6.8

The Complex Numbers

Recall: The square root of a negative number does not exist in the real number system because there is no real number that, when squared, will result in a negative number. For example, − 25 is not a real number because there is no real number whose square is − 25 . Square roots of negative numbers do exist. These numbers exist under another number system called complex numbers. Defn.: The imaginary number i is defined as i = i 2 = − 1. Defn: For any positive number p ,

− 1 . Therefore, squaring both sides give us

− p = i p.

Examples: −2 = i 2 −3 = i 3

These numbers are of the form bi where b is a nonzero real

− 4 = i 4 = 2i

number.

− 25 = i 25 = 5 i

Defn.: A complex number is a number of the form a + bi where a and b are real numbers; a is called the real part and b is called the imaginary part. Examples of Complex Numbers: Complex Number

Real Part

Imaginary Part

5 + 2i

5

2

1 − 7i 3

1 3

−7

8i

0

8

4

4

0

Adding and Subtracting Complex Numbers → To add or subtract complex numbers, add or subtract their real parts and imaginary parts separately.

Addition Rule: Subtraction Rule:

(a + bi ) + (c + di ) = (a + c ) + (b + d )i (a + bi ) − (c + di ) = (a − c ) + (b − d )i

Examples: 1. (2 − 5i ) + (1 + 3i ) Solution: = (2 + 1) + (− 5 + 3)i = 3 + (− 2)i

= 3 − 2i

2. (−1 + 9i ) − (2 − 4i ) Solution: = (−1 − 2 ) + (9 − (− 4 ))i = (− 3) + (9 + 4 ) i = − 3 + 13i Or = − 1 + 9i − 2 + 4i = (− 1 − 2 ) + (9 + 4 )i = − 3 + 13i

(

) (

)

3. 5 − 2 − 4 + − 5 + 7 − 1 Solution: = (5 − 2 ⋅ 2i ) + (− 5 + 7i ) = (5 − 4i ) + (− 5 + 7i ) = (5 − 5) + (− 4 + 7 )i = 3i

e.g.

Perform the indicated operation.

1.

(6 + 5i ) + (− 2 − 6i )

2.

(− 4 + 3i ) − (8 − i )

3.

(1 + 5

) (

−9 + 4 +

− 16

)

Multiplying Radicals with Negative Radicands → The product rule a ⋅ b = ab only works if a and b are real numbers. → To multiply square roots with negative radicands, we must first express the numbers in terms of i , since the product rule of radicals does not in general hold for complex numbers. Examples: Multiply the following: 1. 8⋅ 6 Solution: 8⋅ 6 =

48 =

24 ⋅3 = 2 2 3 = 4 3

2. 8⋅ −6 Solution:

( )

(

)

8 ⋅ − 6 = 8 ⋅ i 6 = i 48 = i 4 3 = 4i 3

3. −8⋅ −6 Solution: − 8 ⋅ − 6 = i 8 i 6 = i 2 48 = −1⋅ 4 3 = − 4 3

( )( )

e.g.

(

)

Multiply the following:

1.

− 5 ⋅ −15

2.

− 24 ⋅ 6

Multiply Complex Numbers → We multiply complex numbers just like we would multiply polynomials. Just remember to replace i 2 with − 1. e.g.

5 (− 2 + 3i ) = −10 + 15i

(8 + 3i )(−1 + 4i )

by the distributive law

= (8)(− 1) + (8)(4i ) + (3i )(− 1) + (3i )(4i ) = − 8 + 32i − 3i + 12i 2 = − 8 + 29i + 12 (− 1) = − 8 + 29i − 12

e.g.

= − 20 + 29i

e.g.

Multiply and simplify.

1. − 3 (6 − 7i )

2.

(5 − i )(4 + 8i )

by FOIL

3.

(− 2 − 9i )(− 2 + 9i )

Multiply a Complex Number by its Conjugate Defn: The conjugate of a + bi is a − bi . The product of a complex number and its conjugate is always a real number.

(a + bi )(a − bi )

2

= (a ) − (bi )

2

= a 2 − b 2i 2 = a 2 − b 2 (− 1) = a2 + b2

a real number

Divide Complex Numbers Rule: To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Write the quotient in the form of a + bi .

e.g.

4 + 5i 3 3 = ⋅ 4 − 5i 4 − 5i 4 + 5i 3 (4 + 5i ) = (4 − 5i )(4 + 5i ) 12 + 15i = 42 + 52 12 + 15i = 16 + 25 12 + 15i 41 12 15 = + i 41 41 =

6 − 2i 6 − 2i −7 − i = ⋅ −7 + i −7 + i −7 − i = = e.g.

= = = =

(6 − 2i )(− 7 − i ) (− 7 + i )(− 7 − i ) − 42 − 6i + 14i + 2i 2

(− 7 )2

+ (1) − 42 + 8i − 2 49 + 1 − 44 + 8i 50 − 44 8 + i 50 50 − 22 4 + i 25 25

2

e.g. 1.

2.

Divide and simplify. 8i 4+i

ans.

2 + 3i 5 − 6i

8 32 + i 17 17

ans. −

8 27 + i 61 61

Simplifying Powers of i → We use the fact that i 2 = − 1 to simplify powers of i . → Let’s write i through i 8 in their simplest forms.

i is in simplest form i 2 = −1 i 3 = i 2 ⋅i = − 1⋅i = − i

( )

2

2

i 4 = i 2 = (− 1) = 1 i 5 = i 4 ⋅ i = 1⋅ i = i i 6 = i 5 ⋅ i = i ⋅ i = i 2 = −1 i 7 = i 6 ⋅ i = − 1⋅ i = − i i 8 = i 7 ⋅ i = − i ⋅ i = − i 2 = − (− 1) = 1 → The pattern repeats so that all powers of i can be simplified to i , − 1, − i , or 1.

e.g.

Simplify each power of i .

1. i 25

2. i 343

3. Divide and simplify:

2i 1 + i 55