Complex Numbers JEE Notes | Revision IIT JEE Notes on Complex Numbers

Complex numbers JEE notes contain all the formulas and important points which are very important for the JEE entrance exam. Students can expect 2-3 questions from this topic for any entrance exam. Since it is not possible to revise complete books when the exams are around the corner, we have come up with complex numbers notes for IIT JEE. You can also download complex numbers IIT JEE notes PDF for free. Complex numbers IIT JEE notes will definitely help students to have a quick, complete revision.

A complex number is a number expressed in the form z = a+ib, where a and b are real numbers and i = √-1 (iota). The real part of z is given by Re(z) = a, and the imaginary part of z is given by Im(z) = b. The complex number is purely imaginary if Re(z) = 0. The complex number is purely real if Im(z) = 0. A complex number z can also be defined as an ordered pair of real numbers and can be denoted by (a, b). If we denote z = (a, b), then the real part of z is a, and the imaginary part of z is b.

Table of Contents

Algebraic Operations with Complex Numbers
Conjugate of a Complex Number
Modulus of a Complex Number
Integral Powers of Iota
Square Root of a Complex Number
Amplitude of a Complex Number
Representation of a Complex Number
De-Moivre’s Theorem
Geometry of Complex Numbers
Solved Examples
Practice Problems
FAQs

Complex Numbers IIT JEE Notes

Algebraic Operations with Complex Numbers

1. Addition: (a+ib) + (c+id) = (a+c) + i(b+d)

2. Subtraction: (a+ib) – (c+id) = (a-c) + i(b-d)

3. Multiplication: (a+ib)(c+id) = (ac-bd) + i(ad+bc)

4. Division: (a+ib)/(c+id) = (ac + bd)/(c² + d²) + i (bc – ad)/(c² + d²) (at least one of c and d is non zero)

Properties of Algebraic Operations

a) z₁z₂ = z₂z₁

b) z₁ + z₂ = z₂ + z₁

c) (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)

d) (z₁z₂)z₃ = z₁(z₂z₃)

e) z+0 = z and z.1 = 1.z = z. (0 and 1 are the identity elements for addition and multiplication, respectively.)

f) Additive inverse of z is -z.

g) Multiplicative inverse of z is 1/z.

h) z₁(z₂+z₃) = z₁z₂ + z₁z₃ and (z₂+z₃) z₁ = z₂z₁ + z₃z₁

Conjugate of a Complex Number

The conjugate of complex number z = a+ib is given by

\(\begin{array}{l}\overline{z}=a-ib\end{array} \)

.

Important Properties of Conjugate of a Complex Number

1	\(\begin{array}{l}\bar{\bar{z}}= z\end{array} \)
2	\(\begin{array}{l}{{z}_{1}}={{z}_{2}}\Leftrightarrow {{\overline{z}}_{1}}={{\overline{z}}_{2}}\end{array} \)
3	\(\begin{array}{l}\overline{\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)}=\frac{{{\overline{z}}_{1}}}{{\overline{z}_{2}}}\end{array} \) , if z₂ ≠ 0
4	\(\begin{array}{l}z+\overline{z}=2\,{Re}(z)\end{array} \)
5	\(\begin{array}{l}z-\overline{z}=2i\,{Im}\,(z)\end{array} \)
6	\(\begin{array}{l}z=\overline{z}\Leftrightarrow z\end{array} \) is purely real
7	\(\begin{array}{l}z+\overline{z}=0\Leftrightarrow z\end{array} \) is purely imaginary.
8	\(\begin{array}{l}z\overline{z}={{[Re\,(z)]}^{2}}+{{[Im(z)]}^{2}}=a^{2}+b^{2}=\left \| z \right \|^{2}\end{array} \)
9	\(\begin{array}{l}\overline{{{z}_{1}}+{{z}_{2}}}={{\overline{z}}_{1}}+{{\overline{z}}_{2}}\end{array} \)
10	\(\begin{array}{l}\overline{{{z}_{1}}-{{z}_{2}}}={{\overline{z}}_{1}}-{{\overline{z}}_{2}}\end{array} \)
11	\(\begin{array}{l}\overline{{{z}_{1}}{{z}_{2}}}={{\overline{z}}_{_{1}}}{{\overline{z}}_{_{2}}}\end{array} \)
12	\(\begin{array}{l}\overline{re^{i\theta }}=re^{-i\theta }\end{array} \)

Modulus of a Complex Number

If z = x+iy, then the modulus of z is denoted by |z| = √(x²+y²).

|z| is a non-negative real number.

