IIT-JEE mains and advance are around the corner. We have created short notes of Complex Numbers for guys so that you start with your preparation!

**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 x^{2} + 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 i^{2} = – 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 :**

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### 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 a^{2} + b^{2} = 0 then a = 0 = b however in complex numbers, z_{1} ^{2} + z_{2} ^{2} = 0 does not imply z_{1} = z_{2} = 0.

### 3. Equality In Complex Number:

Two complex numbers z_{1} = a_{1} + ib_{1} & z_{2} = a_{2} + ib_{2} are equal if and only if their real and imaginary parts are equal respectively

i.e. z_{1} = z_{2 } **—>** Re(z_{1} ) = Re(z_{2} ) and I_{m} (z_{1} ) = I_{m} (z_{2} ).

### 4. Representation Of A Complex Number:

###### (a) Cartesian Form (Geometric Representation) :

### 5. Modulus of a Complex Number :

### 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 IV^{th} quadrant.

**Properties of arguments**

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

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

(iii) arg (z2 ) = 2arg(z) + 2m**π** 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(z_{2} – z_{1} ) = angle of the line segment

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### 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

### 8. Rotation theorem

(i) If P(z_{1} ) and Q(z_{z} ) are two complex numbers such that |z_{1} | = |z_{2} |, then z_{2} = z_{1} e_{iθ} where θ = ∠ POQ

(ii) If P(z_{1} ), Q(z_{2} ) and R(z_{3} ) are three complex numbers and ∠ PQR = θ, then

(iii) If P(z_{1} ), Q(z_{2} ), R(z_{3} ) and S(z_{4} ) are four complex numbers and ∠STQ = θ, then

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### 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 :

(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 :

### 14. Geometrical Properties :

### 15.

Free JEE Exam Preparation Group on Eckovation[/su_button### 16. Ptolemy’s Theorem: