Class 12 Mathematics COMPLEX NUMBERS Notes

Class 12 Mathematics COMPLEX NUMBERS Notes
Class 12 Mathematics Complex Numbers Notes | Nepal eNotes

Unit 4

Algebra

Class 12 Mathematics

Complex Numbers

Class 12 Mathematics – Complex Numbers Notes PDF

On mobile, swipe inside the PDF to read all pages and pinch to zoom.

Contents / Quick Navigation

1. Introduction & Polar Recap 2. De Moivre’s Theorem 3. Proof of De Moivre’s Theorem 4. Powers of Complex Numbers 5. Roots of Complex Numbers 6. Cube Roots of Unity 7. Properties of Cube Roots 8. Euler’s Formula 9. Worked Examples 10. Common Mistakes 11. Important Exam Questions 12. One-Minute Revision 13. Diagram Practice

NEB / CDC Focus

The Class 12 syllabus focus in this chapter is De Moivre’s theorem, its use in finding roots of complex numbers, properties of the cube roots of unity, and Euler’s formula. A short polar-form recap is included only because it is required to apply these results correctly.

1. Introduction and Polar-Form Recap

Complex Number

A complex number is written as z=x+iy, where x,y∈ℝ and i²=-1.

If z=x+iy, then its modulus and argument are

|z| = r = √(x²+y²) arg(z)=θ, where tan θ = y/x, with the correct quadrant chosen. z = r(cos θ + i sin θ)

The form r(cosθ+i sinθ) is called the polar or trigonometric form of a complex number.

Complex Number in Polar Form Real axis Imaginary axis z = x + iy r = |z| x y θ
Figure 1: The polar representation uses the modulus r and argument θ.
Important Note

When finding θ, do not rely only on tan⁻¹(y/x). First identify the quadrant of (x,y).

2. De Moivre’s Theorem

Theorem

If n is an integer, then

(cos θ + i sin θ)n = cos(nθ) + i sin(nθ).

More generally, if z=r(cosθ+i sinθ), then

Power Form
zn = rn[cos(nθ)+i sin(nθ)].

The theorem converts repeated multiplication of a complex number into two simple operations: raise the modulus to the required power and multiply the argument by that power.

Effect of Raising z to the nth Power θ r z = r(cosθ+i sinθ) power n nθ rⁿ zⁿ = rⁿ[cos(nθ)+i sin(nθ)]
Figure 2: Under the nth power, the modulus becomes rⁿ and the argument becomes nθ.

3. Proof of De Moivre’s Theorem

3.1 For Positive Integral n

We prove by mathematical induction.

Step 1: Base case. For n=1,

(cosθ+i sinθ)1=cosθ+i sinθ,

so the theorem is true.

Step 2: Inductive hypothesis. Assume

(cosθ+i sinθ)k=cos(kθ)+i sin(kθ).

Step 3: Prove for k+1.

(cosθ+i sinθ)k+1 =[cos(kθ)+i sin(kθ)](cosθ+i sinθ).

Multiplying, using i²=-1,

=cos(kθ)cosθ-sin(kθ)sinθ +i[sin(kθ)cosθ+cos(kθ)sinθ].

Using angle-addition identities,

=cos[(k+1)θ]+i sin[(k+1)θ].

Therefore the theorem holds for every positive integer n.

3.2 Negative Integral Powers

For n=-m, where m>0,

(cosθ+i sinθ)-m =1/[cos(mθ)+i sin(mθ)] =cos(mθ)-i sin(mθ) =cos(-mθ)+i sin(-mθ).

Thus De Moivre’s theorem holds for integral powers.

Exam Important

In a proof question, show the multiplication step and explicitly use cos(A+B) and sin(A+B).

4. Application: Powers of Complex Numbers

Worked Example 1

Find (1+i)8.

