lecture 1_complex no

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

lecture 1_complex no

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