Complex Numbers - Govt. PG College Una (HP)

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An Introduction
Any number of the form x + iy where x,y are Real
and i=√-1, i.e., i2 = -1 is called a complex number.
 For example, 7 + i10, -5 -4i are examples of
complex numbers.
 The set or complex numbers is denoted by C, i.e.,
C={x+iy: x,y belongs to R(set of Real number) and
i=√-1}.
 The complex number x+yi is usually denoted by z;
x is called real part of z and is written as Re(z) and
y is called imaginary part or z and is written as
Im(z). So, for every z belongs to C.
 Z=Re(z) + i Im(z).
 For example, if z= 3 +5i, then Re(z)=3 and Im
(z)=5.

 Addition
of complex numbers
 Let z1=x1+iy1 and z2=x2+iy2 be any two
complex numbers, then the sum or addition of
z1 and z2 is defined as (x1+x2) + i(y1+y2), and
it is denoted by z1+z2 .
 The sum or two or more complex numbers is
also a complex number.
 Negative of a complex number
 Let z=x+iy be any complex number, then the
number –x-iy is called negative of z and is
denoted by –z.
 Negative of a complex number is again a
complex number.
 Difference
of Complex numbers
 Let z1=x1+iy1 and z2=x2+iy2 be any two
complex numbers, then the difference of z2
from z1 is defined as (x1-x2) + i(y1-y2), and
it is denoted by z1-z2 .
 The difference of two complex numbers is
also a complex number.
 Multiplication
of complex numbers.
 Let z1=x1+iy1 and z2=x2+iy2 be any two
complex numbers, then the product or
multiplication of z1 and z2 is defined as
(x1x2 - y1y2) + i(x1y2 + y1x2) and it is
denoted by z1z2 .
 For example: z1 = 5+3i and z2 = 2+4i then
z1z2= (10-12) +i(20+6)=-2+26i.
Lets understand the entire process with the help of an
example.
 Find the square roots of 3-4i
 Solution: let x+iy be a square root of 3-4i
 Then (x+iy)2 =3-4i
 x2-y2 +2xyi=3-4i ( (a+b)2 =a2+b2+2ab)
 => x2–y2=3 ………………..(i)
 And 2xy=-4, i.e., xy=-2………..(ii)
 We have (x2+y2)2= (x2-y2)2+4x2y2
 32+4(4)=9+16=25
 x2+y2=+5
 but as x2+y2 =5………………..(iii)
 On adding (i) and (iii), we get
 2x2 =8 => x2 =4 => x=+2
 Substituting in (ii), we get
 x=2, y=-1 and x=-2, y=1.
 Therefore, the two square roots of 3-4i are 2-i and

 Let
x be a cube root of unity, then x3=1.
 => x3-1=0=> (x-1)(x2 +x+1)=0
 => either x-1=0 or x2 +x+1=0
 Either x=1or x=(-1+√-3)/2
 Thus, the three cube roots of unity are 1,
(-1+i√3)/2 and (-1-i√3)/2.
Either of the two non-real cube roots of unity is the
square of the other.
 If one of the non-real cube root of unity is denoted
by w(read it as omega), then the other is w2.
Further, to avoid any possible confusion, we shall
take w= (-1+i√3)/2
 Thus, the three cube roots of unity are 1,w and w2.
 Sum of the three cube roots of unity is zero.
 Thus 1+w+w2=0
 Product of the cube roots of unity is one.
 i.e. 1.w.w2 =w3=1
 Either of the two non-real cube roots of unity is
reciprocal of the other. Since w3=1, therefore,
w.w2=1=> w and w2 are reciprocals of each other.

 When
we write the complex number in
polar form, then z=r(cos φ + i sin φ)
 Then according to De-Moivre’s Theorem
zn = [r(cos φ + i sin φ)]n
= rn(cos nφ + i sin nφ)
 If
the Discriminant of any quadratic
equation <0 i.e. –ve number. The roots of
that quadratic equation are not real, that
means the roots of such quadratic
equation are complex number.
 If d<0
where d=b2-4ac
 Then roots are (–b+√b2-4ac)/2a.
 Solve
the equation 3x2+7=0
 Here, the discriminant=02-4x3x7=-84
 Therefore
x=(-0+√-84)/2x3
=(+2(√21)i)/6 => (+(√21)i)/3

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