# COMPLEX NUMBERS CLASS 11 MATHS

## Summary

*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 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 ≠ 0, b ≠ 0), there exists the complex number a/(a ^{2} + b^{2}) + i -b/(a^{2} + b^{2}) , denoted by 1/z or z^{-1}, called the multiplicative inverse of z such that (a + ib) (a/(a^{2} + b^{2}) + i -b/(a^{2} + b^{2}) = 1 + i0 = 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 = √(x ^{2} + y^{2}) (the modulus of z) and cosθ = x/r, sinθ = 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, are given by x = (− b ± √(4ac - b^{2}) i)/2a.## Lecture - 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)

## Lecture - 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)

## Lecture - 3

- Conjugate of a complex number
- Relationship of conjugate with modulus of complex numbers
- NCERT Example 8, Example 12, Example 14, Example 15 and Example 16.

## Lecture - 4

- Square root of complex numbers
- NCERT Supplementary Exercise 5.4 (Q1 to Q6)