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Transcript
```Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
Lesson 8: Complex Number Division
Classwork
Opening Exercises
Use the general formula to find the multiplicative inverse of each complex number.
a.
2 + 3π
b.
β7 β 4π
c.
β4 + 5π
Exercises 1β4
Find the conjugate, and plot the complex number and its conjugate in the complex plane. Label the conjugate with a
prime symbol.
1.
π΄: 3 + 4π
2.
π΅: β2 β π
3.
πΆ: 7
4.
π·: 4π
Lesson 8:
Date:
Complex Number Division
5/11/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.28
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M1
Exercises 5β8
Find the modulus.
5.
3 + 4π
6.
β2 β π
7.
7
8.
4π
Exercises 9β11
Given π§ = π + ππ.
9.
Show that for all complex numbers π§, |ππ§| = |π§|.
Lesson 8:
Date:
Complex Number Division
5/11/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.29
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
M1
10. Show that for all complex numbers π§, π§ β π§Μ = |π§|2 .
11. Explain the following: Every nonzero complex number π§ has a multiplicative inverse. It is given by
1
π§Μ
= | |.
π§
π§
Example 1
2 β 6π
2 + 5π
Exercises 12β13
Divide.
12.
3 + 2π
β2 β 7π
13.
3
3βπ
Lesson 8:
Date:
Complex Number Division
5/11/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.30
Lesson 8
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
Problem Set
1.
2.
Let π§ = 4 β 3π and π€ = 2 β π. Show that
a.
Μ|
|π§| = |π§
b.
| | = | Μ|
c.
If |π§| = 0, must it be that π§ = 0?
d.
Give a specific example to show that |π§ + π€| usually does not equal |π§| + |π€|.
1
π§
1
π§
Divide.
a.
1 β 2π
2π
b.
5 β 2π
5 + 2π
c.
β3
β 2π
β2 β β3π
3.
Prove that |π§π€| = |π§| β |π€| for complex numbers π§ and π€.
4.
Given π§ = 3 + π, π€ = 1 + 3.
5.
a.
Find π§ + π€, and graph π§, π€, and π§ + π€ on the same complex plane. Explain what you discover if you draw line
segments from the origin to those points π§,π€, and π§ + π€. Then draw line segments to connect π€ to π§ + π€,
and π§ + π€ to π§.
b.
Find βπ€, and graph π§, π€, and π§ β π€ on the same complex plane. Explain what you discover if you draw line
segments from the origin to those points π§, π€, and π§ β π€. Then draw line segments to connect π€ to π§ β π€,
and π§ β π€ to π§.
Explain why |π§ + π€| β€ |π§| + |π€| and |π§ β π€| β€ |π§| + |π€| geometrically. (Hint: Triangle inequality theorem)
Lesson 8:
Date:
Complex Number Division
5/11/17
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.31