Ch 4 introduces complex numbers — an extension of real numbers that allows solutions to all quadratic equations. Students learn the imaginary unit i, operations on complex numbers, the Argand plane, and quadratic equations without real roots.
A complex number z = a + bi where a, b are real and i = √(−1). a is the real part, b the imaginary part. Operations: (a+bi) + (c+di) = (a+c) + (b+d)i. (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
Conjugate of z = a+bi is z̄ = a−bi. Modulus |z| = √(a²+b²). On the Argand plane, z is the point (a,b). Multiplicative inverse: z⁻¹ = z̄/|z|².
When discriminant D = b²−4ac < 0, roots are complex: x = (−b ± i√|D|)/2a. Complex roots always occur in conjugate pairs for equations with real coefficients.
Download: https://ncert.nic.in/textbook/pdf/kemh104.pdf | Complete book: https://ncert.nic.in/textbook/pdf/kemh1ps.zip
i is defined as √(−1), so i² = −1. It's called "imaginary" for historical reasons, but complex numbers are just as valid as real numbers and essential in physics, engineering, and mathematics.
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