5. Complex Numbers

COMPLEX NUMBERS CLASS 11 MATHS

A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.

Let z_1= a + ib \text{ and } z_2 = c + id . Then
(i) z_1 + z_2 = (a + c) + i (b + d)
(ii) z_1 z_2 = (ac – bd) + i (ad + bc)

For any non-zero complex number z = a + ib (a \ne 0, b \ne 0), there exists the complex number \frac{a}{a^2+ b^2} + i \frac{-b}{a^2 + b^2} , denoted by \frac{1}{z} or z^{-1}, called the multiplicative inverse of z such that (a + ib)\left ( \frac{a^2}{a^2+b^2}+i \frac{-b}{a^2+b^2} \right ) = 1 + i 0 = 1

For any integer k , i^{4k}=1, i{4k + 1}=i, i^{4k + 2}= – 1, i{4k + 3} = – i

The conjugate of the complex number z = a + ib, denoted by z , is given by z = a – ib.

The polar form of the complex number z = x + iy is r (cosθ + i sinθ), where r = \sqrt{x^2 + y^2} (the modulus of z) and cosθ = \frac{x}{r} , sinθ = \frac{y}{r}. (θ is known as the argument of z. The value of θ, such that – π < θ ≤ π, is called the principal argument of z.

A polynomial equation of n degree has n roots.

The solutions of the quadratic equation ax^2 + bx + c = 0, where a, b, c ∈ R, a ≠ 0, b^2 – 4ac < 0 [/latex], are given by [latex] x = \frac{− b \pm \sqrt{4ac - b2} i}{2a} [/latex].

5. Complex Numbers 1
  • Basics of numbers systems, imaginary numbers, real numbers, iota
  • Meaning of complex numbers
  • Working with Imaginary Numbers (iota)
  • Working with Complex Numbers
  • Addition and Subtraction of Complex Numbers
  • Multiplication of Complex Numbers
  • Division of Complex Numbers
  • Additive Inverse and Multiplicative Inverse of Complex numbers
  • NCERT Exercise 5.1 (Q 1 to Q14)
  • Working with quadratic equations of negative discriminant
  • NCERT Exercise 5.3 (Q1 to Q10)
5. Complex Numbers 2
  • Cartesian Coordinate System and Polar Coordinate System
  • Relationship between cartesian coordinate system and polar coordinate system
  • Argand Diagram
  • Meaning of Modulus and Modulus of Complex Numbers mod(z)=|z|= r
  • Detailed explanation and derivation for argument of complex numbers arg(z), Polar form
  • NCERT Exercise 5.2 (Q1 to Q8)
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