Complex Numbers 2-4

Imaginary Numbers

1. Designed so negative numbers can have square roots.

, i 2 = -1

2. Imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i 2 = -1.

Example

i 1

551515

27492492198 ii

Practice

yy

x

53 .5

5 .4

18 .3

9-- .2

11 .1

2

15||

5

23

3

11

iy

ix

i

i

i

Powers of i

i 1 = i

i 2 = -1

i 3 = -i

i 4 = 1

Higher Powers of iDivide the exponent by 4, then determine

the remainder.

If the remainder is

1 i 1 = i

2 i 2 = -1

3 i 3 = -i

0 i 0 = i 4 = 1

Practice

1. i 20 =

2. i 27 =

3. i 71 - i 49 =

4. i 4444484044844444441 =

5. i -2 =

6. i -27 =

Negative Exponents

11

11

1

1

11-

1

1

1

44-

3-

2-

1-

ii

ii

i

i

-ii

i

ii

i

i

ii

Multiplying Imaginary Numbers

Practice

1010

182

155

33

3646

i

iii 24i

-18

6

-10

Complex Numbers• Complex numbers are the sum of a real

number and an imaginary number.

• They are written in the form a + bi where a is a real number and bi is an imaginary number.

• Complex numbers include real and imaginary numbers since 3 = 3 + 0i or 4i = 0 + 4i

• Imaginary numbers follow the properties we have learned (commutative, associative, distributive…)

Complex Numbers a + bi

Real Numbersb=0

Imaginary Numbersb≠0

PUREImaginaryNumbersa=0

The Big Picture where does everything fit

Pure

복소수

허수

순허수

Addition and Subtraction

• Combine the real parts and combine the imaginary parts.

• (6 + 3i) + (8-2i) = (6+8) + (3 – 2)i = 14 + i

• (6 + 3i) - (8-2i) = (6-8) + (3 – -2)i = -2 + 5i

• (6 - 3i) + (8-2i) = (6+8) + (-3 – 2)i = 14 - 5i

Practice

• 3i – (5 – 2i)=

• (-2 + 8i) – (7+3i)=

• 4 – 10i + 3i – 2 =

-5 + 5i

-9 +5i

2 – 7i

Graphing Complex Numbers

Finding Absolute Values

The absolute value of a complex number | a + bi | is its distance from the origin. So we use the distance formula or simplified as

we ignore the i for the formula

)b a (from 22 iba

Finding Absolute Values

Practice - Find

1)|6 – 4i |

2)|-2 + 5i |

3)|4i| 3) 4

Multiplying Complex Numbers

• Multiply like real numbers and treat i like a variable but i2 = -1

(3 + 2i)(4-7i) = 12 – 21i +8i -14i2 =

12 – 13i -14(-1) = 12-13i +14 =

26 – 13i

Practice

1) (3+2i )(2-i )

2) (2-i )(2+i )

3) (6-5i )(3-2i )

1) 6 – 3i +4i – 2i2 = 6+i+2= 8+ i

2) 4 + 2i -2i – i2 = 4 – (-1) = 5

3) 18 – 12i -15i -10= 8 - 27i

Division

Complex Conjugates

i

i

23

35

13

199

)4(9

)1(61915

23

23

23

35 ii

i

i

i

i

Practice

Equations with complex numbers

• Two complex numbers are equal if their real part is equal and their imaginary part is equal.

• If a+bi = c+di then a=c and b=d

• 5x+1 + (3+2y)i = 2x-2 + (y-6)i

• real part 5x+1 = 2x-2, 3x = -3, x=-1

• imaginary part 3+2y = y-6, y=-9

Conjugates 켤레복소수

In algebra, a conjugate is a binomial formed by negating the second term of a binomial. The conjugate of x + y is x − y,

where x and y are real numbers.

If y is imaginary, the process is termed complex conjugation: the complex conjugate of a + bi is a − bi,

where a and b are real.

Complex Conjugates

i

i

23

35

13

199

)4(9

)1(61915

23

23

23

35 ii

i

i

i

i

Practice

Equations with complex numbers

• Two complex numbers are equal if their real part is equal and their imaginary part is equal.

• If a+bi = c+di then a=c and b=d

• 5x+1 + (3+2y)i = 2x-2 + (y-6)i

• real part 5x+1 = 2x-2, 3x = -3, x=-1

• imaginary part 3+2y = y-6, y=-9

Graphing Points

• You have the real axis (x-axis) and the imaginary axis (y-axis).

• Plot point (a, b)

• You can plot inequalities by shading areas of the graph.

• How would we graph

{a + bi | a≤3 and b ≤2} ?

Graph {a + bi | a≤3 and b ≤2}

What happens when we multiply a complex number, a + bi by i ?

4 + 2i -2 + 4i

-4 -2i

2 – 4i

More about Complex NumbersTo solve an equation involving complex numbers,

equate the real parts and equate the imaginary parts. 3 1 2 2 2

2

1 2 2

2 1 2 2

3 1 4 2 4

2 4 2 4

x y i x yi

y i yi

x yi i yi

i yi

i i

i i

i i

( )

( ) ( )

set up two equations - real and imaginary

real imaginary

3x +1 = 2x ( + 2)

2

y = 2

check : 3(-1) +1+ (2 + 2)

Product of a Complex Number and its Conjugate

• What happens when you multiply a complex number by it’s conjugate?

(3+4i )(3-4i ) =

So (a + bi )(a – bi ) =

9-(-16) = 25

a2 – (b2)(i2) = a2 – (-1)(b2)

= a2 + (b2)

Dividing and Reciprocals

• When you divide by a complex number (a fraction with a complex number as a denominator) you multiply both the numerator and denominator by 1 by multiplying both by the conjugate of the denominator,

Example3 2

2

3 2

2

2

2

6 3 4 2

4 2 2

6 2

4 1

8

5

8

5

1

5

2

2

i

i

i

i

i

i

i i i

i i i

i ii

( )

( )

Finding the Reciprocal

• To find the reciprocal of a complex number, you divide 1 by that complex number.

• The reciprocal of 3 + 2i is

• But now you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate.

• So find the reciprocal of 3 + 2i

1

3 2 i

1

3 2

1

3 2

3 2

3 2

3 2

13

3

13

2

13

i i

i

i

ii