## Saturday, June 7, 2008

### 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 ir

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

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

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

These two are called Eulerian forms of a complex number.

17. Geometrical representations of fundamental operations