**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 = β-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)/(c^{2} + d^{2}) + i (bc – ad)/(c^{2} + d^{2}) (at least one of c and d is non zero)

**Properties of Algebraic Operations**

a) z_{1}z_{2} = z_{2}z_{1}

b) z_{1} + z_{2} = z_{2} + z_{1}

c) (z_{1} + z_{2}) + z_{3} = z_{1} + (z_{2} + z_{3})

d) (z_{1}z_{2})z_{3} = z_{1}(z_{2}z_{3})

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) z_{1}(z_{2}+z_{3}) = z_{1}z_{2} + z_{1}z_{3} and (z_{2}+z_{3}) z_{1} = z_{2}z_{1} + z_{3}z_{1}

### Conjugate of a Complex Number

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

**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 z_{2} β 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| = β(x^{2}+y^{2}).

|z| is a non-negative real number.

**Important Properties of Modulus of a Complex Number**

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

2. -|z| β€ Re(z) β€ |z|

6. |z^{2}| = |z|^{2} or |z^{n}| = |z|^{n}, n belongs to N.

7. |z_{1}z_{2}β¦.z_{n}| = |z_{1}||z_{2}|….|z_{n}|

8. |z_{1} + z_{2}| β€ |z_{1}| + |z_{2}|

9. |z_{1} – z_{2}| β₯ ||z_{1}| – |z_{2}||

10. |z_{1} + z_{2}|^{2} + |z_{1} – z_{2}|^{2} = 2(|z_{1}|^{2} + |z_{2}|^{2})

### Integral Powers of Iota (i)

We know that i = β-1.

- i
^{2}= -1 - i
^{3}= -i - i
^{4}= 1 - The sum of four consecutive powers of i is zero.

### Square Root of a Complex Number

Let z = x + iy

Then,

Where,

(i)

(ii)

(iii)

### 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 ΞΈ and amplitude using the figure below.

**Note:** (a) The principal value of any complex number lies between -Ο <ΞΈ <Ο.

b) The amplitude of any complex number is a many-valued function. If ΞΈ is the argument of a complex number, then (2nΟ + ΞΈ) 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 Ο/2, and if it is multiplied by -i, then the amplitude will be decreased by Ο/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 xoxβ and yoyβ, 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.

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

(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) = ΞΈ = tan^{-1}(y/x).

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

z = r(cos ΞΈ + i sin ΞΈ), where x = r cos ΞΈ, y = r sin ΞΈ and r = β(x^{2}+y^{2}) = |z|.

**Exponential Form:** If z = x+iy is a complex number, then z = re^{iΞΈ }is the exponential form. Here, r is the modulus, and ΞΈ is the amplitude of the complex number.

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

### De-Moivre’s Theorem

If n β Z(set of integers), then (cos ΞΈ + i sin ΞΈ)^{n} = (cos nΞΈ + i sin nΞΈ)

Or (e^{iΞΈ})^{n} = e^{inΞΈ}

De Moivreβs Theorem is used to find the roots of complex numbers.

#### Cube Roots of Unity

The three cube roots of unity are 1, (-Β½ + β3i/2) and (-Β½ – β3i/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 Ο = -Β½ + β3i/2

**Note:** a) 1+Ο+Ο^{2} = 0

b) Ο^{3} = 1

c) Ο^{3n} = 1

d) Ο^{3n+1} = Ο

e) Ο^{3n+2} = Ο^{2}

### Geometry of Complex Numbers

a)** Distance formula:Β **Let z_{1} = x_{1}+iy_{1} and z_{2} = x_{2}+iy_{2} be two complex numbers represented by points P and Q, respectively, in the argand plane, then PQ = β[(x_{2} – x_{1})^{2} + (y_{2}-y_{1})^{2}]

= |(x_{2}-x_{1}) + i(y_{2}-y_{1})|

= |z_{2}-z_{1}|

b)** Section formula: **If the line segment joining A(z_{1}) and B(z_{2}) is divided by the point P(z) internally in the ratio m_{1}:m_{2,} then z = (m_{1}z_{2}+m_{2}z_{1})/(m_{1}+m_{2}).

If P divides AB externally in the ratio m_{1}:m_{2}, then z = (m_{1}z_{2}-m_{2}z_{1})/(m_{1}-m_{2}).

If P is the midpoint of AB, then z = (z_{1}+z_{2})/2.

c) **Condition for collinearity: **Three points z_{1}, z_{2} and z_{3} will be collinear if there exists a relation az_{1} + bz_{2} + cz_{3} = 0 (a, b and c are real), such that a+b+c = 0.

In other words, three points z_{1}, z_{2} and z_{3} are collinear, if

d) **Area of a triangle: **The areaΒ of triangle ABC with vertices A(z_{1}), B(z_{2}) and C(z_{3}) is given by

e)** Equation of a straight line:** Equation of a straight line through z_{1} and z_{2} is given by

f) If ABC is an equilateral triangle having vertices z_{1}, z_{2}, z_{3,} then z_{1}^{2}+ z_{2}^{2}+ z_{3}^{2} = z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1}.

g) If z_{1}, z_{2}, z_{3}, z_{4} are vertices of parallelogram, then z_{1}+z_{3} = z_{2}+z_{4}.

## Solved Examples

**Example 1: **

(i^{19} + (1/i)^{25})^{2} equals

a) i

b) -i

c) 4

d) -4

**Solution:**

(i^{19} + (1/i)^{25})^{2} = (i^{3} + i^{3})^{2}

= (2i^{3})^{2}

= 4i^{6}

= 4i^{2}

= -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)/β13

d) -(3+2i)/β13

**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) β¦.. (1 + ni) = a + ib, then what is 2 Γ 5 Γ 10β¦.(1 + n^{2}) is equal to?

a) a^{2}-b^{2}

b) a^{2}+b^{2}

c) ab

d) none of these

**Solution:**

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

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

Multiplying (i) and (ii),

We get 2 Γ 5 Γ 10 β¦.. (1 + n^{2}) = a^{2} + b^{2}

Hence, option b is the answer.

## Practice Problems

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

2(1 + Ο) (1 + Ο^{2}) + 3(2 + Ο) (2 + Ο^{2}) + β¦ + (n + 1) (n + Ο) (n + Ο^{2}) is

a) ΒΌ n^{2}(n+1)^{2}+n

b) ΒΌ n^{2}(n+1)^{2}-n

c) ΒΌ n(n+1)^{2}-n

d) none of these

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

a) -ΞΈ

b) ΞΈ

c) Ο/2 – ΞΈ

d) Ο – ΞΈ

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 |z^{2} – 1| = |z^{2}| + 1, then z lies on

a) circle

b) ellipse

c) real axis

d) imaginary axis

5. If z and Ο are complex numbers such that |zΟ| = 1, argβ‘(z) – argβ‘(Ο) = (3Ο )/2, then find arg

a) Ο/4

b) -Ο/4

c) 3Ο/4

d) -3Ο/4

## Complex Numbers Rapid Revision for JEE

## Complex Numbers JEE Main Important Topics

## Complex Numbers JEE Main Important Questions

## Complex Numbers JEE Previous Year Questions

## Frequently Asked Questions

### 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.

### What is the modulus of a complex number?

The modulus of a complex number z is denoted by |z| = β(x^{2}+y^{2}).

### 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).

### 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.

### Is the modulus of a complex number always positive?

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

## Comments