IIT JEE Main Maths -Unit 2- Complex numbers as ordered pairs of reals- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 2- Complex numbers as ordered pairs of reals – Study Notes – New syllabus
IIT JEE Main Maths -Unit 2- Complex numbers as ordered pairs of reals – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Complex Numbers and their Representation
- Polar and Exponential Form of Complex Numbers
Complex Numbers and their Representation
A complex number is a number of the form \( z = a + ib \), where:
- \( a \) = real part of \( z \), denoted by \( \text{Re}(z) \)
- \( b \) = imaginary part of \( z \), denoted by \( \text{Im}(z) \)
- \( i \) is the imaginary unit such that \( i^2 = -1 \)
Thus, every complex number can be written as an ordered pair \( (a, b) \).
Set of Complex Numbers: The set of all complex numbers is denoted by \( \mathbb{C} \).
\( \mathbb{C} = \{ a + ib \mid a, b \in \mathbb{R} \} \)
Equality of Complex Numbers:
Two complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \) are equal if and only if their real and imaginary parts are equal, i.e.,
\( a = c \) and \( b = d \).
Representation of Complex Numbers:
A complex number \( z = a + ib \) can be represented geometrically by a point \( P(a, b) \) in the Argand Plane (or complex plane), where:
- The x-axis represents the real part \( a \)
- The y-axis represents the imaginary part \( b \)
Modulus: The distance of the point \( P(a,b) \) from the origin is called the modulus of \( z \), denoted by \( |z| \).
\( |z| = \sqrt{a^2 + b^2} \)
Argument (Amplitude): The angle \( \theta \) made by the line joining \( P(a,b) \) to the origin with the positive x-axis is called the argument (or amplitude) of \( z \).
\( \tan \theta = \dfrac{b}{a} \)
Polar Form:
A complex number can also be expressed as
\( z = r(\cos \theta + i \sin \theta) \)
where \( r = |z| \) and \( \theta = \arg(z) \).
This is known as the polar form or trigonometric form of a complex number.
Example
Find the modulus and argument of the complex number \( z = 3 + 4i \).
▶️ Answer / Explanation
Step 1: Identify \( a = 3 \), \( b = 4 \).
Step 2: Find modulus:
\( |z| = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = 5 \)
Step 3: Find argument:
\( \tan \theta = \dfrac{b}{a} = \dfrac{4}{3} \Rightarrow \theta = \tan^{-1}\left(\dfrac{4}{3}\right) \)
Hence, \( |z| = 5 \) and \( \arg(z) = \tan^{-1}(4/3) \).
Example
Express \( z = -1 + i\sqrt{3} \) in polar form.
▶️ Answer / Explanation
Step 1: Identify \( a = -1 \), \( b = \sqrt{3} \).
Step 2: Find modulus:
\( r = \sqrt{a^2 + b^2} = \sqrt{1 + 3} = 2 \)
Step 3: Find argument:
\( \tan \theta = \dfrac{b}{a} = \dfrac{\sqrt{3}}{-1} = -\sqrt{3} \Rightarrow \theta = 120^\circ \text{ (or } \dfrac{2\pi}{3} \text{ radians)} \)
Step 4: Write in polar form:
\( z = 2(\cos 120^\circ + i\sin 120^\circ) \)
Example
Find the modulus and argument of \( z = -3 – 3i \), and express it in polar form.
▶️ Answer / Explanation
Step 1: \( a = -3 \), \( b = -3 \).
Step 2: Modulus:
\( r = \sqrt{a^2 + b^2} = \sqrt{9 + 9} = 3\sqrt{2} \)
Step 3: Argument:
\( \tan \theta = \dfrac{b}{a} = \dfrac{-3}{-3} = 1 \Rightarrow \theta = 45^\circ \).
