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2.5 Introduction to Complex Numbers. 11/7/2012. Quick Review. If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 8 1 , x = x 1 , -5 = -5 1 Also, any number raised to the Zero power is equal to 1 - PowerPoint PPT Presentation
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2.5 Introduction to Complex Numbers
11/7/2012
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 raised to any power
1oi
iiiii 1123
ii 1
12 i
iiiii 1347
iiiii 1145
111246 iii
111224 iii
111448 iii
Do 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. Raise i to the remainder and determine its value.
22 :Ex i• Step 1. 22÷ 4 has a remainder of 2 • Step 2. i22 = i2
1 22 i51 :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
i 3 = -i
i 0 = 1
i 1 = i
i 2 = -1
Complex Number
Is a number written in the standard form a + biwhere a is the real partand bi is the imaginary part.
State the real and imaginary part of each complex number.
9i
Real Imaginary
0 9i
0
Add/Subtract the real parts, then add/subtract the imaginary parts
Adding and Subtracting
Complex Numbers
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
Subtract Complex Numbers
Write as a complex number in standard form.
Simplify.= 5 + 0i
2i6( (– – 2i1( (–
SOLUTION
Group real and imaginary terms.
2i6( (
= 16 2i2i +– – 2i1( (– – –
Write in standardform.
= 5
-1 + 2i
Checkpoint
Write the expression as a complex number in a + bi form.
Add and Subtract Complex Numbers
1. 2i4( (– + 3i1( (+ ANSWER i5 +
2. i3( (– + 4i2( (+ ANSWER 3i5 +
ANSWER 3i2 +3. 6i4( (+ 3i2( (+–
4. 4i2( (+ 7i2( (+–– ANSWER 3i4– –
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 iremember
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.
Divide Complex Numbers
54 𝑖
multiply top and bottom by i
5 𝑖4 𝑖2
= = -
Complex Conjugates
Two complex numbers of the form a + bi and a - bi
Their product is a real number becauseEx: (3 + 2i)(3 – 2i) using FOIL9 – 6i + 6i - 4i 2
9 – 4i2 i2 = -19 – 4(-1) = 9 + 4 = 13Is used to write quotient of 2 complex numbers in standard form (a + bi)
SOLUTION2i3 +2i1 –
2i3 +2i1 –
2i1 +2i1 +
= •Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i.
+–
Divide Complex Numbers
Write as a complex number in a + bi 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+–15
= Simplify.
51–
58 i+= Write in standard form.
Checkpoint
Write the expression as a complex number in standard form.
Multiply and Divide Complex Numbers
1. i2( (–3i ANSWER 6i3 +
2. ( 2i1 (+ i2( (– ANSWER 3i4 +
3. i2 +i1 –
ANSWER21+23
i
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
Homework
WS 2.5#1-12all, 13-27odd, 31-34all