Schooling · Class 11 · Mathematics · Complex Numbers and Quadratic Equations · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Complex Numbers and Quadratic Equations: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A complex number has the form z = a…
- Two complex numbers are equal if and…
- The conjugate of z = a + ib is z̄ =…
- The modulus |z| = √(a² + b²) gives…
- The argument (or amplitude) θ of z…
- Polar form
- For a quadratic ax² + bx + c = 0…
- Formulas and facts
- i² = −1
- Addition
- Multiplication
- if z = a + ib
- |z| = √(a² + b²)
- |z|² = z × z̄
- Division
- Watch out for
- Writing √(−4) = 2 instead of 2i
- Forgetting that i² = −1 during…
- Using tan⁻¹(b/a) directly without…
- Confusing modulus with absolute…
- Dividing complex numbers by writing…
What this chapter is about
Until now, you have worked only with real numbers—those that fit on the number line. But what happens when you try to solve x² + 1 = 0? No real number squared gives −1. To handle such equations, mathematicians invented a new kind of number called a complex number, built around the imaginary unit i, defined so that i² = −1.
Key ideas
- A complex number has the form z = a + ib, where a and b are real numbers; a is the real part, b is the imaginary part, and i satisfies i² = −1.
- Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.
- The conjugate of z = a + ib is z̄ = a − ib; the product z × z̄ = a² + b², which is always a non-negative real number.
- The modulus |z| = √(a² + b²) gives the distance of the point (a, b) from the origin in the Argand plane.
- The argument (or amplitude) θ of z is the angle the line joining the origin to (a, b) makes with the positive real axis, measured anti-clockwise.
- Polar form: z = r(cos θ + i sin θ), where r = |z| and θ is the argument.
- For a quadratic ax² + bx + c = 0 with real coefficients, the discriminant D = b² − 4ac decides the nature of roots: if D < 0, the roots are complex conjugates of each other.
Formulas and facts to remember
- i² = −1, i³ = −i, i⁴ = 1; powers of i repeat with period 4.
- Addition: (a + ib) + (c + id) = (a + c) + i(b + d).
- Multiplication: (a + ib)(c + id) = (ac − bd) + i(ad + bc).
- Conjugate: if z = a + ib, then z̄ = a − ib.
- Modulus: |z| = √(a² + b²).
- |z|² = z × z̄.
- Division: (a + ib)/(c + id) = [(a + ib)(c − id)] / (c² + d²).
- Polar form: z = r(cos θ + i sin θ), with r = |z| and tan θ = b/a (adjust quadrant).
- Quadratic roots when D < 0: x = [−b ± i√(−D)] / (2a).
Common mistakes
- Writing √(−4) = 2 instead of 2i → remember √(−k) = i√k for positive k.
- Forgetting that i² = −1 during multiplication and leaving i² in the answer → always replace i² with −1.
- Using tan⁻¹(b/a) directly without checking the quadrant → always plot the point mentally and adjust θ to the correct quadrant.
- Confusing modulus with absolute value of real part alone → modulus uses both parts: √(a² + b²).
- Dividing complex numbers by writing (a + ib)/(c + id) and stopping → multiply numerator and denominator by the conjugate c − id first.
Quick revision
- A complex number is a + ib; i² = −1.
- Conjugate flips the sign of the imaginary part; z × z̄ = |z|².
- Modulus |z| = √(a² + b²); argument θ is the angle from the positive real axis.
- Polar form: z = r(cos θ + i sin θ).
- Quadratic with D < 0 has two complex conjugate roots.
- Powers of i cycle: i, −1, −i, 1, i, …