1.4 – Complex Numbers

• Real numbers have a small issue; no symmetry of their roots–

• To remedy this, we introduce an “imaginary” unit, so it does work

• The number i is defined such that

33

12 i

Simplifying Negative Roots

• For a positive number a,• Follow all other rules to simplify the remaining

radical

• Example. Simplify:• •

aia

x20436x

Complex Numbers

• A complex number, a+bi, has the following:– Real part, a– Imaginary part, bi– Only equal if both parts are equal (real/imaginary)– 5 + 10i

• With imaginary numbers, only combine the like terms (real with real, imaginary with imaginary)

• Multiplication, follow same rules as polynomials (FOIL, like terms, etc.)

• Example. Simplify:

• Example. Simplify:

ii 5344

ii 3645

Quotients

• Similar to radical expressions, denominators of fractions cannot contain imaginary numbers or a complex number

• Use the complex conjugate = for given complex number a+bi, the complex conjugate is a-bi

• Example. Simplify the quotient:• Note the denominator contains the complex

number, 4-3i• What is the complex conjugate? 4+3i

i34

2

• Example. Simplify the following:•

i

i

44

36

Roots and Complex Numbers

• When dealing with negative roots, we can simplify using the rules introduced

• Now, we can simplify radicals in a second way

• Example. Simplify:• How can we write ?

2

61

6

Powers of i

• The imaginary number, i, has a particular pattern

• i2= -1• i3 = i2 x i = -1 x i = -i• i4 = i2 x i2 = -1 x -1 = 1• i = i • Pull out powers that are multiples of 4; those

will become 1

• Example. Simplify:

• i15 = i12 x i3 =

• 4i25

• Assignment• Page 61• #1-41 odd