## Saturday, May 24, 2008

### Chapter 5 Complex Numbers

today I studied Chapter Complex Numbers from R D Sharma.
For a first time reading, it seemed to be a hard chapter

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.

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)

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

17. Geometrical representations of fundamental operations