Important Properties of Modulus of a Complex Number

1. |z| ≥ 0 and |z| = 0 iff x = 0 and y = 0 (z = 0)

2. -|z| ≤ Re(z) ≤ |z|

\(\begin{array}{l}3.\ |z| = \left | \overline{z} \right |= |-z| =\left | -\overline{z} \right |\end{array} \)

\(\begin{array}{l}4.\ z\overline{z} =\left | z \right |^{2}\end{array} \)

\(\begin{array}{l}5.\ z^{-1} = \frac{\overline{z}}{\left | z \right |^{2}}\end{array} \)

6. |z²| = |z|² or |zⁿ| = |z|ⁿ, n belongs to N.

7. |z₁z₂….z_n| = |z₁||z₂|….|z_n|

8. |z₁ + z₂| ≤ |z₁| + |z₂|

9. |z₁ – z₂| ≥ ||z₁| – |z₂||

10. |z₁ + z₂|² + |z₁ – z₂|² = 2(|z₁|² + |z₂|²)

\(\begin{array}{l}11.\ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2Re(z_{1}\overline{z_{2}})\end{array} \)

Integral Powers of Iota (i)

We know that i = √-1.

i² = -1
i³ = -i
i⁴ = 1
The sum of four consecutive powers of i is zero.

Square Root of a Complex Number

Let z = x + iy

Then,

\(\begin{array}{l}\sqrt{x+iy}=\left\{ \begin{matrix} \pm \left[ \sqrt{\frac{|z|+x}{2}}+i\sqrt{\frac{|z|-x}{2}} \right]if\,y\,>\,0 \\ \pm \left[ \sqrt{\frac{|z|+x}{2}}-i\sqrt{\frac{|z|-x}{2}} \right]if\,y\,<\,0 \\ \end{matrix} \right.\end{array} \)

Where,

\(\begin{array}{l}|x| =\sqrt{{{x}^{2}}+{{y}^{2}}}\,\end{array} \)

(i)

\(\begin{array}{l}\sqrt{x+iy}+\sqrt{x-iy}=\sqrt{2|z|+2x}\end{array} \)

(ii)

\(\begin{array}{l}\sqrt{x+iy}-\sqrt{x-iy}=i\sqrt{2|z|-2x}\end{array} \)

(iii)

\(\begin{array}{l}\sqrt{i}=\pm \left( \frac{1+i}{\sqrt{2}} \right)\,and\,\sqrt{-i}=\pm \left( \frac{1-i}{\sqrt{2}} \right)\end{array} \)

Amplitude of a Complex Number

If z = x+iy, then amplitude or argument of z is given by amp(z) = tan^-1(y/x).

To find the argument of z, find the quadrant in which the complex number belongs and then find the angle θ and amplitude using the figure below.

Amplitude of a Complex Number

Note: (a) The principal value of any complex number lies between -π <θ <π.

b) The amplitude of any complex number is a many-valued function. If θ is the argument of a complex number, then (2nπ + θ) is also the argument of a complex number.

c) If the complex number is multiplied by i (iota), then the amplitude will be increased by π/2, and if it is multiplied by -i, then the amplitude will be decreased by π/2.

d) Argument of 0 is not defined.

e) In the first and second quadrants, the amplitude is always positive. In the third and fourth quadrants, the amplitude is always negative.

