IIT JEE Main Maths -Unit 2- Argand diagram representation- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 2- Argand diagram representation – Study Notes – New syllabus
IIT JEE Main Maths -Unit 2- Argand diagram representation – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Argand Diagram Representation
Argand Diagram Representation
The Argand diagram is the graphical representation of complex numbers in a plane. Each complex number \( z = a + ib \) is represented by a unique point \( P(a, b) \) in the plane called the Argand plane or complex plane.
Structure of Argand Plane:
- The horizontal axis (x-axis) represents the real part of the complex number.
- The vertical axis (y-axis) represents the imaginary part of the complex number.
- The origin \( O(0,0) \) represents the complex number \( 0 + 0i = 0 \).
Hence, a complex number \( z = a + ib \) corresponds to the point \( P(a, b) \) in the plane or the vector \( \overrightarrow{OP} \).
Geometrical Representation:
The complex number \( z = a + ib \) can be represented as a vector from the origin \( O(0, 0) \) to the point \( P(a, b) \).
The modulus and argument of the complex number correspond to the magnitude and direction of this vector.
- Modulus: \( |z| = OP = \sqrt{a^2 + b^2} \)
- Argument: \( \arg(z) = \theta = \tan^{-1}\left(\dfrac{b}{a}\right) \)
The argument \( \theta \) depends on the quadrant in which the point \( (a, b) \) lies:
| Quadrant | Sign of \( a, b \) | Range of \( \theta \) |
|---|---|---|
| I | \( a > 0, b > 0 \) | \( 0 < \theta < \dfrac{\pi}{2} \) |
| II | \( a < 0, b > 0 \) | \( \dfrac{\pi}{2} < \theta < \pi \) |
| III | \( a < 0, b < 0 \) | \( \pi < \theta < \dfrac{3\pi}{2} \) |
| IV | \( a > 0, b < 0 \) | \( \dfrac{3\pi}{2} < \theta < 2\pi \) |
Example
Represent the complex number \( z = 2 + 3i \) on the Argand diagram and find its modulus and argument.
▶️ Answer / Explanation
Step 1: \( a = 2, b = 3 \Rightarrow P(2, 3) \)
Step 2: Modulus: \( |z| = \sqrt{2^2 + 3^2} = \sqrt{13} \)
Step 3: Argument: \( \tan \theta = \dfrac{b}{a} = \dfrac{3}{2} \Rightarrow \theta = \tan^{-1}\left(\dfrac{3}{2}\right) \)
The point lies in the first quadrant. So, \( \theta = \tan^{-1}\left(\dfrac{3}{2}\right) \).
Conclusion: \( z \) is represented by the point \( P(2,3) \) and \( |z| = \sqrt{13}, \, \arg(z) = \tan^{-1}(3/2) \).
Example
Find the modulus and argument of \( z = -4 + 4i \) and represent it in the Argand plane.
▶️ Answer / Explanation
Step 1: \( a = -4, b = 4 \Rightarrow P(-4, 4) \)
Step 2: Modulus: \( |z| = \sqrt{(-4)^2 + 4^2} = 4\sqrt{2} \)
Step 3: Argument: \( \tan \theta = \dfrac{b}{a} = \dfrac{4}{-4} = -1 \Rightarrow \theta = 135^\circ \text{ (since 2nd quadrant)} \)
Step 4: Representation: \( z = 4\sqrt{2}(\cos 135^\circ + i\sin 135^\circ) \)
Example
Locate \( z = -2 – 2i \) on the Argand diagram and find its polar form.
▶️ Answer / Explanation
Step 1: \( a = -2, b = -2 \Rightarrow P(-2, -2) \)
Step 2: Modulus: \( |z| = \sqrt{(-2)^2 + (-2)^2} = 2\sqrt{2} \)
Step 3: Argument: \( \tan \theta = \dfrac{b}{a} = 1 \Rightarrow \theta = 45^\circ \)
Since both \( a, b < 0 \), \( z \) lies in the third quadrant → \( \theta = 180^\circ + 45^\circ = 225^\circ \text{ (or } \dfrac{5\pi}{4}) \)
Step 4: Polar form: \( z = 2\sqrt{2}(\cos 225^\circ + i\sin 225^\circ) \)