Pc Complex Numbers

Complex Numbers

a + bi

Recall from Math 1010 that we can add, subtract, multiply or divide complex numbers. After performing these operations if we’ve simplified everything correctly we should always again get a complex number (although the real or imaginary parts may be zero). Below is an example of each to refresh your memory. ADDING

SUBTRACTING

MULTIPLYING

(3 – 2i) + (5 – 4i) = 8 – 6i (3 – 2i) - (5 – 4i) 3 – 2i - 5 + 4i = -2 +2i (3 – 2i) (5 – 4i) = 15 – 12i – 10i+8i2

=15 – 22i +8(-1) = 7 – 22i

Combine real parts and combine imaginary parts

Be sure to distribute the negative through before combining real parts and imaginary parts FOIL and then combine like terms. Remember i 2 = -1

Notice when I’m done simplifying that I only have two terms, a real term and an imaginary one. If I have more than that, I need to simplify more.

DIVIDING

FOIL

21) ( 15 + 12 i − 10 i − 8 − 3 − 2i 5 + 4i = 15 + 12i − 10i − 8i ⋅ = 21) ( 25 + 20 i − 20 i − 16 − 5 − 4i 5 + 4i 25 + 20i − 20i − 16i Combine like terms

i = −1 2

Recall that to divide complex numbers, you multiply the top and bottom of the fraction by the conjugate of the bottom.

23 + 2i 23 2 = = + i 41 41 41 We’ll put the 41 under each term so we can see the real part and the imaginary part

This means the same complex number, but with opposite sign on the imaginary term

Let’s solve a couple of equations that have complex solutions to refresh our memories of how it works.

x + 25 = 0 2

-25

-25

x = ± − 25 2

x = ± 25 i = ±5 i Use the quadratic formula

x − 6 x + 13 = 0 2

x=

− ( − 6) ±

( − 6) 2(1)

2

The negative under the square root becomes i

− b ± b 2 − 4ac x= 2a

− 4(1)(13)

6 ± − 16 6 ± 16 i = = 2 2

Square root and don’t forget the ±

6 ± 36 − 52 = 2 6±4i = 3 ± 2i = 2