Chapter 4 — Complex Numbers and Quadratic Equations

Complex numbers extend the real number system to solve equations like x² = -1 by introducing the imaginary unit i. The chapter covers complex number algebra, modulus, conjugate, and the Argand plane representation.

Complex Numbers and Quadratic Equations (Chapter 4) introduces the complex number system to solve equations with no real solutions. A complex number is a + ib where a is the real part and b is the imaginary part. The chapter develops complete algebra for complex numbers including addition, subtraction, multiplication, and division, establishes that powers of i cycle with period 4 (i⁴ᵏ=1, i⁴ᵏ⁺¹=i, i⁴ᵏ⁺²=-1, i⁴ᵏ⁺³=-i), proves algebraic identities like (z₁+z₂)²=z₁²+2z₁z₂+z₂², and introduces modulus |z|=√(a²+b²) and conjugate z̄=a-ib. The Argand plane geometrically represents complex numbers as points, with the x-axis as the real axis and y-axis as the imaginary axis.

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)