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Complex Numbers and De Moivre’s Theorem — Syllabus

Telangana Open School (TOSS) · Class 12 · Mathematics

What Complex Numbers and De Moivre’s Theorem covers in Telangana Open School (TOSS) Class 12 Mathematics: 4 topics, for the 2026-27 session.

47 questions30 flashcards15 formulas & key relations5 concepts

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4 Topics · Telangana Open School (TOSS) Class 12 Mathematics · 2026-27

Topics in Complex Numbers and De Moivre’s Theorem

1

Introduction to Complex Numbers

  • The equation \(x^2 + 1 = 0\) has no real solution because the square of any real number is non-negative.
  • To solve such equations, we introduce a new number \(i = \sqrt{-1}\), called iota.
  • A complex number is any number of the form \(z = a + bi\), where \(a\) and \(b\) are real numbers and \(i = \sqrt{-1}\).
2

Algebra of Complex Numbers

  • Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
  • Subtraction: \((a + bi) - (c + di) = (a - c) + (b - d)i\)
  • Multiplication: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\)
3

Modulus, Argument, and Polar Form

  • Modulus of \(z = a + bi\) is \(|z| = \sqrt{a^2 + b^2}\).
  • Argument \(\theta\) is the angle made with the positive real axis: \(\theta = \tan^{-1}(b/a)\).
  • Polar form: \(z = r(\cos\theta + i\sin\theta)\), where \(r = |z|\).
4

De Moivre’s Theorem and Applications

  • De Moivre’s Theorem: \((\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)\) for any integer \(n\).
  • Used to find powers and roots of complex numbers easily.
  • \(n^{\text{th}}\) roots of unity are solutions to \(x^n = 1\), given by \(\cos(2k\pi/n) + i\sin(2k\pi/n)\), \(k = 0,1,\dots,n-1\).

Key Concepts

Central concept: Complex Numbers

Definition of Complex NumberPowers of iConjugate of a Complex NumberModulus of a Complex NumberGeometrical Representation (Argand Plane)Equality of Complex NumbersAlgebraic OperationsProperties of Operations

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Frequently Asked Questions

What are the important topics in Complex Numbers and De Moivre’s Theorem for Telangana Open School (TOSS) Class 12 Mathematics?
Key topics in Complex Numbers and De Moivre’s Theorem include Introduction to Complex Numbers, Algebra of Complex Numbers, Modulus, Argument, and Polar Form, De Moivre’s Theorem and Applications. Study these first, then practise questions on each for the Telangana Open School (TOSS) Class 12 board exam.
How should I revise Complex Numbers and De Moivre’s Theorem for the Telangana Open School (TOSS) Class 12 board exam?
Learn the core ideas first, then work through the 47 practice questions on Complex Numbers and De Moivre’s Theorem. Revise definitions regularly and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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All Telangana Open School (TOSS) Class 12 Mathematics Syllabus

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