Complex Numbers 2015 Imagine That!. Warm-Up Find all solutions to the polynomial. Copyright © by...

Complex Numbers2015

Imagine

That!

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Warm-Up

Find all solutions to the polynomial.

5 4 3 22 7 8 8 8 0x x x x x


Warm-Up

Find all solutions to the polynomial.

2 1 0x

2.4 Complex Numbers

Students will use the imaginary unit i to write complex numbers.

Students will add, subtract, and multiply complex numbers.

Students will use complex conjugates to write the quotient of two complex numbers in standard form.


The letter i represents the numbers whose square is –1.

1i = Imaginary unit

If a is a positive real number, then the principal square root of

negative a is the imaginary number i .a

a a= i

Examples: 4 4= i = 2i

36 36= i = 6i

The number a is the real part of a + bi, and b is the imaginary part.

A complex number is a number of the form a + bi, where a and

b are real numbers and i = .1

i2 = –1


Examples of complex numbers:

Real Part Imaginary Parta bi+

2 7i+20 3i–

Real Numbers: a + 0i Pure Imaginary Numbers: 0 + bi

a + bi form16 50+ i=

24 + 5i=

225 •+ i16= Simplify using the product property of radicals.

Simplify: = i = i = 3i90 90 109 • 101.

= i64 64 = 8i2.

+16 503.


To add or subtract complex numbers:

1. Write each complex number in the form a + bi.

2. Add or subtract the real parts of the complex numbers.

3. Add or subtract the imaginary parts of the complex numbers.


(10 + ) + (21 – )55

= (10 + i ) + (21 – i )55 1i =

= 31

Group real and imaginary terms.

a + bi form

= (10 + 21) + (i – i )55

(11 + 5i) + (8 – 2i )

= 19 + 3i

Group real and imaginary terms.

a + bi form

= (11 + 8) + (5i – 2i )

Adding Complex Numbers


(– 21 + 3i ) – (7 – 9i)

= (– 21 – 7) + [(3 – (– 9)]i

= (– 21 – 7) + (3i + 9i)

= –28 + 12i

(11 + ) – (6 + )16 9

= (11 + i ) – (6 + i )16 9

= (11 – 6) + [ – ]i16 9

= (11 – 6) + [ 4 – 3]i

= 5 + i

Group real and

imaginary terms.

Group real and

imaginary terms.

a + bi form

a + bi form

Subtracting Complex Numbers


a)

b)

c)

d)

You Try: Adding and Subtracting Complex Numbers

( ) ( )3 2 3 i i

2 4 2i i ( )

3 2 3 5 ( ) ( )i i

( ) ( ) ( )3 2 4 7 i i i

5 2i

4

2i

0


To find the product of two complex numbers ,

distribute and combine like terms.

1. Use the FOIL method to find the product.

2. Replace i2 by – 1.

3. Write the answer in standard form: a + bi.

(a + bi)(c + di )


= 5i2 5

= 5 (–1) 5

= –5 5

2. 7i (11– 5i) = 77i – 35i2

= 35 + 77i

3. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i2

= 12 + 4i – 21i2

= 12 + 4i – 21(–1)= 12 + 4i + 21= 33 + 4i

Examples: 251. 25 5 = i i 5• •

= 5i i 5•

= 77i – 35 (– 1)


Examples: Multiplying Complex Numbers

a)

b)

c)

4 16

( )( )2 4 3 i i

( )( )3 2 3 2 i i

8

11 2i

13


The complex numbers a + bi and a - bi are called

complex conjugates.

Example: (5 + 2i)(5 – 2i) = (52 – 4i2)

= 25 – 4 (–1)

= 29

The product of complex conjugates is the real number a2 + b2.

(a + bi)(a – bi) = a2 – b2i2

= a2 – b2(– 1)

= a2 + b2


Replace i2 by –1 and simplify.

Dividing Complex Numbers

A rational expression, containing one or more complex numbers,

is in simplest form when there are no imaginary numbers

remaining in the denominator.

Multiply the expression by .i

i

Write the answer in the form a + bi.6

7

2

1 i

6

73

i

Example:7 3

6

ii

•

ii7 3

6

i

i7 3

6

i 2i

2i

) (6

) (3 7 i –1

–1


Replace i2 by –1 and simplify.

Multiply the numerator and

denominator by the conjugate of 2 + i.

Write the answer in the form

a + bi.

In 2 + i, a = 2 and b = 1.

a2 + b2 = 22 + 12

5

13 i

22

2

12

3 6 510

i i i

5

1

5

13 i

Simplify: 2

35

ii

i

2

35 i 2

2•

i

i

14

) (3 10

i –1


Examples: Dividing Complex Numbers

Write the quotient in standard form a + bi.2 3

4 2

i

i

1 8 1 4

10 10 5

ior i


Example 5: Plotting Complex Numbers

Plot each complex number in the complex plane.

a) b) c) d)2 3 i 1 2i 4 3iy

x–2

2


Complex Numbers

Engineers use imaginary numbers to study stresses on beams and to study resonance. Complex numbers help us study the flow of fluid around objects, such as water around a pipe. They are used in electric circuits, and help in transmitting radio waves. So, if it weren’t for i, we might not be able to talk on cell phones, or listen to the radio! Imaginary numbers also help in studying infinite series. Lastly, every polynomial equation has a solution if complex numbers are used. Clearly, it is good that i was created.

http://rossroessler.tripod.com/

Applications of Complex Numbers

•Control theory•Improper integrals•Fluid dynamics•Dynamic equations•Electromagnetism and electrical engineering•Signal analysis•Quantum mechanics•Relativity•Geometry – Fractals – Triangles•Algebraic number theory - Analytic number theory

Control Systems

Input Output

Feedback

Process


HOMEWORK

• Section 2.4, pg 133: 15-61 odd 65-71 odd

Complex Numbers 2015 Imagine That!. Warm-Up Find all solutions to the polynomial. Copyright © by...

Documents

Transcript of Complex Numbers 2015 Imagine That!. Warm-Up Find all solutions to the polynomial. Copyright © by...