5.4 Complex Numbers5.4 Complex Numbers

Until now, you have always been told that you can’t take the square root of a negative number.

If you use imaginary units, you can!

OBJ: To solve quadratic equations with complex solutions & perform operations with complex numbers

Imaginary UnitImaginary Unit

• The imaginary unit is ¡

• ¡= • It is used to write the square root of

a negative number

1

Property of the square root of Property of the square root of negative #snegative #s

If r is a positive real number, then:

r riEx 1:

3 4

then,1- If i12 iii 3

14 iii 5

16 iii 7

18 i

*For larger exponents, divide the exponent by 4, then use the remainder

as your exponent instead.

Ex 2: ?23 i3 ofremainder a with 5

4

23

.etc

ii - which use So, 3

ii 23

Ex 3:Ex 3:2)3(i

22 )3(i)3*3(1

)3(13

26103 Solve 2 x

363 2 x122 x

122 x

12ix 32ix

a. b.

Complex NumbersComplex NumbersA complex number has a real part & an

imaginary part.

bia

Real part Imaginary part

Like: 5+4iLike: 5+4i

Adding and SubtractingAdding and Subtracting(add or subtract the ”matching” parts)(add or subtract the ”matching” parts)

Ex 4: )33()21( ii )32()31( ii

i52

Ex 5: )73()32( ii )73()32( ii

i41

Ex: 6 )32()3(2 iii )32()23( iii

i21

HomeworkHomework

MultiplyingMultiplying((Treat the i’s like variables, then simplify any Treat the i’s like variables, then simplify any that are not to the first power)that are not to the first power)

Ex 7: )3( ii 23 ii

)1(3 i

i31

Ex 8: )26)(32( ii 2618412 iii

)1(62212 i62212 i

i226

i

i

i

i

21

21

21

113

)21)(21(

)21)(113(

ii

ii

2

2

4221

221163

iii

iii

)1(41

)1(2253

i

41

2253

i

5

525 i

5

5

5

25 i

i 5

Ex 9:

DividingDividing(Multiply the numerator and the

denominator by the “conjugate”)

Ex 10:

i

i

21

35

i

i

21

21

2

2

4221

63105

iii

iii

5

131 i

ior5

13

5

1

Absolute Value of a Complex #Absolute Value of a Complex #

If you have a complex number the absolute value can be found using:

) ( bia

22 ba

Ex 11Ex 11::a. i52

22 )5()2(

254 29

b. i622 )6()0(

360

366