Complex Numbers

Complex Numbers

• Complex number is a number in the form z = a+bi, where a and b are real numbers and i is imaginary. Here a is the real part and b is the imaginary part of z.

• Where

• Examples: If r is a positive real number, then

r ri

3 3i 4 4i i2

2

3 2

4 2 2

5 4

1

1 1 1

1

...

i

i i i i

i i i

i i i i i

Geometry

Plot the points 3 + 4i and –2 – 2i in the complex plane.

Imaginary axis

Real axis

2

4

– 2

– 2

2

(3, 4) or

3 + 4i

(– 2, – 2) or

– 2 – 2i

Operations on Complex Numbers

)26)(32( ii 2618412 iii

)1(62212 i

62212 ii226

26103 Solve 2. 2 x

363 2 x122 x

122 x

12ix 32ix

)3( ii 23 ii )1(3 i

i31

3

5 2

i

i2

2

3 5 2

5 2 5 2

15 6

25 46 15 6 15

29 29 29

i i

i i

i i

ii

i

16 49 16 49 4 7 28 Which one is true?

or

Absolute ValueThe absolute value of the complex number z = a + bi is the distance between the origin (0, 0) and the point (a, b).

Example:Plot z = 3 + 6i and find its absolute value.

2 2| | a bi a b

2 23 6z 9 36

453 5

Imaginary axis

Real axis

4

4

– 2– 4

2

6

8

3 5 units

z = 3 + 6i

Trigonometric Representation of Complex NumberTo write a complex number z = x + yi in trigonometric form, let be the angle from the positive real axis (measured counter clockwise) to the line segment connecting the origin to the point (x, y).

x = r cos y = r sin

Imaginary axis

Real axis

yr

x

z = (x, y)

r =

z = x + yi = r (cos + i sin )

The number r is the modulus of z, and is the argument of z.

Example: 1 cos sin2 3 3

z i

modulus argument

tan =

1111 sincos irz 2222 sincos irz

1 2 1 1 1 2 2 2cos sin cos sinz z r i r i

212121 sincos irr

Let and

zz

rr

i1

2

1

21 2 1 2 cos sin

then,0 If 2 z

ExampleWrite the complex number z = –7 + 4i in trigonometric form.

Imaginary axis

Real axis

z = –7 + 4i

7 4z r i 2 2( 7) 4 65

cos sinz r θ i θ

65(cos150.26 sin150.26 ) i

65z 150.26°

180 29.74 150.26

1 4tan 29.747

tan = = arctan ()

3cos4

22

2 23.752 2

z i

Standard form

Write the complex number in standard form a + bi.

3 33.75 cos sin4 4

i Example:

23sin4 2

15 2 15 28 8

i

De Moivre’s Theorem

Expanding a power of a complex number in rectangular form is tedious.

...

na bi

a bi a bi a bi a bi

The best way to expand one of these is using Pascal’s

triangle and binomial expansion.

It’s much nicer in trig form. We just raise the r to the power and multiply theta by the exponent.

cos sin

cos sinn n

z r i

z r n i n

3 3

3

5 cos20 sin 20

5 cos3 20 sin 3 20

125 cos60 sin 60

Example

z i

z i

z i

Nth Root of a Complex Number cos sin

360 360 2 2cos sin cos sinn n n

z r i

k k k kz r i or r i

n n n n

; k =0,1,…,n-1

Find the 4th root of 81 cos80 sin80z i

and θ = 80 n=4

Put K=0,1,2,3 into the above equation we get 4 roots as follows:

1

2

3

4

3 cos20 sin 20

3 cos 20 90 sin 20 90 3 cos110 sin110

3 cos 110 90 sin 110 90 3 cos200 sin 200

3 cos 200 90 sin 200 90 3 cos290 sin 290

z i

z i i

z i i

z i i

Assignment: Find the 6th root of unity. That is, solve

Hint: 1=1+i.0=1(cos0+isin0)