But since \( z \) lies in the third quadrant, \( \theta = 180^\circ + 45^\circ = 225^\circ \text{ (or } \dfrac{5\pi}{4} \text{ radians)} \)
Step 4: Polar form:
\( z = 3\sqrt{2}(\cos 225^\circ + i\sin 225^\circ) \)
Polar and Exponential Form of Complex Numbers
Every complex number \( z = a + ib \) can be represented in different forms. In the polar form, it is expressed in terms of its modulus and argument.
1. Polar Form
Let \( z = a + ib \). Then:
\( r = |z| = \sqrt{a^2 + b^2} \)
\( \theta = \arg(z) = \tan^{-1}\!\left(\dfrac{b}{a}\right) \)
Hence, the polar form is:
\( z = r(\cos \theta + i \sin \theta) \)
Here:
- \( r \) → modulus (distance from origin)
- \( \theta \) → argument (angle with positive x-axis, measured anticlockwise)
2. Exponential Form (Euler’s Form)
By Euler’s formula: \( e^{i\theta} = \cos \theta + i \sin \theta \)
So the exponential form of a complex number is:
\( z = r e^{i\theta} \)
This form is especially useful in multiplication, division, and powers of complex numbers.
3. Conversion Between Forms
(i) Cartesian → Polar: Given \( z = a + ib \), find \( r = \sqrt{a^2 + b^2} \) and \( \theta = \tan^{-1}\!\left(\dfrac{b}{a}\right) \).
(ii) Polar → Cartesian: Given \( z = r(\cos \theta + i \sin \theta) \), then \( a = r \cos \theta \) and \( b = r \sin \theta \).
Example
Convert \( z = 1 + i \) into polar and exponential form.
▶️ Answer / Explanation
Step 1: Find modulus: \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \)
Step 2: Find argument: \( \theta = \tan^{-1}\!\left(\dfrac{1}{1}\right) = 45^\circ = \dfrac{\pi}{4} \)
Step 3: Polar form: \( z = \sqrt{2}(\cos \dfrac{\pi}{4} + i \sin \dfrac{\pi}{4}) \)
Step 4: Exponential form: \( z = \sqrt{2} e^{i\pi/4} \)
Example
Express \( z = -1 + i\sqrt{3} \) in exponential form.
▶️ Answer / Explanation
Step 1: Modulus: \( r = \sqrt{(-1)^2 + (\sqrt{3})^2} = 2 \)
Step 2: Argument: \( \tan \theta = \dfrac{\sqrt{3}}{-1} = -\sqrt{3} \)
Since \( a = -1 \) and \( b = +\sqrt{3} \), the point lies in the 2nd quadrant. So \( \theta = \pi – \dfrac{\pi}{3} = \dfrac{2\pi}{3} \).
Step 3: Exponential form: \( z = 2 e^{i(2\pi/3)} \)
Example
The complex number \( z = -3 – 3i \) is given. Find its polar and exponential forms.
▶️ Answer / Explanation
Step 1: Modulus: \( r = \sqrt{(-3)^2 + (-3)^2} = 3\sqrt{2} \)
Step 2: Argument: \( \tan \theta = \dfrac{-3}{-3} = 1 \Rightarrow \theta = 45^\circ \)
Since \( a, b < 0 \), \( z \) lies in the 3rd quadrant. Therefore, \( \theta = \pi + \dfrac{\pi}{4} = \dfrac{5\pi}{4} \).
Step 3: Polar form: \( z = 3\sqrt{2}(\cos \dfrac{5\pi}{4} + i \sin \dfrac{5\pi}{4}) \)
Step 4: Exponential form: \( z = 3\sqrt{2} e^{i(5\pi/4)} \)
Notes and Study Materials
- Concepts of Complex Numbers
- Complex Numbers Master File
- Complex Numbers Revision Notes
- Complex Numbers Formulae
- Complex Numbers Reference Book
- Complex Numbers Past Many Years Questions and Answer
Examples and Exercise
IIT JEE (Main) Mathematics ,”Complex Numbers” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Complex numbers as ordered pairs of reals.Representation of complex numbers in the form (a+ib) and their representation in a plane, Argand diagram. Algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number.Triangle inequality.
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