COMPLEX NUMBERS

Unit 4Radicals

Complex/Imaginary Numbers

WHAT IS?

25 25

WHY?

5

2552

There is no real number whose square is -25 so we have to use an imaginary number

125

15 i5

i 1“i” is an imaginary number. “i” is equal to the square root of -1

BASICALLY: any time you see a negative under a SQUARE ROOT an “i” gets pulled out.

Simplifying Radicals with Imaginary Numbers

ALWAYS pull the “i” out first before multiplying together.

503281 yx

36

25i yxyi9

182i

23 2i

23

2251 32811 yx

3161

Adding & Subtracting Complex Numbers

A complex number is a number with “i” in it. Complex numbers can be written in the form :

bia Real part

Imaginary part

To add or subtract complex numbers combine the real parts and combine the imaginary parts separately.

Adding & Subtracting Complex Numbers

)34()65( ii )42()35( ii

i31 i77

Multiplying Complex Numbers

You multiply complex numbers like you would binomials. (Double Distribute, Box, FOIL…etc)

)53)(32( ii )52)(6( ii

23012 ii )1(3012 i

i1230

i106 2159 ii )1(1516 i

i21

Dividing Complex Numbers

Remember that we don’t want to leave a radical in the denominator.

To simplify a quotient, multiply by the conjugate of the denominator.

Conjugate – change only the middle sign

)43( i)42( i

)53(

CONJUGATE =

CONJUGATE =

CONJUGATE =

)43( i)42( i

)53(

Rationalize the Denominator

i

i

41

32

Simplify

i

i

41

412

2

161

12382

i

iii

17

1110 i

i

i

5

27

i

i

5

52

2

25

1035

i

ii

25

3510

i

5

72

i

5

72 iImaginary # song

i

1i

2i

3i

0i

1

i

1

12i

22i

33i

Since “i” raised to a power follows a pattern you can easily find the answer by dividing the exponent by 4 and using the remainder to simplify.

What about higher exponents? 4-7?

4 goes into 12, 3 times with a remainder of zero.

10 i

4 goes into 22, 5 times with a remainder of 2

12 i

4 goes into 33, 8 times with a remainder of 1

11 i