Complex Numbers

Section 0.7

What if it isn’t Real??

We have found the square root of a positive number like = 4,

Previously when asked to find the square root of a negative number like

we said there is not a real solution.

To find the square root of a negative number we need to learn about complex numbers

16

16

Imaginary unit

The imaginary unit is represented by

What would i² be??

i 1

i2 1

Simplify the following

25

i5

125

12

134

32i

Simplify the following

7 36 7 6i

This can not be simplified any further. Your solution is a complex number that contains a real part (the 7) and an imaginary part (the 6i).

Defining a Complex Number Complex numbers in standard form

are writtena + bia is the real part of the complex number and bi

as the imaginary part of the solution.

If a = 0 then our complex number will only have the imaginary part (bi) and is called a pure imaginary number.

Imaginary Number example:

Complex Number example:

Adding and Subtracting Complex Numbers

To add and subtract, simply treat the “i” like a typical variable.

ii 123

i2

Adding and subtracting complex numbers.

ii 375

i22

ii 3676

i10

Multiplying complex numbers ii 42

242 ii

)1(42 i42 ii24

ii 5125ii

)1(5 i

5i

i5

Always write in the form a + bi (real part first, imaginary second)

Multiply

(2 + 3i)(2 – i)

4 + 4i – 3(-1)

4 + 4i + 3

7 + 4i

Complex Conjugate

The product of complex conjugates is a real number (imaginary part will be gone)

(a + bi) and (a – bi) are conjugates.(a + bi)(a – bi)= a² - abi + abi - b²i²=a² - b²(-1)=a² + b²

z = 2 + 4iFind z ( the conjugate of z) and then multiply z times zz = 2 – 4i

zz = (2 + 4i)(2 – 4i)

= 4 – 16 i²

= 4 + 16

= 20

Write the quotient in standard form

i

i1 3

i i

i

3

1 9

2

2

i 3

1 9

3

10

i 3

10

1

10i

Multiply numerator and denominator by conjugate

Simplify remembering i² = -1

Write in standard form a + b = a + b

c c c

i

i

i

i

31

31

31

Write in Standard Form

i

i

31

2

i

i

i

i

31

31

31

2

2

2

91

362

i

iii

)1(91

)1(372

i

91

372

i

10

71 i i

10

7

10

1

Powers for i

-1

i

i3

i4

i5

i6

i7

i8

i13

i2

i19

i

i i2 ii i2 2 1 1 ( )

i

i i4 2

i i4 3

i i4

i

i

i

1

1

-1

24i

ii 34

344 ii