RD Sharma Class 11 Solutions Chapter 13 - Complex Numbers (Ex 13.1) Exercise 13.1

The solutions simplify higher powers of iota (i) by using its cyclic nature, which repeats every four powers. The standard approach is to divide the exponent by 4 and check the remainder (r).

• If r = 1 (e.g., i⁵), the value is i.
• If r = 2 (e.g., i⁶), the value is -1.
• If r = 3 (e.g., i⁷), the value is -i.
• If r = 0 (e.g., i⁸), the value is 1.

The additive inverse of a complex number is the number that, when added to the original number, results in zero (0 + 0i). For any complex number z = a + ib, its additive inverse is -z. To find it, you simply negate both the real and the imaginary parts.

• Step 1: Identify the real part (a) and the imaginary part (b) of the given complex number.
• Step 2: Change the sign of the real part from 'a' to '-a'.
• Step 3: Change the sign of the imaginary part from 'b' to '-b'.

It is necessary because a complex number is defined by two independent components: a real part and an imaginary part. A single equation involving complex numbers, like z₁ = z₂, is not a single algebraic equation but a compact way of writing two separate equations for real numbers. By equating the real parts, you ensure the numbers have the same position on the real axis, and by equating the imaginary parts, you ensure they have the same position on the imaginary axis. You cannot solve for two variables (like x and y) from a single equation without this separation, as the real and imaginary dimensions are orthogonal and do not mix.

To successfully solve problems in Exercise 13.1, a student must master three foundational concepts from the chapter:

• The concept of iota (i): Understanding that i = √-1 and knowing the cyclic values of its powers (i, -1, -i, 1) is crucial for simplification.
• Standard Form (a + ib): The ability to express any complex expression in the standard a + ib form by separating real and imaginary terms.
• Equality of Complex Numbers: The principle that two complex numbers are equal only if their corresponding real parts are equal and their imaginary parts are equal. This is the key to solving for unknown variables.

The basic operations in Exercise 13.1, such as addition and subtraction, are algebraic representations of geometric operations on the Argand Plane. For example:

• Addition (z₁ + z₂): Mastering the algebraic addition ( (a+c) + i(b+d) ) is the prerequisite for understanding vector addition of complex numbers on the Argand plane, where the resultant complex number is found using the parallelogram law.
• Standard Form (a + ib): Consistently converting to this form helps you identify the coordinates (a, b) needed to plot the complex number on the Argand plane.