  1. 1+i=√2(cos π/4+i sin π/4).
  2. By De Moivre, (1+i)8=(√2)8[cos(2π)+i sin(2π)].
  3. (√2)8=16.
  4. cos2π=1 and sin2π=0.

Answer: 16.

Worked Example 2

Evaluate (√3+i)6.

Here r=2 and θ=π/6.

(√3+i)6 =26[cosπ+i sinπ] =64(-1) =-64.

Shortcut

For high powers, convert to polar form first. Direct repeated multiplication is usually much longer.

5. Application: Roots of Complex Numbers

Let z=r(cosθ+i sinθ). To find its nth roots, solve wn=z.

nth-Root Formula
wk = r1/n [cos((θ+2kπ)/n)+i sin((θ+2kπ)/n)], k=0,1,2,…,n−1.

Therefore every non-zero complex number has exactly n distinct nth roots. They have equal modulus r1/n and are equally spaced in argument by 2π/n.

Geometry of nth Roots w₀w₁w₂ w₃w₄w₅ Example shown for six equally spaced roots.
Figure 3: The n roots lie on a circle and differ in argument by 2π/n.

5.1 Square Roots of a Complex Number

For n=2, there are two roots separated by π radians.

Worked Example: Square roots of i

i=cos(π/2)+i sin(π/2).

The square roots are

w0=cos(π/4)+i sin(π/4)=(1+i)/√2,

w1=cos(5π/4)+i sin(5π/4)=-(1+i)/√2.

Square Roots of i (1+i)/√2 −(1+i)/√2
Figure 4: The two square roots are diametrically opposite.

6. Cube Roots of Unity

The cube roots of unity are the solutions of

z³=1.

Since 1=cos(2kπ)+i sin(2kπ), the three roots are

1,   ω = -1/2 + (√3/2)i,   ω² = -1/2 – (√3/2)i.

Here ω=cos(2π/3)+i sin(2π/3) and ω²=cos(4π/3)+i sin(4π/3).

Cube Roots of Unity 1 ω ω² The three roots lie 120° apart and form an equilateral triangle.
Figure 5: Geometric representation of 1, ω and ω² on the unit circle.

7. Properties of Cube Roots of Unity

PropertyResult
Cube of each root1³=ω³=(ω²)³=1
Sum of roots1+ω+ω²=0
Product of non-real rootsω·ω²=ω³=1
Reciprocal1/ω=ω² and 1/ω²=ω
Conjugatesω̄=ω² and \overline{ω²}=ω
Powers repeatωⁿ depends on n mod 3

7.1 Proving 1+ω+ω²=0

Factor

z³-1=(z-1)(z²+z+1).

For the non-real cube roots ω and ω², we have

ω²+ω+1=0.

7.2 Cyclic Powers

ω⁰=1, ω¹=ω, ω²=ω², ω³=1, ω⁴=ω, ω⁵=ω², …
Powers of ω Repeat Every 3 1 ω ω² multiply by ω each step
Figure 6: Reducing an exponent modulo 3 gives the value of any power of ω.
Worked Example

Evaluate ω2026.

Since 2026=3×675+1,

ω2026=ω.

8. Euler’s Formula

Euler’s Formula
eiθ = cosθ + i sinθ.

Replacing θ by -θ,

e-iθ = cosθ – i sinθ.

Adding and subtracting these two equations gives

cosθ = (eiθ+e-iθ)/2 sinθ = (eiθ-e-iθ)/(2i)

Therefore the polar form z=r(cosθ+i sinθ) may also be written in exponential form as

z = reiθ.
Euler’s Formula on the Unit Circle eⁱθ cosθ sinθ θ
Figure 7: On the unit circle, eⁱθ has coordinates (cosθ, sinθ).

8.1 Euler’s Identity

Putting θ=π gives

eiπ + 1 = 0.

8.2 De Moivre from Euler’s Formula

Since cosθ+i sinθ=eiθ,

(cosθ+i sinθ)n =(eiθ)n =einθ =cos(nθ)+i sin(nθ).

