p(z) is defined by the angle which OP makes with the positive direction of x-axis.

2. Argument of z generally refers to the principal argument of z (i.e. the argument lying in (–π, π) unless the context requires otherwise.

3. Hence, the argument of the complex number z = a + ib = r (cos θ + i sin θ) is the value of θ satisfying r cos θ = a and r sin θ = b.

4. The angle θ is given by θ = tan^{-1} |b/a|.

5. The value of argument in various quadrants is given below:

- If OP = |z| and arg (z) = θ, then obviously z = r (cos θ + i sin θ) and is called the polar form of complex number z.
- |(z-z
_{1})/(z-z_{2})| =1 the locus of point representing z is the perpendicular bisector of line joining z_{1}and z_{2}. - -|z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z|
- If a and b are real numbers and z
_{1 }and z_{2}are complex numbers then

|az_{1}+ bz_{2}|^{2}+ |bz_{1}- az_{2}|^{2}= (a^{2}+ b^{2}) (|z_{1}|^{2}+ (|z_{2}|^{2}) - The distance between the complex numbers z
_{1}and z_{2}is given by |z_{1 }- z_{2}|. - In parametric form, the equation of line joining z
_{1}and z_{2}is given by z = tz_{1}+ (1-t)z_{2}. - If A(z
_{1}) and B(z_{2}) are two points in the argand plane, then the complex slope μ of the straight line AB is given by μ = (z_{1}- z_{2})/ (1 - 2). **Two lines having complex slopes μ**_{1}and μ_{2}are:**1. Parallel iff μ**_{1 }= μ_{2}**2. Perpendicular iff μ**_{1}= - μ_{2}or μ_{1}+ μ_{2 }= 0- If A(z
_{1}), B(z_{2}), C(z_{3}) and D(z_{4}) are four points in the argand plane, then the angle θ between the lines AB and CD is given by θ = arg{(z_{1}- z_{2})/ (z_{3}– z_{4})} **Some basic properties of complex numbers:**

I. ||z_{1}| - |z_{2}|| = |z_{1}+z_{2}| and |z_{1}-z_{2}| = |z_{1}| + |z_{2}| iff origin, z_{1}, and z_{2}are collinear and origin lies between z_{1 }and z_{2}.

II. |z_{1}+ z_{2}| = |z_{1}| + |z_{2}| and ||z_{1}| - |z_{2}|| = |z_{1}-z_{2}| iff origin, z_{1 }and z_{2 }are collinear and z_{1}and z_{2}lie on the same side of origin.

III. The product of n^{th}roots of any complex number z is z(-1)^{n-1}.

IV. amp(z^{n}) = n amp z

V. The least value of |z - a| + |z - b| is |a - b|.

**Demoivre's Theorem: The theorem can be stated in two forms:****Case I:**If n is any integer, then

(i) (cos θ + i sin θ)^{n}= cos nθ + i sin nθ

(ii) (cos θ_{1 }+ i sin θ_{1}) . (cos θ_{2}+ i sin θ_{2}) ......... (cos θ_{n}+ i sin θ_{n})

= cos (θ_{1 }+ θ_{2}+ θ_{3}.................. + θ_{n}) + i sin (θ_{1}+ θ_{2}+ .............. θ_{n})**Case II:**For p and q such that q ≠ 0, we have

(cosθ + isinθ)p/q = cos((2kπ + pq)/q) + isin((2kπ+pq/q) where k = 0,1,2,3,.....,q^{-1}- Demoivre’s formula does not hold for non-integer powers.
- Main application of Demoivre’s formula is in finding the nth roots of unity. So, if we write the complex number z in the polar form then,

z = r(cos x + isin x)

Then z^{1/n}= [r (cos x +i sinx )]^{1/n}

= r^{ 1/n}[ cos (x+2kπ/n) + i sin (x+2kπ/n)]

Here k is an integer. To get the n different roots of z one only needs to consider values of k from 0 to n –1 - Continued product of the roots of a complex quantity should be determined using theory of equations.
- The modulus of a complex number is given by |z| = √x
^{2}+y^{2}. - The only complex number with modulus zero is the number (0, 0).
**The following figures illustrate geometrically the meaning of addition and subtraction of complex numbers:**

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