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Complex Numbers and De Moivre’s Theorem

Telangana Open School (TOSS) · Class 12 · Mathematics

Quick revision notes for Complex Numbers and De Moivre’s Theorem — Telangana Open School (TOSS) Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

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Key Topics to Revise

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 \(\cos(2k\pi/n) + i\sin(2k\pi/n)\), \(k = 0,1,\dots,n-1\).

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Full Notes

Key Concepts

A number of the form \(zDefined as \(i = \sqrt{For \(z = a + bi\)The modulus \(|z| = \sqrt{a^2 +A complex number can be written

Frequently Asked Questions

What are the important topics in Complex Numbers and De Moivre’s Theorem for Telangana Open School (TOSS) Class 12 Mathematics?
Complex Numbers and De Moivre’s Theorem covers several key topics that are frequently asked in Telangana Open School (TOSS) Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Complex Numbers and De Moivre’s Theorem — Telangana Open School (TOSS) Class 12 Mathematics?
Understand the core concepts first, then work through the 47 practice questions available for this chapter. Revise formulas and 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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