# 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 z1 = a + ib and z2 = c + id . Then
(i) z1 + z2 = (a + c) + i (b + d)
(ii) z1 z2 = (ac – bd) + i (ad + bc)

For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists the complex number a/(a2 + b2) + i -b/(a2 + b2) , denoted by 1/z or z-1, called the multiplicative inverse of z such that (a + ib) (a/(a2 + b2) + i -b/(a2 + b2) = 1 + i0 = 1

For any integer k, i4k = 1, i4k + 1 = i, i4k + 2 = – 1, i4k + 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 = √(x2 + y2) (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 ax2 + bx + c = 0, where a, b, c ∈ R, a ≠ 0, b2 – 4ac < 0, are given by x = (− b ± √(4ac - b2) 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)

## Lecture - 5

• NCERT Miscellaneous Exercise (Q1 to Q10)

## Lecture - 6

• NCERT Miscellaneous Exercise (Q11 to Q17)