# Short Notes of Complex Numbers for IIT-JEE mains and advance

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INDEX

1. The complex number system

2. Algebraic Operations

3. Equality In Complex Number

4. Representation Of A Complex Number

5. Modulus of a Complex Number

6. Argument of a Complex Number

7. Conjugate of a complex Number

8. Rotation theorem

9. Demoivre’s Theorem

10. Cube Root Of Unity

11. n th Roots of Unity

12. The Sum Of The Following Series Should Be Remembered

13. Logarithm Of A Complex Quantity

14. Geometrical Properties

15.  Reflection points for a straight line , Inverse points w.r.t. a circle

16. Ptolemy’s Theorem

### 1. The complex number system

There is no real number x which satisfies the polynomial equation x2 + 1 = 0. To permit solutions of this and similar equations, the set of complex numbers is introduced. We can consider a complex number as having the form a + bi where a and b are real number and i, which is called the imaginary unit, has the property that i2 = – 1. It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z which is denoted by (Re z) and ‘b’ is called as imaginary part of z which is denoted by (Im z).

Any complex number is : (i) Purely real, if b = 0    (ii) Purely imaginary, if a = 0  (iii) Imaginary, if b not equal to 0

NOTE :

### 2. Algebraic Operations:

Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative. e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless. In real numbers if a2 + b2 = 0 then a = 0 = b however in complex numbers,                 z1 2 + z2 2 = 0 does not imply z1 = z2 = 0.

### 3. Equality In Complex Number:

Two complex numbers z1 = a1 + ib1 & z2 = a2 + ib2 are equal if and only if their real and imaginary parts are equal respectively
i.e. z1 = z2    —>   Re(z1 ) = Re(z2 ) and Im (z1 ) = Im (z2 ).

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### 6. Argument of a Complex Number :

Argument of a non-zero complex number P(z) is denoted and defined by arg(z) = angle which OP makes with the positive direction of real axis. If OP = |z| = r and arg(z) = θ, then obviously z = r(cosθ + isinθ), called the polar form of z. In what follows, ‘argument of z’ would mean principal argument of z(i.e. argument lying in (–π ,π] unless the context requires otherwise. Thus argument of a complex number z = a + ib = r(cosθ + isinθ) is the value of θ satisfying rcosθ = a and rsinθ = b. Thus the argument of z =θ, π –θ , – π +θ , –θ ,θ  = tan–1 |a/ b| , according as z = a + ib lies in I, II, III or IVth quadrant.

Properties of arguments

(i) arg(z1 z2 ) = arg(z1 ) + arg(z2 ) + 2mπ for some integer m.

(ii) arg(z1 /z2 ) = arg (z1 ) – arg(z2 ) + 2mπ for some integer m.

(iii) arg (z2 ) = 2arg(z) + 2mπ for some integer m.

(iv) arg(z) = 0           <—>       z is real, for any complex number z ≠ 0

(v) arg(z) = ± π/2   <—>      z is purely imaginary, for any complex number z ≠ 0

(vi) arg(z2 – z1 ) = angle of the line segment

### 7. Conjugate of a complex Number

Conjugate of a complex number z = a + b is denoted and defined by z¯ = a – ib. In a complex number if we replace i by – i, we get conjugate of the complex number. z¯ is the mirror image of z about real axis on Argand’s Plane

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### 8. Rotation theorem

(i) If P(z1 ) and Q(zz ) are two complex numbers such that |z1 | = |z2 |, then z2 = z1 e  where θ = ∠ POQ

(ii) If P(z1 ), Q(z2 ) and R(z3 ) are three complex numbers and ∠ PQR = θ, then

(iii) If P(z1 ), Q(z2 ), R(z3 ) and S(z4 ) are four complex numbers and  ∠STQ = θ, then

### 10. Cube Root Of Unity :

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### 12.The Sum Of The Following Series Should Be Remembered :

(i) cos  θ+ cos 2 θ + cos 3 θ +….. + cos n θ = sin (nθ/2) / sin(θ/2)    cos ( n + 1/2 ) θ

(ii) sin θ + sin 2 θ + sin 3 θ +….. + sin n θ =  sin (nθ/2)/sin(θ/2)  sin (n+1/2) θ

NOTE : If θ = (2π/n) then the sum of the above series vanishes.

### 13. Logarithm Of A Complex Quantity :

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### 15.

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