Download complex numbers - Siby Sebastian

Consider the quadratic Equation
X2 + 1 = 0
What is its solution ?
X2 = - 1 or x = ± −𝟏
But this number is not known to us.
It is a tool
to solve an
equation.
It has been
used to
solve
equations
for the last
200 years
or so.
It is
defined to
be i such
that ;
i2 = - 1
Or in other
words;
−𝟏 = i
What is i ?
i is an imaginary number
unreal
complex
Or a complex number
Or an unreal number
The terms are inter-changeable
imaginary
Definition
• A complex number is an expression of the form Z = a+ i b,
where a and b are real numbers and i is a mathematical
symbol for −𝟏 which is called the imaginary unit.
• For example, −3.5 + 2i is a complex number.
• The real number a of the complex number z = a + bi is called
the real part of z and the real number b is the imaginary part.
If b = 0, the number a + bi = a is a real number.
Example: 5= 5+ i 0
If a = 0, the number a + bi is called an
imaginary number
Example: -2i= 0+ (-2)i
A number such as 3i is a purely imaginary number
A number such as 6 is a purely real number
6 + 3i is a complex number
Z=x + iy is the general form of a complex number
If x + iy = 6 – 4i then x = 6 and y = – 4
The ‘real part’ of 6 – 4i is 6
The imaginary part of 6 - 4i is - 4
NATURAL NUMBERS
INTEGERS
RATIONAL NUMBERS
IRRATIONAL NUMBERS
REAL NUMBERS
COMPLEX NUMBERS
• Real numbers and imaginary numbers are
subsets of the set of complex numbers.
Imaginary
Numbers
Rational
Real Irrational
Numbers
Complex Numbers
Complex numbers
do not have order!
Practice Time!!!!
−𝟒
1. Simplify
2. Evaluate 3i x -4i
Addition of Complex Numbers
• Let z1=a+ib and z2=c+id be any
two complex numbers. Then
the sum of those two complex
numbers is defined as :
z1 +z2 = (a+bi) + (c+di)
= (a+c) + (b+d)i
Addition of two complex numbers can be done
geometrically by constructing a parallelogram
Practice Time!!!!
Simplify
• (2+3i ) + (4 -3i)
• (-3+4i) + (-2- i10)
Properties of addition
The closure law
• The sum of two
complex
numbers is
complex
number
The commutative
law
• For any two
complex
numbers a and
b, a+b=b+a
The associative
law
• For any three
complex
numbers a,b,c;
(a+b)+c=a+(b+c)
Properties of addition
The existence of
additive Identity
• There exists the complex
number 0+i 0(denoted as
0), called the additive
identity or the zero
complex number.
• (2+i10 ) + 0 = 2+i10
The existence of
additive Inverse
• To every complex number
z=a+ib, we have the
complex number –a+i(-b)
(denoted as –z), called the
additive inverse of z. We
observe that z+(-z)=0(the
additive identity).
Difference of two complex numbers
• Given any two complex numbers z1 and z2,
the difference z1 - z2 is defined as follows :
• z1 - z2 = z1 +(-z2)
Simplify
(3i+2i) – ( -2 + i3)
Multiplication of two complex numbers
• Multiplying complex numbers is similar to
multiplying polynomials and combining like
terms.
• Let a+ib and c+id be any two complex
numbers. Then the product of those two
complex numbers is defined as follows:
• (a+ib) (c+id) = (ac – bd) + i(ad + bc)
PROPERTIES OF MULTIPLICATION
The closure law
• The product of two complex numbers is a
complex number
The commutative law
• For any two complex numbers a and b,
ab=ba
The associative law
• For any three complex numbers a,b,c;
(a b) c = a (b c).
The existence of
multiplicative identity
The distributive law
• There exists a complex number 1+i0 (denoted as 1),
called the multiplicative identity such that a.1 =a, for
every complex number a
• For any three complex numbers a,b,c;
• (1) a(b+c) = a.b+a.c
• (2) (a+b)c = a.c+b.c
• Practice Time!!!!
Simplify
• (2+3i)(4-3i)
• (-4+2i)(7-12i)
Division of two complex numbers
• Given any two complex numbers a and b ,
𝒂
where b≠ 0, the quotient is defined by
𝒃
a
1
 a
b
b
• Simplify
𝟐+𝟑𝒊
𝟒−𝒊𝟗
• The following identifies are true for complex
numbers
POWERS OF i
ii
i 
2

1

2
 1
i 3  1 i  i
i 4  1 1  1
In general, for any integer k,
i4k = 1, i4k+1 = i, i4k+2 = -1.
The modules and the conjugate of a complex
number
• Let z = a + ib be a complex number. Then
the modulus of z, is denoted by IzI, is
defined to be the non negative real
number 𝒂𝟐 + 𝒃𝟐 , i.e., IzI = 𝒂𝟐 + 𝒃𝟐
and the conjugate of z, is denoted as z ,
is the complex number z = a- ib.
• USEFUL RESULTS
Consider z  a  bi and z  a  bi (Conjugate)
z  z  2a
z  z  2bi
z  (a  bi )(a  bi )
2
 a 2  2bi  b 2
z  (a  bi )(a  bi )
2
 a 2  2abi  b 2
• USEFUL RESULTS
Consider z  a  bi and z  a  bi (Conjugate)
zz  (a  bi )(a  bi )
 a 2  b2
 z
2
z (a  bi ) (a  bi )


z (a  bi ) (a  bi )

a 2  2abi  b 2
a 2  b2
USEFUL RESULTS
Consider z1  a  bi and z2  c  di
1.
2.
 z1  z2   z1  z2
z1 z2   z1 z2 
• An Argand
diagram is a plot
of complex
numbers as points
on a complex
plane.
Argand plane
Argand plane
• The complex plane use
the x-axis as the real
axis and y-axis as
the imaginary axis.
• The dashed circle represents
the complex modulus |Z| of
z and the
angle theta represents
its complex argument.
z = x + iy
IZI
Argand (1806) is credited with the discovery, of
the Argand diagram (also known as the Argand
plane) .
Historically, the geometric representation of
a complex number as a point in the plane was
important because it made the whole idea of a
complex number more acceptable. In particular,
this visualization helped "imaginary" and
"complex" numbers become accepted in
mainstream mathematics as a natural extension
to negative numbers along the real line.
y
2 + 3i
3
2
1
x
1
2
3
We can represent complex numbers as a
point.
Polar representation of a
Complex number
• Let the point P
represent the non zero
complex number
z = x + iy. Let the
directed line segment
OP be the length r and
Ө be the angle which
OP makes with the
positive direction of xaxis
• We may note that the
point P is uniquely
determined by the
ordered pair of real
numbers (r, Ө), called the
polar coordinates of the
point P.
• We consider the origin as
the pole and the positive
direction of the x-axis as
the initial line.
• We have ,
x = r cos Ө,
y = r sin Ө and therefore ,
z = r(cos Ө + i sin Ө).
The latter is said to be
the polar form of the
complex number.
• Here
is the modules of z and Ө
is called the argument of
z which is denoted by arg
z.
• For any complex number z ≠ 0, there
corresponds only one value of Ө in 0 ≤ Ө < 2𝝅
However any other interval of length 2 𝝅
for example - 𝝅 < Ө ≤ 𝝅 , can be such an
interval. We shall take the value of Ө such
that - 𝝅 < Ө ≤ 𝝅 , called principle argument of
z and is denoted by arg z, unless specified
otherwise. Figures in the next slide.