Welcome to this module on algebraic operations with complex numbers. Understanding these operations is fundamental for success in competitive exams like the JEE, where complex numbers frequently appear in calculus and algebra problems. We'll cover addition, subtraction, multiplication, and division, building a strong foundation for more advanced topics.

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying the equation $i^2 = -1$. The term '$a$' is called the real part, and '$b$' is called the imaginary part of the complex number.

Just like real numbers, complex numbers can be added, subtracted, multiplied, and divided. These operations follow specific rules that ensure consistency and allow us to manipulate complex number expressions effectively.

To add or subtract complex numbers, we add or subtract their corresponding real and imaginary parts separately. If we have two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, then:

$z_1 + z_2 = (a + c) + (b + d)i$ $z_1 - z_2 = (a - c) + (b - d)i$

Multiplying complex numbers is similar to multiplying binomials, remembering that $i^2 = -1$. For $z_1 = a + bi$ and $z_2 = c + di$:

$z_1 \times z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2$ Since $i^2 = -1$, this becomes: $z_1 \times z_2 = (ac - bd) + (ad + bc)i$

To divide complex numbers, we use the concept of the conjugate. The conjugate of a complex number $a + bi$ is $a - bi$. We multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

For $z_1 = a + bi$ and $z_2 = c + di$, the division $\frac{z_1}{z_2}$ is calculated as:

(\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i)

Consistent practice with these algebraic operations is crucial for competitive exams. Many problems involve combining these operations, so ensure you are comfortable with each step. Look for opportunities to apply these concepts in calculus problems involving derivatives or integrals of functions that might implicitly involve complex numbers or their properties.