Ch. 5. Complex Numbers - 1

1. Introduction
Sqrt(-1) = i
“i” is called imaginary unity

2. Integral powers of IOTA (i)
i³ = i*i² = i*(-1) = -i

To find iⁿ divide n by 4 to get 4m+r where m is the quotient and r is the remainder.

i⁴ = 1

iⁿ will be equal to i^r

3. Imaginary quantities

Square root of -3, -5 etc are called imaginary quantities

4. complex numbers
Number of the form a+ib (ex: 4+i3) is called a complex number.

A is called real part (Re(z)) and b is called imaginary part (Im(z)).

5. Equality of complex numbers

6. Addition of complex numbers

7. Subtraction of complex numbers

8. Multiplication of complex numbers

(a1+ib1) (a2+ib2) by multiplying and simplifying we get

(a1a2 – b1b2) + i(a1b2+a2b1)

Multiplicative inverse of a+ib = a/(a² + b²) - ib/(a² + b²))

9. Division of complex numbers

z1/z2 = z1* Multiplicative inverse of z2

10. Conjugate of a complex number
conjugate of z (= a+ib) = a-ib (is termed as z bar)

11. Modulus of a complex number

|z| = |a+ib| = SQRT(a² +b²)

Properties of Modulus

If z is a complex number, then

(i) |z| = 0 <=> z = 0
(ii) |z| = |conjugate of z| = |-z| = |-conjugate of z|
(iii) -|z| ≤Re(z) ≤|z|
(iv) -|z| ≤Im(z) ≤|z|
(v) z*congulage of z = |z|²



12, Reciprocal of a complex number

Multiplicative inverse and reciprocal are same

13. Square root of a complex number

14. Representation of a complex number

Graphical – Argand plane
Trigonometric
Vector
Euler

15. Argument or amplitude of a complex number

16. Eulerian form of a complex number

    e^θ  = cosθ + i sinθ  and e^-θ = cos θ  - i sin θ

These two are called Eulerian forms of a complex number.

17. Geometrical representations of fundamental operations

Addition
Subtraction

17a. Modulus and argument of multiplication of two complex numbers

18. Modulus and argument of division of two complex numbers

19. Geometrical representation of conjugate of a complex number

20. Some important results on modulus and argument