5.45.4Complex Complex NumbersNumbers

By: L. Keali’i AliceaBy: L. Keali’i Alicea

GoalsGoals1) Solve quadratic equations with

complex solutions and perform operations with complex numbers.

2) Apply complex numbers to fractal geometry.

Imaginary UnitImaginary Unit• 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!

• The imaginary unit is ¡.• ¡= • It is used to write the square root of

a negative number.

1

Property of the square root Property of the square root of negative numbersof negative numbers

• If r is a positive real number, then

r ri

Examples:

3 3i 4 4i i2±

then,1- If i

12 i

ii 3

14 i

ii 5

16 i

ii 7

18 i

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

as your exponent instead.

Example: ?23 i3 ofremainder a with 5

4

23

.etcii - which use So, 3

ii 23

ExamplesExamples2)3( 1. i

22 )3(i)3*3(1

)3(13

26103 Solve 2. 2 x

363 2 x122 x

122 x

12ix 32ix

Complex NumbersComplex Numbers• A complex number has a real part &

an imaginary part.• Standard form is:

bia

Real part Imaginary part

Example: 5+4iExample: 5+4i

The Complex planeThe Complex plane

Imaginary Axis

Real Axis

Graphing in the complex planeGraphing in the complex plane

i34 .

i52 .i22 .

i34

.

i- axis

“Real” number axis

Adding and SubtractingAdding and Subtracting(add or subtract the real parts, then (add or subtract the real parts, then add or subtract the imaginary parts)add or subtract the imaginary parts)

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

i52

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

i41

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

i21

MultiplyingMultiplyingTreat the i’s like variables, then Treat the i’s like variables, then change any that are not to the change any that are not to the

first powerfirst power

Ex: )3( ii 23 ii

)1(3 i

i31

Ex: )26)(32( ii 2618412 iii

)1(62212 i62212 i

i226

i

i

i

iEx

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

Absolute Value of a Absolute Value of a Complex NumberComplex Number

• The distance the complex number is from the origin on the complex plane.

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

) ( bia

22 ba

ExamplesExamples1. i52

22 )5()2(

254 29

2. i622 )6()0(

360

366

Which of these 2 complex numbers is closest to the origin? -2+5i

AssignmentAssignment• 5.4 A (All)