Chapter 4 — Complex Numbers and Quadratic Equations

Chapter 4 of the Class 11 Maths NCERT textbook, "Complex Numbers and Quadratic Equations", extends the real number system to solve equations like x² = -1 by introducing the imaginary unit i, covering complex-number algebra, modulus, conjugate, and the Argand plane.

• Why the number system is extended — The chapter motivates complex numbers as the fix for equations with no real solution, defining a number a + ib with a real and an imaginary part so that quadratics which stumped the real system finally become solvable.
• A full algebra for complex numbers — It develops addition, subtraction, multiplication and division for complex numbers and shows familiar algebraic identities still hold — establishing that this new number set behaves as a complete, consistent arithmetic system.
• Geometry in the Argand plane — Modulus and conjugate are introduced alongside the Argand plane, where each complex number becomes a point, giving arithmetic operations a visual, geometric meaning of distance and reflection.

1. 01Complex number z = a + ib has real part a and imaginary part b; two complex numbers are equal only when both parts match
2. 02Four basic operations: z₁+z₂=(a+c)+i(b+d), z₁z₂=(ac-bd)+i(ad+bc), division via multiplicative inverse z⁻¹=z̄/|z|²
3. 03Powers of imaginary unit i repeat in cycles of 4: i²=-1, i³=-i, i⁴=1, then repeats; i⁻¹=-i, i⁻²=-1, i⁻³=i, i⁻⁴=1
4. 04Square roots of negative numbers: √(-a)=√(a)i for positive a; but √(ab)≠√a·√b when both a,b<0 (contradiction to i²=-1)
5. 05Modulus |z|=√(a²+b²) measures distance from origin; conjugate z̄=a-ib is mirror image across real axis in Argand plane
6. 06Key algebraic identities hold for complex numbers: (z₁-z₂)²=z₁²-2z₁z₂+z₂², (z₁+z₂)³=z₁³+3z₁²z₂+3z₁z₂²+z₂³, z₁²-z₂²=(z₁+z₂)(z₁-z₂)