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

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 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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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 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

 

 

9. Demoivre’s Theorem:

10. Cube Root Of Unity :

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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 :

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14. Geometrical Properties :

15.

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16. Ptolemy’s Theorem:

 

 

 

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