11th Class Mathematics Chapter 1 Notes: Complex Numbers (Punjab Board)

Chapter at a glance

This chapter is the first unit of the 11th class Mathematics book and has five exercises in the book. Your teacher may also teach extra topics such as the polar form, so keep your own textbook and class notes beside these pages.

Key concepts and definitions

• Imaginary unit: i = √−1, so i² = −1. Then i³ = −i and i⁴ = 1.
• Complex number: a number of the form z = a + bi, where a and b are real numbers. a is the real part, Re(z), and b is the imaginary part, Im(z). Note that b itself is real; the imaginary part is not bi.
• Every real number is a complex number with b = 0. A number with a = 0 and b ≠ 0 is called purely imaginary.
• Equality: a + bi = c + di if and only if a = c and b = d.
• Conjugate: the conjugate of z = a + bi is z = a − bi. Only the sign of the imaginary part changes.
• Modulus: |z| = √(a² + b²), the distance of the point (a, b) from the origin.
• Additive inverse: −z = −a − bi. Multiplicative inverse: 1 ÷ z = z ÷ (a² + b²), for z not equal to 0.
• Argand diagram: the plane in which a + bi is the point (a, b). The horizontal axis is the real axis and the vertical axis is the imaginary axis.

Formulas and properties

Important derivations step by step

Powers of i

1. i² = −1, i³ = i² × i = −i, i⁴ = i² × i² = 1.
2. Any power can be split as i⁴ᵏ⁺ʳ = (i⁴)ᵏ × iʳ = 1 × iʳ.
3. So only the remainder r after dividing the power by 4 matters.
4. For a negative power, 1 ÷ i = i ÷ i² = −i. Use this, or write i⁻ⁿ = 1 ÷ iⁿ.

Product of a number and its conjugate

(a + bi)(a − bi) = a² − (bi)² = a² − b²i² = a² + b². This is a real number, which is why the conjugate is used to clear i from a denominator.

Division

1. Multiply the numerator and the denominator by the conjugate of c + di.
2. Top: (a + bi)(c − di) = ac − adi + bci − bdi² = (ac + bd) + (bc − ad)i.
3. Bottom: (c + di)(c − di) = c² + d².
4. Divide the real part and the imaginary part separately by c² + d².

Square root of a + bi

1. Let (x + yi)² = a + bi. Then x² − y² = a and 2xy = b.
2. Take the modulus on both sides: x² + y² = √(a² + b²). Call this value m.
3. Add and subtract: x² = (m + a) ÷ 2 and y² = (m − a) ÷ 2.
4. Choose the signs so that the product xy has the same sign as b.

Complex roots of a quadratic

For x² − 2x + 5 = 0: the discriminant is 4 − 20 = −16, so x = (2 ± √−16) ÷ 2 = (2 ± 4i) ÷ 2 = 1 ± 2i. The two roots are conjugates of each other when the coefficients are real.

Meaning on the Argand diagram

• Adding two numbers works like adding two vectors (the parallelogram rule).
• The conjugate is the mirror image in the real axis.
• The modulus is the length of the line from the origin to the point.

Solved numericals

Numerical 1: powers of i

i²⁷: 27 = 4 × 6 + 3, so i²⁷ = i³ = −i. i⁻¹⁰ = 1 ÷ i¹⁰; 10 = 4 × 2 + 2, so i¹⁰ = i² = −1, and i⁻¹⁰ = −1. i¹⁰⁰ = 1, since 100 is a multiple of 4. Also i + i² + i³ + i⁴ = i − 1 − i + 1 = 0.

Numerical 2: multiply and divide

(3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i.

(2 + 3i) ÷ (1 − i): multiply top and bottom by 1 + i. Top: (2 + 3i)(1 + i) = 2 + 2i + 3i + 3i² = −1 + 5i. Bottom: 1² + 1² = 2. Answer: −1/2 + (5/2)i. Check: (−1/2 + 5i/2)(1 − i) = −1/2 + i/2 + 5i/2 + 5/2 = 2 + 3i.

Numerical 3: conjugate, modulus and inverse

Let z = 3 − 4i. The conjugate is 3 + 4i. The modulus is √(9 + 16) = 5. The multiplicative inverse isz ÷ |z|² = (3 + 4i) ÷ 25 = 3/25 + (4/25)i.

Numerical 4: finding x and y

Find real x and y if (x + yi)(2 − 3i) = 4 + i. Expand: (2x + 3y) + (2y − 3x)i = 4 + i. Equate the parts: 2x + 3y = 4 and −3x + 2y = 1. Multiply the first by 3 and the second by 2: 6x + 9y = 12 and −6x + 4y = 2. Add: 13y = 14, so y = 14/13. Then 2x = 4 − 42/13 = 10/13, so x = 5/13. Check in the first equation: 2(5/13) + 3(14/13) = 52/13 = 4. Answer: x = 5/13, y = 14/13.

Numerical 5: square root of 3 + 4i

Here a = 3, b = 4, and m = √(9 + 16) = 5. x² = (5 + 3) ÷ 2 = 4 and y² = (5 − 3) ÷ 2 = 1. So x = ±2 and y = ±1. Since b = 4 is positive, x and y have the same sign. The roots are ±(2 + i). Check: (2 + i)² = 4 + 4i − 1 = 3 + 4i.

