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§7.7 Complex

Numbers

Review §

 Any QUESTIONS About

• §7.6 → Radical Equations

 Any QUESTIONS About HomeWork

• §7.6 → HW-

7.6 MTH 55

The “Number” i

• i is the unique number for which

i^2 = −1 and so

i = − 1

 Thus for any positive number p we can now define the square root of a negative number using the product-rule as follows . − p = − 1 p = i p or pi

Imaginary Numbers

• An imaginary number is a number that

can be written in the form bi , where b is

a real number that is not equal to zero

• Some

Examples (^5 ) 73

37 ⋅ −

i

i i

 i is called the “imaginary unit”

ReWriting Imaginary Numbers

• To write an imaginary number in

terms of the imaginary unit i :

− n

1. Separate the radical into two

factors − 1 ⋅ n.

2. Replace − 1 with i

3. Simplify n.

Example  Imaginary Numbers

• Express in terms of i :

a) (^) − 9 b) − − 48

 SOLUTION

a)

b) −^ −^48 = −^ − ⋅1 16 3^ ⋅

− 9 = − ⋅1 9 = − 1 9 = i ⋅3, or 3. i

= − − 1 16 3 = − ⋅ i 4 ⋅ 3 = − 4 i 3

Complex Number Examples

• The following are examples of Complex numbers 7 2

1 2 3

11

i

i

i

−

Here a = 7, b =2.

Here a = 2, b = −1/ 3.

Here a = 0, b = 11.

Example  Complex Add & Sub

• Add or subtract and simplify a + bi

(−3 + 4 i ) − (4 − 12 i )

 SOLUTION: We subtract complex numbers just like we subtract polynomials. That is, add/sub LIKE Terms → Add Reals & Imag’s Separately

• (−3 + 4 i ) − (4 − 12 i ) = (−3 + 4 i ) + (−4 + 12 i )
• = −7 + 16 i

Example  Complex Add & Sub

• Add or subtract and simplify to a + bi

a) (^) (3 + 2 ) i + (7 +b) 8 ) i (10 − 2 ) i − (9 + i )

 SOLUTION

a)

b)

(3 + 2 ) i + (7 + 8 ) i = (3 + 7) + (2 i +8 ) i = 10 + (2 + 8) i = 10 + 10 i

Combining real and imaginary parts

(10 − 2 ) i − (9 + i ) = (10 − 9) + −( 2 i − i ) = 1 − 3 i

Caveat Complex-Multiplication

• CAUTION
• With complex numbers, simply

multiplying radicands is incorrect

when both radicands are negative:

− 3 ⋅ − 5 ≠ 15.

 The Correct Multiplicative Operation

( ) ( ) ( ( ) ) ( ( ) ) ( )( ) ( 1 1 )( 3 5 ) ( 1 ) ( 3 5 ) ( 1 ) 15 15

3 5 1 3 1 5 1 3 1 5 2 = − ⋅ − ⋅ = − ⋅ = − = −

− ⋅ − = − ⋅ ⋅ − ⋅ = − ⋅ − ⋅

Example  Complex Multiply

• Multiply & Simplify to a + bi form

a) b) c)

 SOLUTION

a)

− 2 ⋅ − 10 2 i^^ (^5 +^3 i ) (^2 +^ i^ )(^4 −^3 i )

= i ⋅ 2 ⋅ ⋅ i 10

= i^2 ⋅ 20 = − ⋅1 2 5 = − 2 5

Example  Complex Multiply

• Multiply & Simplify to a + bi form

a) b) c)

 SOLUTION : Use F.O.I.L.

c)

( 2 +^ i^ )( 4 −^3 i )

( )( )

2 2 + i 4 − 3 i = 8 − 6 i + 4 i − 3 i

= 8 − 2 i + 3

= 11 − 2 i

Complex Number CONJUGATE

• The CONJUGATE of a complex number a

+ bi is a – bi , and the conjugate of a – bi

is a + bi

• Some Examples

31 2 Conjugate 31 2

13 Conjugate 13 i i

i i − ⇒ +

−π + ⇒ −π −