If `z_1 = x_1 + iy_1 ` and `z_2 = x_2 + iy_2` are two complex numbers, then

(i) Addition of complex numbers is `z_1 + z_2 = (x_1 + x_2 ) + i (y_1 + y_2)` Its additive identity is `0 +0i.`

(ii) Subtraction of complex numbers is `z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2)`

(iii) Multiplication of complex numbers is `z_1z_2 = ( x_1x_2 - y_1y_2 ) + i ( x_1y_2 + x_2y_1)` Its multiplicative identity is `1 + 0i.`

(iv) Division of complex number is `z_1/z_2 = ((x_1x_2 + y_1y_2 ) + i(x_2y_1 - x_1y_2))/(x_2^2 +y_2^2)`

Complex numbers `z = a+ ib` and `bar z =a- ib` are called conjugate to each other.

If `z, z_1` and `z_2` are complex numbers, then

(i) `(bar(bar z)) = z`
(ii) `bar(z_1 + z_2) = bar(z_1) + bar(z_2)`
(iii) `bar(z_1 -z_2) =bar(z_1) -bar(z_2)`
(iv) `bar(z_1z_2) = bar(z_1) * bar (z_2)`
(v) `(bar (z_1/z_2) ) = bar(z_1)/bar(z_2)`
(vi) `z + bar z = 2 Re(bar z) = 2 Re (z)`
(vii) `z - bar z = 2 i Im (z)`
(viii) `z + bar z =0 => z` is purely imaginary.
(ix) `z = bar z <=> z` is purely real.
(x) `z_1 bar(z_2) + bar (z_1) z_2 = 2 Re (bar(z_1) z_2) = 2 Re (z_1 bar(z_2) )`
(xi) If `z = f (z_1)`, then `barz = f (bar (z_1))`

Let `z =a+ ib,` then the modulus of z is the positive real number `sqrt(a^2 +b^2)` and is denoted by `|z|.`

i.e `|z| = sqrt(a^2 +b^2) = sqrt(|Re(z)|^2 + | Im(z)|^2), | z| >= 0 AA z in C`

(i) `|z| >= 0 => |z| = 0 ` iff `z =0` i.e `Re (z) = Im(z) = 0`
(ii) `z barz = |bar z|^2`
(iii) `|z_1z_2 | = |z_1| |z_2|`
(iv) `|z_1/z_2| = (|z_1|)/(|z_2|)`
(v) `|z^n| = | z|^n`
(vi) `|z_1 + z_2| <= | z_1| + | z_2|`
(vii) `|z_1 - z_2 | >= |z_1| -|z_2|`
(viii) `|z_1| - |z_2 | <= | z_1 pm z_2| <= |z_1| + |z_2|`
(ix) `| z_1 + z_2 |^2 = |z_1|^2 +| z_2|^2 + 2 Re (z_1 bar(z_2))`
(x) `|z| = | barz| = | -z| = | - bar z|`
(xi) `z_1bar(z_2) + bar(z_1)z_2 = 2 |z_1||z_2| cos ( theta_1 - theta_2)`
where `theta_1 = arg (z_1)` and `theta_2 = arg (z_2)`
(xii) `|z_1 + z_2|^2 + |z_1 -z_2|^2 = 2 { |z_1|^2+ |z_2|^2 }`

For a complex number `z = x + iy,` the argument or amplitude of `z (!=0) = tan^-1(y/x)`

i.e. solution of the system of equations ` cos theta = x/sqrt(x^2 +y^2), sin theta = y/sqrt(x^2 +y^2)` and is denoted by `arg(z).`

The value of `theta` is found by solving these equations and `theta` is called the argument or amplitude of `z.` If `- pi < theta <= pi`, then `theta` is called the principal argument of ` z .`

(i) If `x < 0, y < 0,` then principal `arg (z)` lies between `- pi` and `-pi/2`
(ii) If `x =0, y < 0`, then principal `arg (z)` is `-pi/2`.
(iii) If `x > 0, y < 0,` then principal `arg (z)` lies between `- pi/2` and `0.`
(iv) If `x > 0, y = 0,` then principal `arg (z)` is `0 .`
(v) If `x > 0, y > 0,` then principal `arg (z)` lies between `0 ` and `pi/2`.
(vi) If `x = 0, y > 0,` then principal `arg (z)` is `pi/2`.
(vii) If `x < 0, y > 0,` then principal `arg (z)` is `pi`.

Properties of `arg (z) :`

(i) `arg (z_1z_2 ) = arg(z_1) + arg (z_2)`
(ii) `arg (z_1/z_2) = arg(z_1) - arg(z_2)`
(iii) `arg (z)` is not defined, if `z =0`
(iv) `arg (z^n) = n \ \arg (z)`
(v) `arg (z) + arg (barz) = 2 pi`
(vi) If `arg(z_2/z_1) = theta`, then `arg (z_1/z_2) = 2 k pi - theta, k in I`

Expression `r(cos theta + i sin theta)` is called the polar form of the complex number ` x + iy.`

Here , `r = sqrt(x^2 + y^2)`

`cos theta = x/sqrt(x^2 + y^2)`

`=> sin theta = y /sqrt(x^2 +y^2)`

In any triangle, sum of any two sides is greater than the third side and difference of any two sides is less than the third side.