Common mistakes

• Writing i² = 1. It is −1.
• Writing the imaginary part as bi instead of b.
• Using √(a² − b²) for the modulus. It is √(a² + b²).
• Leaving i in the denominator instead of multiplying by the conjugate.
• Changing the sign of the real part when finding the conjugate. Only the imaginary part changes sign.
• Using √−a × √−b = √(ab). For example √−4 × √−9 = −6, not 6.
• Forgetting that (a + bi)² = a² − b² + 2abi, and writing a² + b².
• Giving only one square root. A complex number has two, and they differ in sign.

How to approach typical questions

• Simplify an expression: expand brackets, replace every i² by −1, then collect the real terms and the imaginary terms separately.
• Divide: multiply the top and bottom by the conjugate of the denominator. Never leave i in the denominator.
• Find real x and y: expand, write the real part and the imaginary part, and equate each pair. You get two simple equations.
• Prove a property: write z = a + bi (and w = c + di if needed), work out both sides separately, and show that they are equal.
• Find a square root: set up x² − y² = a, 2xy = b and x² + y² = modulus. Decide the signs from b, and always check by squaring your answer.
• Solve a quadratic: find the discriminant first. If it is negative, the roots are a complex conjugate pair.

Keep your working in steps. In a written paper, a correct method with a small sign slip still earns most of the marks, while an answer with no steps may earn very little.

Questions to practise

Short questions

1. Define the imaginary unit and write i², i³ and i⁴.
2. Write the real and imaginary parts of 4 − 9i and of 6i.
3. When are two complex numbers equal?
4. Define the conjugate and the modulus of a complex number.
5. Show that z + z is real and z − z is purely imaginary.
6. Find the multiplicative inverse of 1 + 2i.
7. Prove that z z = |z|².
8. Show that i⁻⁷ = i.
9. What does the point representing −3 + 2i look like on the Argand diagram?

Long questions

1. Prove that the conjugate of the product of two complex numbers is the product of their conjugates, and that |z₁z₂| = |z₁||z₂|.
2. Find the square roots of a given complex number using the method of equating real and imaginary parts.
3. Solve a quadratic equation with a negative discriminant and show that its roots are conjugates.
4. Show addition, subtraction and conjugates of complex numbers on an Argand diagram.

Numericals to try

1. Simplify i⁴⁵ + i⁻³.
2. Find (2 − i)(3 + 4i) and (1 + i) ÷ (1 − i).
3. Find |(3 + 4i) ÷ (5 − 12i)|.
4. Find the square roots of −5 + 12i.
5. Solve x² + 2x + 10 = 0.

Practice MCQs

Try these first. The answer key is at the end of the article.

1. The value of i⁴ is
   A) −1
   B) i
   C) 1
   D) −i
2. The conjugate of 2 − 5i is
   A) −2 + 5i
   B) 2 − 5i
   C) 2 + 5i
   D) −2 − 5i
3. The modulus of 3 + 4i is
   A) 7
   B) 5
   C) 25
   D) √7
4. The imaginary part of 5 − 7i is
   A) 5
   B) −7i
   C) 7
   D) −7
5. (1 + i)² is equal to
   A) 2
   B) 1 + 2i
   C) 2i
   D) 0
6. 1 ÷ i is equal to
   A) i
   B) 1
   C) −1
   D) −i
7. For z = 1 + 2i, the product of z and its conjugate is
   A) 5
   B) 3
   C) 1 + 4i
   D) −3
8. The value of i¹³ is
   A) 1
   B) i
   C) −i
   D) −1

Preparation strategy

1. Write the powers of i cycle and the main properties on one card and read it daily.
2. Do each exercise of the book once, then redo only the questions you got wrong after three days.
3. For every answer, substitute back or use the modulus check, as in the solved examples.
4. Practise signs slowly. Most lost marks in this chapter are sign errors.
5. Finish with short questions that ask for proofs, because they carry easy marks if the steps are in order.

More help: the 11th class Mathematics textbook guide and the 11th class Mathematics pairing scheme.

Diagram ideas

• Argand diagram. Alt text: "Graph with a real axis and an imaginary axis, showing the point 3 + 4i joined to the origin by a line of length 5."
• Conjugate mirror. Alt text: "Argand diagram with the points 3 + 4i and 3 − 4i placed as mirror images in the real axis."
• Addition of complex numbers. Alt text: "Parallelogram on an Argand diagram showing z₁, z₂ and their sum."
• Cycle of powers of i. Alt text: "Circle with four points marked i, −1, −i and 1, with arrows showing the repeating cycle."
• Method flow chart. Alt text: "Flow chart for dividing complex numbers: multiply by the conjugate, simplify the denominator, split into real and imaginary parts."

MCQ answer key

1. C. i⁴ = (i²)² = (−1)² = 1.
2. C. The conjugate changes the sign of the imaginary part only: 2 + 5i.
3. B. √(3² + 4²) = √25 = 5.
4. D. In a + bi the imaginary part is b, so it is −7 (not −7i).
5. C. (1 + i)² = 1 + 2i + i² = 1 + 2i − 1 = 2i.
6. D. 1 ÷ i = i ÷ i² = i ÷ (−1) = −i.
7. A. (1 + 2i)(1 − 2i) = 1² + 2² = 5.
8. B. 13 = 4 × 3 + 1, so i¹³ = i¹ = i.

Frequently asked questions