5.7 Complex Numbers
12/4/2013
Quick ReviewIf a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 81 , x = x1 , -5 = -51 Also, any number raised to the Zero power is equal to 1Ex: 30 = 1 -40 = 1
Exponent Rule:When multiplying powers with the same base, you add the exponent.x2 • x3 = x5
y • y7 = y8
The square of any real number x is never negative, so the equation x2 = -1 has no real number solution.
To solve this x2 = -1 , mathematicians created an expanded system of numbers
using the IMAGINARY UNIT, i.
1i
12 i
Simplifying i given any powers
1oi
iiiii 1123
ii 1
12 i
iiiii 1347
iiiii 1145
111246 iii
111224 iii
111448 iiiDo you see the pattern yet?
The pattern repeats after every 4.So you can find i raised to any power by dividing the exponent by 4 and see what the remainder is. Based on that remainder, you can determine it’s value.
22 :Ex i• Step 1. 22÷ 4 has a remainder of 2 • Step 2. i22 = i2
1 22 i
50 :Ex i• Step 1. 51 ÷ 4 has a remainder of 3 • Step 2. i51 = i3
ii 51
Checkpoint Find the value of
1. i 15
2. i 20
3. i 61
4. i 122
Complex Number
Is a number written in the standard form a + biwhere a is the real partand bi is the imaginary part.Ex: 3 + 2iAdd/Subtract the real parts, then add/subtract the imaginary parts
Adding and Subtracting
Complex Numbers
Example 2 Add Complex Numbers
Write as a complex number in standard form.
2i3(
(
+ i1(
(
–+
SOLUTION
Group real and imaginary terms.
2i3(
(
+ i1(
(
–+ = 13 + 12i i+ –
Write in standard form.
= 4 + i
Example 3 Subtract Complex Numbers
Write as a complex number in standard form.
= 5 + 0i
2i6(
(
– – 2i1(
(
–
SOLUTION
2i6(
(
=– – 2i1(
(
–
Write in standardform.
= 5
-1 + 2i
6 – 1 – 2i + 2i
Checkpoint
Write the expression as a complex number in standard form.
Add and Subtract Complex Numbers
6. 2i4(
(
– + 3i1(
(
+ ANSWER i5 +
7. i3((
– + 4i2((
+ ANSWER 3i5 +
ANSWER 3i2 +8. 6i4(
(
+ 3i2(
(
+–
9. 4i2(
(
+ 7i2(
(
+–– ANSWER 3i4– –
Checkpoint
Write the expression as a complex number in standard form.
Add and Subtract Complex Numbers
11. 2i1(
(
– + 5i4(
(
+ ANSWER 3i5 +
ANSWER 3i3 +12. i2((
– ((
– 4i1– –
Example 4 Multiply Complex Numbers
Write the expression as a complex number in standard form.
a. b.1( 3i
(
+–2i 3i6(
(
+ 3i4(
(
–
SOLUTION
Multiply using distributive property.
1( 3i
(
+–2i = 2i 6i 2– +a.
1
(
(–2i 6–= + Use i 2 1.= –
6 2i––= Write in standard form.
1
:2 i
remember
Example 4 Multiply Complex Numbers
Multiply using FOIL. b. 3i6(
(
+ 3i4(
(
– 24 18i– 12i+ 9i 2–=
24 6i– – 9i 2= Simplify.
24 6i– – 1
(
(–9= Use i 2 1.= –
6i33 –= Write in standard form.
Homework
WS 5.7 Do problems 13-38 only
Complex Conjugates
Two complex numbers of the form a + bi and a - bi
Their product is a real number because(3 + 2i)(3 – 2i) using FOIL9 – 6i + 6i -4i2
9 – 4i2 i2 = -19 – 4(-1) = 9 + 4 = 13Is used to write quotient of 2 complex numbers in standard form (a + bi)
SOLUTION
2i3 +
2i1 –
2i3 +
2i1 –
2i1 +
2i1 += •
Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i.
+–
Example 5 Divide Complex Numbers
Write as a complex number in standard form.2i3 +
2i1 –
Multiply using FOIL. 1
2i3 6i+ + 4i 2+
2i2i+ – 4i 2–=
3 8i+ 1
(
(–4+
1 – 1
(
(–4= Simplify and use i 2 1.= –
8i+–1
5= Simplify.
51–
58 i+= Write in standard form.
a + bi
Checkpoint
Write the expression as a complex number in standard form.
Multiply and Divide Complex Numbers
i2 +
i1 –ANSWER
2
1+2
3i
Properties of Square Root of Negative Number
r
rir
1
12
i
ir 1
Ex:
Example 1 Solve a Quadratic Equation
Solve the equation.
=7x 2 49–a. b. =3x 2 5– 29–
SOLUTION
=7x 2 49–a.
Divide each side by 7.=x 2 7–
Take the square root of each side.=x +– 7–
Write in terms of i.=x +– 7i
Example 1 Solve a Quadratic Equation
Write original equation. b. =3x 2 29–5–
Add 5 to each side.=3x 2 24–
Divide each side by 3.=x 2 8–
Write in terms of i.=x +– 8i
Take the square root of each side. =x +– 8–
Simplify the radical. =x +– 2i2
22248
Checkpoint
Solve the equation.
Solve a Quadratic Equation
1. =x 2 3– ANSWER 3,i 3i–
2. =x 2 20–
3. =x 2 3 2+ –
ANSWER 5,2 52–i i
ANSWER 5,i 5i–
Graphing Complex Number
Real axis
Imaginary axis
Ex: Graph 3 – 2i
3
2
To plot, start at the origin, move 3 units to the right and 2 units down
3 – 2i
Ex: Name the complex number represented by the points.
A
D
C
B
Answers:A is 1 + iB is 0 + 2i = 2iC is -2 – iD is -2 + 3i
rriiririri 2
rrrandiiiSince 1 2
rrri or 1 2