9. Worked Examples

Example 1: Find cube roots of 8

Write 8=8[cos(2kπ)+i sin(2kπ)].

Its cube roots have modulus 2 and arguments 2kπ/3, k=0,1,2.

2,   -1+i√3,   -1-i√3.

Example 2: Fourth roots of 16

The roots are

2[cos(kπ/2)+i sin(kπ/2)],   k=0,1,2,3.

Thus the roots are 2, 2i, -2, -2i.

Fourth Roots of 16 2 2i −2 −2i
Figure 8: The fourth roots are equally spaced by 90°.
Example 3: Use cube-root properties

Evaluate (1-ω)(1-ω²).

(1-ω)(1-ω²)=1-(ω+ω²)+ω³.

Since ω+ω²=-1 and ω³=1,

=1-(-1)+1=3.

Example 4: Simplify using 1+ω+ω²=0

Find 1/(1-ω)+1/(1-ω²).

=[(1-ω²)+(1-ω)]/[(1-ω)(1-ω²)] =[2-(ω+ω²)]/3 =[2-(-1)]/3=1.

Example 5: Find roots of −8i

Write -8i=8[cos(3π/2)+i sin(3π/2)].

The cube roots are

2[cos((3π/2+2kπ)/3)+i sin((3π/2+2kπ)/3)], k=0,1,2.

Therefore the arguments are π/2, 7π/6, 11π/6.

Example 6: Euler-form simplification

Simplify ei5π/3.

ei5π/3=cos(5π/3)+i sin(5π/3)=1/2-(√3/2)i.

Example 7: Derive a trigonometric identity

Use De Moivre’s theorem to expand (cosθ+i sinθ)³.

cos3θ+i sin3θ =cos³θ+3i cos²θ sinθ-3cosθ sin²θ-i sin³θ.

Equating real and imaginary parts:

cos3θ=cos³θ-3cosθ sin²θ=4cos³θ-3cosθ,

sin3θ=3cos²θ sinθ-sin³θ=3sinθ-4sin³θ.

10. Step-by-Step Problem Strategy

  1. Convert the complex number to polar form r(cosθ+i sinθ).
  2. Check the quadrant carefully before fixing the argument.
  3. For powers, apply rⁿ[cos(nθ)+i sin(nθ)].
  4. For nth roots, use all values k=0,1,…,n-1.
  5. For cube roots of unity, use 1+ω+ω²=0 and ω³=1 before doing long algebra.
  6. For very large powers of ω, reduce the exponent modulo 3.
  7. For Euler-form questions, use eⁱθ=cosθ+i sinθ.
  8. Check the final answer by raising roots back to the required power whenever practical.

11. Common Mistakes and Warnings

Mistake 1: Wrong quadrant

The principal arctangent alone may give the wrong argument.

Mistake 2: Forgetting the modulus

For z=r(cosθ+i sinθ), the power contains rⁿ.

Mistake 3: Finding only one root

An nth-root problem requires all n roots, usually with k=0,…,n-1.

Mistake 4: Missing 2kπ

All possible arguments of a complex number are θ+2kπ.

Mistake 5: Confusing ω² with −ω

Use 1+ω+ω²=0, so ω²=-1-ω.

Mistake 6: Treating ω as 1

The three cube roots are 1,ω,ω²; only one of them is real.

Mistake 7: Degrees/radians mixing

Use one angle system consistently, especially in Euler’s formula.

Mistake 8: Ignoring periodicity

Arguments differing by 2π represent the same direction.

