Lecture No 01
Semester: 1st A
Engineering Technology
Subject Title
Mathematics-I
Recommended BOOKS
• Calculus by Thomas Finney
• Schaum’s series, Calculus, Schaum’s Series(Latest Edition)
• Schaum’s series, Complex, Variables Va Schaum’s series,
(Latest Edition)
• Antom, H. Calculus and Analytic Geometry, John Wiley
and Sons (Latest Edition)
• Talpur, Calculus and Analytic Geometry, Ferozsons (Latest
Edition)
• Yousuf, S.M. Mathematical Methods, Ilmi Kutab Khana
(Latest Edition)
COURSE CONTENTS
Weeks Planned Topics
Week 1: Complex numbers
Week 2: Argand diagram,
Week 3: De Moivre‟s theorem,
Week 4: Hyperbolic and inverse hyperbolic functions.
Week 5: Algebra of vectors and matrices, systems of linear equations.
Week 6: Derivative as slope, as rate of change (graphical representation).
Extreme values, tangents and normals, curvature and radius of
Week 7:
curvature.
Week 8: Differentiation as approximation.
Partial derivatives and their application to extreme values and
Week 9:
approximation
Week 10: Integration by substitution and by parts,
Week 11: Integration by parts
integration and definite integration as area under curve (graphical
Week 12:
representation). Reduction formulae
Week 13: Double integration and its applications
Week 14: Polar and Cartesian coordinates
Week 15: Polar curves, radius of curvature
Week 16: cycloid, hypocycloid, epicycloids and involutes of a circle
Today’s Topic
Complex Number
Lecture Contents
At the end of this lecture, students will be able to understand:
History
Number System
Complex Numbers
Operations on Complex Numbers
History
Complex numbers were first introduced by G.
Cardano
R. Bombelli introduced the symbol 𝑖.
A. Girard called “solutions impossible”.
C. F.Gauss called “complex number”
(ComplexNumber) 6
Number System
Imaginary
Real Numbers
Number
Irrational Rational
Number Number
Natural Whole
Integer
Number Number
(ComplexNumber) 7
Complex Numbers
• A complex number is a number that can b express in
the form of"a+b𝒊".
• Where a and b are real number and 𝑖 is an imaginary.
• In this expression, a is the real part and b is the
imaginary part of complex number.
(ComplexNumber) 8
Complex Number
When we combine the real and
imaginary number then
complex number isform.
Real Imaginary Complex
Number Number Number
Complex Number
• A complex number has a real part and an imaginary part,
But either part can be 0 .
• So, all real number and Imaginary number are also
complex number.
(ComplexNumber) 10
Complex Numbers
Complex number convert our visualization into physical things.
(ComplexNumber) 11
COMPLEX NUMBERS
A complex number is a number consisting
of a Real and Imaginary part.
It can be written in the form
i 1
COMPLEX NUMBERS
Why complex numbers are introduced???
Equations like x2=-1 do not have a solution within
the real numbers
x 2 1
x 1
i 1
i 1
2
COMPLEX CONJUGATE
The COMPLEX CONJUGATE of a complex number
z = x + iy, denoted by z* , is given by
z* = x – iy
The Modulus or absolute value
is defined by
z x y 2 2
COMPLEX NUMBERS
Equal complex numbers
Two complex numbers are equal if their
real parts are equal and their imaginary
parts are equal.
If a + bi = c + di,
then a = c and b = d
ADDITION OF COMPLEX NUMBERS
(abi)(cdi) (ac)(bd)i
Imaginary Axis
EXAMPLE
z2
z sum
z1
(2 3i) (1 5i)
z2
(2 1) (3 5)i Real Axis
3 8 i
SUBTRACTION OF COMPLEX
NUMBERS
(abi)(cdi) (ac)(bd)i
Imaginary Axis
Example z1 z2
( 2 3 i ) (1 5 i ) z2
zdiff
( 2 1 ) (3 5)i z2
Real Axis
1 2i
MULTIPLICATION OF COMPLEX
NUMBERS
(a bi)(c di) (acbd)(ad bc)i
Example
(2 3i)(1 5i)
( 2 1 5 ) (10 3)i
13 13i
DIVISION OFACOMPLEX
NUMBERS
a bi
a bi c di
c di c di c di
ac adi bci bdi 2
c d
2 2
ac bd bc ad i
c d
2 2
EXAMPLE
6 7 i
6 7 i 1 2 i
1 2 i 1 2 i 1 2 i
6 12i 7i 14i 2 6 1 4 5i
1 2
2 2
1 4
2 0 5i 2 0 5i
4 i
5 5 5
Assignment #1
Solve the following
(1+3i)+(2+5i)
(1+3i)-(2+5i)
(1+3i)x(2+5i)
(1+3i)÷(2+5i)