Representation of a Complex Number

Cartesian Representation

The complex number z = x+iy is represented by a point P whose coordinates are referred to the rectangular axis xox’ and yoy’, which are called real and imaginary axes, respectively. A complex number z is represented by a point in a plane. There exists a complex number corresponding to every point in this plane. Such a plane is called the Argand plane or Argand diagram.

Cartesian Representation of Complex Number

(i) Distance of any complex number from the origin is called the modulus of a complex number. Thus,

\(\begin{array}{l}\left | z \right |=\sqrt{x^{2}+y^{2}}\end{array} \)

.

(ii) Angle of any complex number with a positive direction of the x-axis is called amplitude or argument of z. amp(z) = arg(z) = θ = tan^-1(y/x).

Polar Form: If z = x+iy is a complex number, then the polar form is given by,

z = r(cos θ + i sin θ), where x = r cos θ, y = r sin θ and r = √(x²+y²) = |z|.

Exponential Form: If z = x+iy is a complex number, then z = re^iθis the exponential form. Here, r is the modulus, and θ is the amplitude of the complex number.

Vector Representation: If z = x+iy is a complex number, then its vector representation is given by

\(\begin{array}{l}z=\overrightarrow{OP}\end{array} \)

.

De-Moivre’s Theorem

If n ∈ Z(set of integers), then (cos θ + i sin θ)ⁿ = (cos nθ + i sin nθ)

Or (e^iθ)ⁿ = e^inθ

De Moivre’s Theorem is used to find the roots of complex numbers.

Cube Roots of Unity

The three cube roots of unity are 1, (-½ + √3i/2) and (-½ – √3i/2). 1 is a real number, and the other two are complex conjugate numbers, and they are called imaginary cube roots of unity.

We write ω = -½ + √3i/2

Note: a) 1+ω+ω² = 0

b) ω³ = 1

c) ω³ⁿ = 1

d) ω³ⁿ⁺¹ = ω

e) ω³ⁿ⁺² = ω²

Geometry of Complex Numbers

a) Distance formula: Let z₁ = x₁+iy₁ and z₂ = x₂+iy₂ be two complex numbers represented by points P and Q, respectively, in the argand plane, then PQ = √[(x₂ – x₁)² + (y₂-y₁)²]

= |(x₂-x₁) + i(y₂-y₁)|

= |z₂-z₁|

b) Section formula: If the line segment joining A(z₁) and B(z₂) is divided by the point P(z) internally in the ratio m₁:m_2, then z = (m₁z₂+m₂z₁)/(m₁+m₂).

If P divides AB externally in the ratio m₁:m₂, then z = (m₁z₂-m₂z₁)/(m₁-m₂).

If P is the midpoint of AB, then z = (z₁+z₂)/2.

c) Condition for collinearity: Three points z₁, z₂ and z₃ will be collinear if there exists a relation az₁ + bz₂ + cz₃ = 0 (a, b and c are real), such that a+b+c = 0.

In other words, three points z₁, z₂ and z₃ are collinear, if

\(\begin{array}{l}\begin{vmatrix} z_{1} & \bar{z_{1}}&1 \\ z_{2} & \bar{z_{2}}& 1\\ z_{3}& \bar{z_{3}}& 1 \end{vmatrix}=0\end{array} \)

d) Area of a triangle: The area of triangle ABC with vertices A(z₁), B(z₂) and C(z₃) is given by

\(\begin{array}{l}\Delta =\frac{1}{4}\left | \begin{vmatrix} z_{1} & \bar{z_{1}}&1 \\ z_{2} & \bar{z_{2}}& 1\\ z_{3}& \bar{z_{3}}& 1 \end{vmatrix} \right |\end{array} \)

e) Equation of a straight line: Equation of a straight line through z₁ and z₂ is given by

\(\begin{array}{l}\frac{z-z_{1}}{z_{2}-z_{1}}=\frac{\bar{z}-\bar{z_{1}}}{\bar{z_{2}}-\bar{z_{1}}}\end{array} \)

\(\begin{array}{l}\Rightarrow \begin{vmatrix} z & \bar{z}&1 \\ z_{1} & \bar{z_{1}}& 1\\ z_{2}& \bar{z_{2}}& 1 \end{vmatrix}=0\end{array} \)

f) If ABC is an equilateral triangle having vertices z₁, z₂, z_3, then z₁²+ z₂²+ z₃² = z₁z₂ + z₂z₃ + z₃z₁.

g) If z₁, z₂, z₃, z₄ are vertices of parallelogram, then z₁+z₃ = z₂+z₄.

