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 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.Ashish Kumar Let's Learn
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)