Review of Complex Numbers

REVIEW OF COMPLEX NUMBERS

The Imaginary Number i :

Complex numbers are always written in the form a + bi (a is the real part and b is the imaginary part)

Examples: Write as complex numbers.

a) =

b) =

c) =

d) =

e) =

f) =

g) =

h) =

i) =

The conjugate of a + bi is a – bi .

Examples: Find the conjugate of the following complex numbers.

a) 2 + 6i à 2 – 6i

b) –3 – 4i à –3 + 4i

c) 15i à –15i

To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

Examples: Divide. Be sure to write each answer in standard form.

a)

b)

Powers of i :

i = i i5 = i

i2 = –1 i6 = –1

i3 = –i i7 = –i

i4 = 1 i8 = 1

So, the pattern for the powers of i is: { i, –1, –i, 1 }

Examples: Simplify

a) i12 = 1 (count through the pattern until you get to 12)

b) i67 = –i (count through the pattern until you get to 67)

c) 4 + i3 = 4 + (–i) = 4 – i

d) 4i4 – 2i2 + i = 4(1) – 2(–1) + i = 4 + 2 + i = 6 + i

When solving the quadratic equation, , if b2 – 4ac < 0, then there will be 0 real solutions (because the 2 answers are complex)

Examples: Solve

a)

a = 1, b = –2, c = 5

b)

a = 6, b = 4, c = 1

Notice that complex solutions ALWAYS come in complex conjugate pairs!!

Example: If 5i is a solution to a quadratic equation, then so is – 5i

If –4 – 6i is a solution to a quadratic equation, then so is –4 + 6i