Solved Examples

Example 1:

(i¹⁹ + (1/i)²⁵)² equals

a) i

b) -i

c) 4

d) -4

Solution:

(i¹⁹ + (1/i)²⁵)² = (i³ + i³)²

= (2i³)²

= 4i⁶

= 4i²

= -4

Hence, option d is the answer.

Example 2:

The multiplicative inverse of z = 3-2i is

a) (3+2i)/13

b) -(3+2i)/13

c) (3-2i)/√13

d) -(3+2i)/√13

Solution:

Given z = 3-2i

z^-1 = 1/z

= 1/(3-2i)

Multiply the numerator and denominator with (3+2i)

=> (3+2i)/(3-2i)(3+2i)

= (3+2i)/(9+4)

= (3+2i)/13

Hence, option a is the answer.

Example 3:

If (1 + i)(1 + 2i)(1 + 3i) ….. (1 + ni) = a + ib, then what is 2 × 5 × 10….(1 + n²) is equal to?

a) a²-b²

b) a²+b²

c) ab

d) none of these

Solution:

We have (1 + i)(1 + 2i)(1 + 3i) ….. (1 + ni) = a + ib …..(i)

(1 – i) (1 – 2i) (1 – 3i) ….. (1 – ni) = a – ib …..(ii)

Multiplying (i) and (ii),

We get 2 × 5 × 10 ….. (1 + n²) = a² + b²

Hence, option b is the answer.

Practice Problems

1. If ω is an imaginary cube root of unity, then the value of the expression is

2(1 + ω) (1 + ω²) + 3(2 + ω) (2 + ω²) + … + (n + 1) (n + ω) (n + ω²) is

a) ¼ n²(n+1)²+n

b) ¼ n²(n+1)²-n

c) ¼ n(n+1)²-n

d) none of these

2. If z is a complex number of unit modulus and argument θ, then arg(1+z)/(1+z bar) equals

a) -θ

b) θ

c) π/2 – θ

d) π – θ

3. If ((1+i)/(1-i))^x = 1, then

a) x = 4n, where n is any positive integer

b) x = 2n, where n is any positive integer

c) x = 4n+1, where n is any positive integer

d) x = 2n+1, where n is any positive integer

4. If |z² – 1| = |z²| + 1, then z lies on

a) circle

b) ellipse

c) real axis

d) imaginary axis

5. If z and ω are complex numbers such that |zω| = 1, arg⁡(z) – arg⁡(ω) = (3π )/2, then find arg

\(\begin{array}{l}\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega}\end{array} \)

a) π/4

b) -π/4

c) 3π/4

d) -3π/4

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Frequently Asked Questions

Q1

How to find the conjugate of a complex number?

To find the conjugate of a complex number, we have to change the sign of the imaginary part of the complex number. For example, the conjugate of 2+6i is 2-6i.

Q2

What is the modulus of a complex number?

The modulus of a complex number z is denoted by |z| = √(x²+y²).

Q3

What is meant by the amplitude of a complex number?

If z = x+iy, then amplitude or argument of z is given by amp(z) = tan^-1(y/x).

Q4

What is the multiplicative identity of complex numbers?

The multiplicative identity of complex numbers is (1+0i).
For example, we have z = 2+3i.
(2+3i)(1+0i) = 2+3i.

Q5

Is the modulus of a complex number always positive?

Yes, the modulus of a complex number is a non-negative real number.

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