This document defines complex numbers and outlines their algebra. A complex number z is defined as an ordered pair (x,y) where x is the real part and y is the imaginary part. The four basic operations on complex numbers are defined: addition, subtraction, multiplication, and division. Complex numbers can also be represented geometrically on an Argand diagram, with the real part on the x-axis and imaginary part on the y-axis. Polar form represents a complex number in terms of magnitude and angle. De Moivre's theorem and nth roots of complex numbers are also discussed.

1. COMPLEX
NUMBER
PREPARED BY :
PARMAR RONAKKUMAR VINODBHAI

2. DEFINITION
A complex number z = x+iy is defined as an
order pair (x,y), where x & y are real number
and i =
Y = imaginary part of z
X = real part of z
If x=0 & y≠0 then complex number is called
purely imaginary number.
If x≠0 & y=0 then complex number is called
purely real number.

3. ALGEBRA OF COMPLEX NUMBER
z1 = x1+iy1 z2 = x2+iy2
(1) Equality Of Complex Number:-
z1 ↔ z2 when x1 = x2 & y1 = y2
(2) Addition of complex number:-
z1 + z2 = (x1+iy1) + (x2+iy2)
z1 + z2 = (x1+x2) + i (y1+y2)

4. CONTINUED…
(3) Subtraction Of Complex Number:-
z1 – z2 = (x1+iy1) – (x2+iy2)
z1 – z2 = (x1 - x2) + i (y1 – y2)
(4) Multiplication Of Complex Number:-
z1.z2 = (x1+iy1) (x2+iy2)
z1.z2 = (x1x2 + ix1y2 + iy1x2 – y1y2)
z1.z2 = (x1x2 – y1y2) + i (x1y2 + x2y1)

5. CONTINUED…
(5) Division Of Complex Number:-
=
 
=
Conjugate of complex number:

6. ARGAND DIAGRAM / COMPLEX PLANE
A complex number (z) can also be
represented as an order pair (x,y) and thus
as a point in a plane, known as complex
plane or argand diagram, where x-axis is
called real axis and y-axis is called
imaginary axis.

7. NUMERICAL :-

8. POLAR FORM OF A COMPLEX NUMBER
Where,  = Angle is between real axis
and imaginary axis in anti-
clockwise direction

9. CONTINUED…
Where,  = argument of z

10. NUMERICAL :-

11. NUMERICAL :-

12. DE MOIVRE’S THEOREM
Statement : If n be a rational number, the
value or one of the values of
is
this extends to,

13. NUMERICAL :-

14. nth ROOT OF COMPLEX
NUMBER
It includes,
Where, k = 0,1,2,….,n-1
(General argument = +2k)

15. NUMERICAL :-

16. THANK
YOU