1st Year Math Chapter 1 Exercise 1.1 Notes: Complex Numbers (Punjab Board 2026-27)
Complete concept notes, formulas, solved examples, and practice questions for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Chapter 1 of the Mathematics 11 textbook (Punjab Education, Curriculum, Training and Assessment Authority — PECTAA, formerly PTB) is Complex Numbers, the opening unit of the 14-chapter 1st Year (ICS Part 1 / FSc Part 1 / XI) Mathematics course. Exercise 1.1 introduces the foundational ideas the rest of the chapter builds on: what a complex number is, the imaginary unit, real and imaginary parts, the conjugate, the modulus, and how complex numbers are represented on the complex plane.
What Does Exercise 1.1 Cover?
Exercise 1.1 is where complex numbers are first defined and worked with directly. Before this point, square roots of negative numbers had no solution within real numbers; this exercise introduces the number system that resolves that gap and gives students practice performing basic operations — addition, subtraction, and multiplication — on complex numbers.
Key Concepts
1. What Is a Complex Number?
A complex number is a number of the form z = a + bi, where a and b are real numbers and i is the imaginary unit. Here, a is called the real part of z, written Re(z), and b is called the imaginary part, written Im(z).
2. The Imaginary Unit i
The imaginary unit i is defined by the property i² = -1, which means i = √-1. This single definition is what allows square roots of negative numbers to be expressed and worked with.
3. Real and Imaginary Parts
For z = a + bi: the real part is a (a real number on its own), and the imaginary part is b (also a real number — note that the imaginary part is the coefficient of i, not b i itself).
4. Conjugate of a Complex Number
The conjugate of z = a + bi, written z̄ (z bar), is a – bi — the same real part, with the sign of the imaginary part reversed. Conjugates are especially useful for simplifying expressions and dividing complex numbers.
5. Modulus of a Complex Number
The modulus (or absolute value) of z = a + bi, written |z|, measures its distance from the origin and is calculated as |z| = √(a² + b²).
6. The Complex Plane (Argand Diagram)
A complex number a + bi can be plotted as the point (a, b) on a two-dimensional plane called the complex plane or Argand diagram, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Powers of i
Powers of i repeat in a cycle of four, which makes simplifying higher powers straightforward once the pattern is known:
| Power | Value |
| i¹ | i |
| i² | -1 |
| i³ | -i |
| i⁴ | 1 |
For any higher power, divide the exponent by 4 and use the remainder to find the equivalent value from this cycle (e.g., i¹⁰ = i^(4×2+2) = i²= -1).
Basic Operations on Complex Numbers
For z = a + bi and w = c + di, the core operations follow directly from treating i as a variable and then simplifying using i² = -1:
| Operation | Formula |
| Addition | z + w = (a + c) + (b + d)i |
| Subtraction | z − w = (a − c) + (b − d)i |
| Multiplication | z × w = (ac − bd) + (ad + bc)i |
| Conjugate | z̄ = a − bi |
| Modulus | |z| = √(a² + b²) |
Solved Examples
Example 1: Simplify (3 + 2i) + (5 − 4i).
Add the real parts: 3 + 5 = 8.
Add the imaginary parts: 2 + (−4) = −2.
Answer: 8 − 2i
Example 2: Multiply (2 + 3i)(1 − 2i).
Expand using distribution: (2)(1) + (2)(−2i) + (3i)(1) + (3i)(−2i).
This gives: 2 − 4i + 3i − 6i².
Since i² = −1, replace −6i² with +6.
Combine real parts (2 + 6 = 8) and imaginary parts (−4i + 3i = −i).
Answer: 8 − i
Example 3: Find the conjugate and modulus of z = −3 + 4i.
Conjugate: reverse the sign of the imaginary part → −3 − 4i.
Modulus: |z| = √((−3)² + 4²) = √(9 + 16) = √25.
Answer: Conjugate = −3 − 4i, Modulus = 5
Sample MCQs
1. The imaginary unit i is defined such that i² equals:
a) 1 b) −1 c) 0 d) i
Answer: b) −1
2. In the complex number 3 − 4i, the imaginary part is:
a) 3 b) −4 c) 4i d) −4i
Answer: b) −4
3. The conjugate of 5 + 2i is:
a) 5 − 2i b) −5 + 2i c) −5 − 2i d) 2 + 5i
Answer: a) 5 − 2i
4. The modulus of the complex number 3 + 4i is:
a) 3 b) 4 c) 5 d) 7
Answer: c) 5
5. The value of i⁴ is:
a) i b) −1 c) −i d) 1
Answer: d) 1
Important Short Questions
- Define a complex number and identify its real and imaginary parts with an example.
- State the value of i² and explain how it is used to simplify higher powers of i.
- Define the conjugate of a complex number and give one example.
- What is the modulus of a complex number, and how is it calculated?
- Explain how a complex number is represented on the complex plane (Argand diagram).
Important Long Questions
- If z = 4 + 3i and w = 1 − 2i, find z + w, z − w, and z × w, showing each step.
- Simplify i¹⁷ by using the repeating cycle of powers of i, and explain the method used.
- Find the conjugate and modulus of z = 5 − 12i, and verify that z × z̄ equals |z|².
- Plot the complex numbers 2 + 3i and −1 − 2i on an Argand diagram and describe their positions relative to the origin.
How to Approach Exercise 1.1 Effectively
- Always simplify i² to −1 as the very first step in any multiplication — this is where most calculation errors happen.
- Keep real and imaginary parts in separate columns when adding or subtracting, to avoid mixing them up.
- Memorize the four-value cycle of powers of i (i, −1, −i, 1) so higher powers can be simplified quickly using the remainder method.
- Practice finding the conjugate and modulus as a pair — they’re often asked together and use the same a, b values.
- Sketch a quick Argand diagram for any question that mentions position, distance, or the complex plane, even if not explicitly asked to draw one.
FAQs
Q: What is Exercise 1.1 in 1st Year Math about?
A: It covers the foundational ideas of complex numbers — their definition, the imaginary unit i, real and imaginary parts, conjugates, modulus, and basic operations like addition, subtraction, and multiplication.
Q: Why is i² equal to −1?
A: This is the defining property of the imaginary unit i; it’s what allows square roots of negative numbers to be expressed as real multiples of i.
Q: What is the difference between the imaginary part and an imaginary number?
A: The imaginary part of a + bi is the real number b (the coefficient), while an imaginary number specifically refers to a number of the form bi on its own, with no real part.
Q: Is this exercise mostly calculation-based or definition-based?
A: Both — expect short questions asking for definitions (conjugate, modulus, imaginary unit) as well as problems requiring you to add, subtract, or multiply complex numbers step by step.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 1: Complex Numbers, Exercise 1.1.