Complex NumbersComplex Numbers

Consider the quadratic equation x2 + 1 = 0.

Solving for x , gives x2 = – 1

12 x

1x

Complex Numbers

Since there is not real number whose Since there is not real number whose square is -1, the equation has no real square is -1, the equation has no real solution. French mathematician solution. French mathematician Rene Descartes (1596 -1650) Rene Descartes (1596 -1650) proposed that proposed that i i be defined such that,be defined such that,

1i

1i

Complex Numbers

12 iNote that squaring both sides yields:therefore

and

so

and

iiiii *1* 13 2

1)1(*)1(* 224 iii

iiiii *1*45

1*1* 2246 iiii

And so on…

Finding Powers of Finding Powers of ii

The successive powers of i rotate through the four values of i, -1, -i, and 1.

in = i if n = 1, 5, 9, …

in = -1 if n = 2, 6, 10, …

in = -i if n = 3, 7, 11, …

in = 1 if n = 4, 8, 12, …

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

Real NumbersImaginary Numbers

Real numbers and imaginary numbers are subsets of the set of complex numbers.

Complex Numbers

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

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

Adding and SubtractingAdding and Subtracting

To Add or Subtract Complex Numbers

1. Change all imaginary numbers to bi form.

2. Add (or subtract) the real parts of the complex numbers.

3. Add (or subtract) the imaginary parts of the complex numbers.

4. Write the answer in the form a + bi.

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

MultiplyingMultiplying

To Multiply Complex Numbers

1. Change all imaginary numbers to bi form.

2. Multiply the complex numbers as you would multiply polynomials.

3. Substitute –1 for each i2.

4. Combine the real parts and the imaginary parts. Write the answer in a + bi form.

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

CAUTION!CAUTION!

?24

2424 ii

22 2i 22

824

DividingDividing

To Divide Complex Numbers

1. Change all imaginary numbers to bi form.

2. Rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

3. Substitute –1 for each i2.

DividingDividing

Examples:

i

i

34

34

i

i

i

i

34

34

34

34

2316

3434

i

ii

916

92416 2ii25

247 i

53

5

53

5

i

53

53

53

5

i

i

i

259

)53(5

i

i

59

)5515 i

14

)5515 i

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