Lecture 1_Complex No
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Transcript of Lecture 1_Complex No
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8/3/2019 Lecture 1_Complex No
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After successfully completing this chapter,you should be able to:
a) define complex numbers
b)
solve operations with complex numbersc) solve quadratic equations involving complex
numbers
d) show complex numbers on an Argand
diagrame) find the modulus and argument of a
complex number
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Lecture content:1 COMPLEX NUMBER
1.1 Definition
1.2 Operations with Complex Numbers
1.2.1 Addition and Subtraction1.2.2 Multiplication
Introducing complex conjugate
1.2.3 Division
1.3 Solving Quadratic Equations1.4 Argand Diagram
1.5 Modulus and Argument
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Complex Numbers1.1 Definition
The complex numbers consist of numbers of the form
,
where a and b are real numbers and i = 1
is called the real part and is called the imaginary
part.
Eg. Re ( i) = ; Im ( i) = .
0i are called real numbers.
0 i are called imaginary numbers.
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Eg.
2+3i 3-5i -4+7i -9i
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About i i= 1
i = 1
i = ?
i. i = 1 i= - I i = ?
i. i = 1 1 = 1
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1.2 Operations with complex numbers1.2.1 Addition and Subtraction
By the usual rules of algebra,
i i = i
Eg.
If = 2 3i and = 2 3i, find in the form i,
(a) (b)
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Eg.
If = 2 3i and q = 2 3i, find in the form a bi ,(a) = (2 3i) (2 3i)
= 2 2 3 3 i
= 4
(b) = (2 3i) (2 3i)
= 2 2 3 3 i
= 6i
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1.2.2 Multiplication
By the usual rules for multiplying out brackets,
i i = i i i
= i
Eg.
If = 2 3i and q = 2 3i, find in the form a bi,
(a) (b)
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1.2.2 Multiplication
(a) = 2 3i 2 3i
= 4 6i 6i 9i
= 4 9 = 13(b) = 2 3 2 3
= 4 6i 6i 9i
= 5 12i
An important special case is =
The result is a real number.
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Introducing complex conjugateIf is a complex number, then the complex conjugateof is denoted by .
If = i, then = i.
Eg. What is the conjugate of the complex number
below?(a) 24i (b) 2 (c) 4i (d) 2 4i
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1.2.3 Division
Eg.
1)+
=
=
i
2)
=
=
i
3)+
=
+
+
+=
+
7=
7+
7i
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Problem:
If 2 i i = 1 3i, find and by using
division method.
Ans: 1
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1.3 Solving quadratic equationsEg. Solve the quadratic equation 4 13 = 0.
= 4
2
=4 4 52
2
=4 36
2
=
= 2 3i.
Solve the quadratic equation 40 = 10.
Ans: 5 15i.
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1.4 Argand Diagram
One of the ways of representing a complex number
geometrically is using the Argand diagram.
This is named after John-Robert Argand (1786-1822), a Parisianbookkeeper and mathematician.
There are two axis: the real axis and the imaginary axis.
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Argand Diagram (cont)
Eg. Represent = 3+3i on an Argand diagram.
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Argand Diagram (cont)
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1.5 Modulus and Argument
: (modulus)
: argument, where <
Clearly,
= , > 0
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Find the modulus and argument for the following
complex numbers
(a) 24i
(b) 2
(c) 4i(d) 24i
(e) 2 4i