Introduction
Complex numbers are a crucial topic in the JEE Main Mathematics syllabus. They extend the real number system $\mathbb{R}$ to the complex number system $\mathbb{C}$, which allows for the solution of equations that have no real solutions. This study note will delve into the fundamental concepts, operations, and properties of complex numbers, ensuring that each idea is clearly explained and supported by examples.
Definition of Complex Numbers
A complex number is a number of the form $z = a + bi$, where:
- $a$ and $b$ are real numbers.
- $i$ is the imaginary unit, defined by the property $i^2 = -1$.
Real and Imaginary Parts
- The real part of $z$ is $a$, denoted as $\text{Re}(z) = a$.
- The imaginary part of $z$ is $b$, denoted as $\text{Im}(z) = b$.
Consider the complex number $z = 3 + 4i$. Here, $\text{Re}(z) = 3$ and $\text{Im}(z) = 4$.
Algebra of Complex Numbers
Addition and Subtraction
- Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
- Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$
Add $z_1 = 2 + 3i$ and $z_2 = 1 + 4i$: $$ z_1 + z_2 = (2 + 3i) + (1 + 4i) = 3 + 7i $$
Multiplication
- Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
Multiply $z_1 = 1 + 2i$ and $z_2 = 3 + 4i$: $$ z_1 \cdot z_2 = (1 + 2i)(3 + 4i) = (1 \cdot 3 - 2 \cdot 4) + (1 \cdot 4 + 2 \cdot 3)i = -5 + 10i $$
Division
- Division: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$
Divide $z_1 = 1 + i$ by $z_2 = 1 - i$: $$ \frac{1 + i}{1 - i} = \frac{(1 + i)(1 + i)}{(1 - i)(1 + i)} = \frac{1 + 2i + i^2}{1 - i^2} = \frac{1 + 2i - 1}{1 + 1} = \frac{2i}{2} = i $$
TipWhen dividing complex numbers, always multiply by the conjugate of the denominator to simplify the expression.
Conjugate of a Complex Number
The conjugate of a complex number $z = a + bi$ is $\bar{z} = a - bi$. It is useful in various operations, especially in division.
ExampleThe conjugate of $z = 3 + 4i$ is $\bar{z} = 3 - 4i$.
Modulus of a Complex Number
The modulus of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$. It represents the distance of the complex number from the origin in the complex plane.
ExampleFind the modulus of $z = 3 + 4i$: $$ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 $$
Argument of a Complex Number
The argument of a complex number $z = a + bi$, denoted as $\arg(z)$, is the angle $\theta$ such that $\tan(\theta) = \frac{b}{a}$.
NoteThe argument is typically measured in radians and can have multiple values differing by $2\pi$.
ExampleFind the argument of $z = 1 + i$: $$ \arg(z) = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4} $$
Polar Form of Complex Numbers
A complex number can be represented in polar form as $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$ and $\theta = \arg(z)$. This is also written as $z = re^{i\theta}$ using Euler's formula.
ExampleConvert $z = 1 + i$ to polar form: $$ r = \sqrt{1^2 + 1^2} = \sqrt{2}, \quad \theta = \frac{\pi}{4} $$ So, $z = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$ or $z = \sqrt{2} e^{i\pi/4}$.
De Moivre's Theorem
De Moivre's theorem states that for any real number $\theta$ and any integer $n$: $$ (\cos \theta + i \sin \theta)^n = \cos (n\theta) + i \sin (n\theta) $$
ExampleUsing De Moivre's theorem to find $(1 + i)^4$: $$ z = \sqrt{2} e^{i\pi/4}, \quad z^4 = (\sqrt{2} e^{i\pi/4})^4 = 2^2 e^{i\pi} = 4(\cos \pi + i \sin \pi) = -4 $$
Roots of Complex Numbers
The $n$th roots of a complex number $z = re^{i\theta}$ are given by: $$ z_k = r^{1/n} e^{i(\theta + 2k\pi)/n}, \quad k = 0, 1, 2, \ldots, n-1 $$
ExampleFind the cube roots of $8$: $$ z = 8 = 8 e^{i \cdot 0}, \quad \text{roots are} \quad 2 e^{i \cdot 0}, \quad 2 e^{i \frac{2\pi}{3}}, \quad 2 e^{i \frac{4\pi}{3}} $$
Complex Plane and Geometric Interpretation
The complex plane, also known as the Argand plane, is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of complex numbers.
Conclusion
Understanding complex numbers is essential for solving various mathematical problems, especially in JEE Main Mathematics. Mastery of their properties, operations, and geometric interpretations will provide a strong foundation for tackling complex problems.
Common MistakeA common mistake is to forget the imaginary unit $i$ when performing operations. Always ensure to handle $i^2 = -1$ correctly.
TipPractice converting between rectangular and polar forms of complex numbers, as this skill is frequently tested in exams.
By breaking down each concept and practicing with examples, you can develop a solid understanding of complex numbers and their applications.