12. Formula Sheet

Core Formulas
z=r(cosθ+i sinθ)=reiθ (cosθ+i sinθ)ⁿ=cos(nθ)+i sin(nθ) zⁿ=rⁿ[cos(nθ)+i sin(nθ)] wk=r1/n[cos((θ+2kπ)/n)+i sin((θ+2kπ)/n)] k=0,1,…,n−1 1+ω+ω²=0 ω³=1 ω·ω²=1 1/ω=ω²,   1/ω²=ω eiθ=cosθ+i sinθ cosθ=(eiθ+e-iθ)/2 sinθ=(eiθ-e-iθ)/(2i)

13. Important Exam Questions

Short-Answer Questions

  1. State De Moivre’s theorem.
  2. Write the polar form of a complex number.
  3. Write the formula for the nth roots of a non-zero complex number.
  4. What are the three cube roots of unity?
  5. Prove that 1+ω+ω²=0.
  6. Show that 1/ω=ω².
  7. State Euler’s formula.
  8. Express cosθ and sinθ in exponential form.
  9. Evaluate a large power such as ω100.

Long-Answer / Proof Questions

  1. State and prove De Moivre’s theorem for positive integral powers.
  2. Derive the formula for the nth roots of a complex number using De Moivre’s theorem.
  3. Find all nth roots of a specified complex number and represent them geometrically.
  4. Establish important properties of the cube roots of unity.
  5. Use De Moivre’s theorem to derive triple-angle identities.
  6. Explain Euler’s formula and derive the exponential forms of sine and cosine.

Numerical Questions

  1. Evaluate a high power of a complex number using De Moivre’s theorem.
  2. Find the square, cube or fourth roots of a given complex number.
  3. Simplify expressions involving ω and ω².
  4. Evaluate powers of ω with very large exponents.
  5. Convert between trigonometric and exponential forms using Euler’s formula.

Diagram Questions

  1. Represent a complex number in the Argand plane and show its modulus and argument.
  2. Plot the cube roots of unity on the unit circle.
  3. Show geometrically that nth roots are equally spaced on a circle.
  4. Illustrate Euler’s formula on the unit circle.

14. One-Minute Revision

Quick Revision
  • A complex number in polar form is z=r(cosθ+i sinθ).
  • Its modulus is r=√(x²+y²).
  • De Moivre: (cosθ+i sinθ)ⁿ=cos(nθ)+i sin(nθ).
  • For z=r(cosθ+i sinθ), zⁿ=rⁿ[cos(nθ)+i sin(nθ)].
  • A non-zero complex number has exactly n distinct nth roots.
  • The roots have equal modulus and are equally spaced by 2π/n.
  • The cube roots of unity are 1,ω,ω².
  • 1+ω+ω²=0.
  • ω³=1.
  • Powers of ω repeat every 3.
  • 1/ω=ω² and 1/ω²=ω.
  • Euler’s formula is eⁱθ=cosθ+i sinθ.
  • z=reⁱθ is the exponential form.
  • Always include 2kπ before dividing an argument when finding roots.
  • Check the quadrant before fixing the argument.

15. Diagram Practice

  • Argand-plane diagram showing x, y, r and θ.
  • De Moivre angle-multiplication diagram.
  • Equally spaced nth roots on a circle.
  • Square roots as diametrically opposite points.
  • Cube roots of unity forming an equilateral triangle.
  • Cyclic powers of ω.
  • Euler’s formula on the unit circle.
  • Fourth roots placed at right angles on the complex plane.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
Source handling: The original Nepal eNotes PDF remains embedded above. The typed section follows the verified NEB/CDC syllabus and is designed as a searchable, responsive study companion. Where the PDF viewer does not expose handwritten page text, the typed section is a syllabus-aligned reconstruction and is not claimed to be a word-for-word transcription.

Original Nepal eNotes page: Class 12 Mathematics Complex Numbers Notes.
KEEP LEARNING View All Posts

Discussion

Share a helpful question, idea, or explanation with other students.

Leave a Comment

Write a clear question, answer, or helpful explanation.
Your email will not be published.

Download Our Offline App

Study class-wise notes even when internet is not available. Get the app from Play Store.

Nepal eNotes offline app preview
Get it on Google Play