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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) â€“ 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 (â€“Ï€, Ï€) unless the context requires otherwise.

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

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

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

- If OP = |z| and arg (z) = Î¸, then obviously z = r (cos Î¸ + i sin Î¸) 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| â‰¤ Re(z) â‰¤ |z| and -|z| â‰¤ Im(z) â‰¤ |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 Î¼ of the straight line AB is given by Î¼ = (z_{1}- z_{2})/ (1 - 2). **Two lines having complex slopes Î¼**_{1}and Î¼_{2}are:**1. Parallel iff Î¼**_{1 }= Î¼_{2}**2. Perpendicular iff Î¼**_{1}= - Î¼_{2}or Î¼_{1}+ Î¼_{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 Î¸ between the lines AB and CD is given by Î¸ = arg{(z_{1}- z_{2})/ (z_{3}â€“ 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 Î¸ + i sin Î¸)^{n}= cos nÎ¸ + i sin nÎ¸

(ii) (cos Î¸_{1 }+ i sin Î¸_{1}) . (cos Î¸_{2}+ i sin Î¸_{2}) ......... (cos Î¸_{n}+ i sin Î¸_{n})

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

(cosÎ¸ + isinÎ¸)p/q = cos((2kÏ€ + pq)/q) + isin((2kÏ€+pq/q) where k = 0,1,2,3,.....,q^{-1}- Demoivreâ€™s formula does not hold for non-integer powers.
- Main application of Demoivreâ€™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+2kÏ€/n) + i sin (x+2kÏ€/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 â€“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| = âˆšx
^{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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