(i) |`z_1 + z_2| <= |z_1| + |z_2|`

(ii) `|z_1 + z_2| >= ||z_1|- |z_2||`

(iii) `|z_1 - z_2| <= |z_1| + |z_2|`

(iv) `|z_1 -z_2| >= ||z_1| - |z_2||`

If n is an integer, positive, negative or a rational number, then

`(cos theta + i sin theta)^n =cos ntheta + i sin ntheta`

Other Forms of De-Moivre's Theorem :

(i) `(cos theta + i sin theta)^(-n) = cos n theta - i sin n theta`

(ii) `(cos theta- i sin theta)^n = cos n theta - i sin n theta`

(iii) `(cos theta - i sin theta)^(-n) = cos n theta + i sin n theta`

If `z = a+ ib` is a compIex number, such that `sqrt(a + ib) = x + iy`, where x,y are real number , then `a + ib = (x + iy)^2 => a + ib = (x^2 - y^2) + i(2xy)` and square roots of `z` by equating real and imaginary part can be given

`sqrtz = pm [ sqrt(1/2 (sqrt(a^2+b^2) + a ) ) + i sqrt(1/2 (sqrt(a^2 +b^2) - a )) ] , b > 0`

`= pm [ sqrt(1/2 (sqrt(a^2+b^2) + a ) ) - i sqrt(1/2 (sqrt(a^2 +b^2) - a )) ] , b < 0`

nth Roots of Unity :

`(cos theta + i sin theta)^(1//n) = cos ((2kpi)/n) + i sin (( 2k pi )/n) = e^((2 pi k )/n i)`

where, `k = 0, 1, 2, ... , (n -1)`

Thus, `nth` roots of unity are `i = 1 , alpha , alpha^2, alpha^3 ,... alpha^(n -1)`

where , `alpha = e^(i 2 pi//n) = cos \ \(2pi)/n + i sin \ \ (2 pi ) /n`

`text(Cube Roots of Unity :)`

Cube roots of unity are `1, omega, omega^2` where

`omega = (-1 + sqrt3 i )/2 ; omega^2 = (-1 -sqrt3i )/2` and `omega^3 =1`

Properties of Roots of Unity :

(i) Sum of the roots of unity in square roots, cube roots, 4th roots or nth roots is zero.

(ii) They are in GP of common ratio `d = e^(2pi i//r)`, where `r =2,3,4 ... (n-1)`

(iii) Sum of their `p`th power is zero.

i.e `1+ alpha^p + alpha^(2p) + .. + alpha ^((n -1)p) = (1 - (alpha^n)^p)/(1 - alpha^p) = 0`

`:. alpha^n = 1 , p!= kn`

If `p = kn`, then sum of their pth powers is n

i.e. if `p = kn` , thne `alpha^p = alpha^(kn) = (alpha^n)^k =1`

`because` Each root is 1

`:.` Sum ` =1 + 1 + 1 ... + 1 = n`

1. If `z_1` and `z_2` are two complex numbers, then

(i) `|z_1 -z_2|` is the distance between the points affixes `z_1` and `z_2`.

(ii) `(mz_2 + nz_1)/( m +n)` is the affix of the point dividing the line joining the points with affixes `z_1` and `z_2` in the ratio `m : n` internally, where `m != n`.

(iii) `(mz_2 - nz_1 )/(m -n)` is the affix of the point dividing the line joining the points with affixes `z_1` and `z_2 `in the ratio `m : n ` externally, where `m != n`

(iv) If the affixes of the vertices of the triangle are `z_1, z_2` and `z_3` , then the affix of its centroid is `(z_1 + z_2 + z_3 ) /3`

2. Three points with affixes `z_1, z_2 , z_3` are collinear, if

`|( z_1 , barz_1 ,1 ), ( z_2 , barz_2, 1),(z_3, barz_3, 1) | = 0`

3. (i) `|z -z_1| = r` represents the circle with centre `z_1` and radius `r.`

(ii) `| z - z_1 | < r` represents the interior of the circle with centre `z_1` and radius ` r.`

4. `|(z -z_1 ) / (z -z_2) | = k ` represents a circle, if `k != 1` and if `k =1` , then it represent a straight line which is the perpendicular bisector of the line segment joining points with affixes `z_1` and `z_2`.

5. `(z-z_1) (bar z- barz_2) + (bar z - barz_1) (z - z_2) = 0` represents the circle with line joining points with affixes `z_1` and `z_2` as a diameter.

6. If `z_1, z_2` and `z_3` are the affixes of the points `A, B, C` respectively , then the angle between `AB` and `AC` is given by `arg((z_3 - z_1 )/( z_2 - z_1 ) )`