complex number updated
TRANSCRIPT
WELCOME TO OUR PRESENTATION
Presented By :Jakir Hasan Saniatul Haque
Presented To :Md. Arifuzzaman (AZ)Lecturer (Mathematics)Department Of Natural SciencesDaffodil International University
COMPLEX NUMBERS• An ordered pair of real number generally
written in the form “a+ib”
• Where a and b are real number and is an imaginary.
• In this expression, a is the real part and b is the imaginary part of complex number.
COMPLEX NUMBER
Real Number
Imaginary Number
Complex Number
Definition of pure imaginary numbers:
Any positive real number b,
where i is the imaginary unit and bi is called the pure imaginary number.
b2 b2 1 bi
Definition of pure imaginary numbers:
i 1
i2 1 i is not a variable
it is a symbol for a specific number
Simplify each expression.
1. 81 81 1 9i
2. 121x5 121x4 1 x 11x2i x
3. 200x 100 1 2x 10i 2x
Any number in form a+bi, where a and b are real numbers and i is imaginary unit.
Definition of Complex Numbers
COMPLEX NUMBER
Complex number extend the concept of one-dimensional number line to the two-dimensional complex plan.
• Horizontal axis use for real part. • Vertical axis for the imaginary part.
Equations like x2=-1 do not have a solution within the real numbers
12 x
1x
1i
12 i Real no:
Imaginary no:
Why complex numbers are introduced???
THE POWERS OF i: then,1- If i
12 i ii 3 14 i ii 5
16 i ii 7 18 i .etc
iii
)53()12()51()32(
i83
Example
Addition : Complex number added by adding real part in real and imaginary part in imaginary.
(a + b) + (c + d ) = (a + c) + (b + d)
Fundamental Operations with complex number
Subtraction: Similarly, subtraction is defined
(a + b) - (c + d ) = (a - c) + (b - d) .
ii
ii
21)53()12()51()32(
Example
Multiplication:
The multiplication of two complex number is define by the following formula:
(a + b).(c + d ) =(ac - bd) + (b c + ad) Square of the imaginary unit is -1.
²== -1
ii
ii
1313)310()152(
)51)(32(
Example
Division:Division can be defined as:
𝑎+𝑏𝑖 𝑐+𝑑𝑖 =¿ () + (
EXAMPLE i
i2176
i
iii
2121
2176
22
2
21147126
iii
415146
i
5520 i
55
520 i
i4
How complex numbers can be applied to “The Real World”???
Examples of the application of complex numbers: 1) Electric field and magnetic field.2) Application in ohms law.
3) In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes
4) A complex number could be used to represent the position of an object in a two dimensional plane.