Download Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6 Name: Operations

Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6
Name:
Operations with Complex Numbers Sample Answers
Date:
Period:
Adding Complex Numbers
Enter #1–5 on your calculator. Record the solution below and discuss with a partner how you think the
calculator is adding the two complex numbers.
1.
(3 + 4i) + (2 + 5i) =
5 + 9i
2.
(1 – 6i) + (3 – 2i) =
4 – 8i
3.
(2 + 5i) + (6 – 8i) =
8– 3i
4.
(4 – 3i) + (–5 – 7i) = –1 – 10i
5.
(–2 + 3i) + (1 – 2i) = –1 + i
6.
Based on the equivalent expressions produced by the calculator, what process is used to add two
complex numbers? Be sure to think about the Commutative, Associative, and Distributive Properties.
Grouping symbols are removed and the commutative property allows me to reorder the terms. I can use
the associative property to regroup terms so that the real terms are grouped together and the imaginary
terms are grouped together. I then add the real terms and the imaginary terms to result in one complex
number.
Subtracting Complex Numbers
Enter #7–12 on your calculator. Record the solution below and discuss with a partner how you think the
calculator is subtracting the two complex numbers.
7.
(3 + 4i) – (2 + 5i) =
1–i
8.
(1 – 6i) – (3 – 2i) =
–2 – 4i
9.
(2 + 5i) – (6 – 8i) =
–4 + 13i
10.
(4 – 3i) – (–5 – 7i) = 9 + 4i
11.
(–2 + 3i) – (1 – 2i) = –3 + 5i
12.
Based on the equivalent expressions produced by the calculator, what process is used to subtract two
complex numbers? Be sure to think about the Commutative, Associative, and Distributive Properties.
The subtraction sign can be viewed as adding –1 times the second quantity. I then distribute the –1 to all
terms in the second quantity. Then I follow the same process as for adding using commutative and
associative properties to regroup real and imaginary terms together.
Adapted from:
Texas Instruments: Complex Numbers
Page 1 of 3
Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6
Name:
Operations with Complex Numbers Sample Answers
Date:
Period:
Multiplying Complex Numbers
Enter #13–19 on your calculator. Record the solution below and discuss with a partner how you think the
calculator is multiplying the two complex numbers.
13.
(3 + 4i)(2 + 5i) = –14 + 23i
14.
(1 – 6i)(3 – 2i) = –9 – 20i
15.
Why is there not a term containing i2 in the solutions to #13 and #14?
When distributing the imaginary term in the first complex number to the imaginary term in the second
complex number, the product contains i 2 . But i 2 = –1, a real number, so this is regrouped with the other
real number terms.
16.
(2 + 5i)(6 – 8i) = 52 + 14i
17.
(–2 + 3i)(1 – 2i) = 4 + 7i
18.
(4 – 3i)(–5 – 7i) = –41 – 13i
19.
Based on the equivalent expressions produced by the calculator, what process is used to multiply two
complex numbers? Be sure to think about the Commutative, Associative, and Distributive Properties.
I distribute each term in the first complex number to both of the terms in the second complex number.
This results in 4 terms, two imaginary and two real that can then be regrouped with the commutative
and associative properties.
Use the strategies determined in #1 – 19 to add, subtract, or multiply the complex numbers below.
20.
(2  4i )  (2  4i)
4
21.
(2  4i )  (2  4i)
–8i
22.
(2  4i)(2  4i)
20
23.
(7  2i )  (7  2i)
14
24.
(7  2i )  (7  2i)
4i
25.
(7  2i)(7  2i)
53
26.
(4  4i )  (4  4i)
–8
27.
(4  4i )  (4  4i)
8i
28.
(4  4i)(4  4i)
32
29.
What do you notice?
Adding and multiplying complex conjugate pairs always result in real numbers. Subtracting always
results in a pure imaginary number.
Adapted from:
Texas Instruments: Complex Numbers
Page 2 of 3
Curriculum 2.0 Algebra 2: Unit 2-Topic 1, SLT 6
Name:
Operations with Complex Numbers Sample Answers
Date:
30.
Period:
Why does this happen?



Adding: Because the imaginary terms of complex conjugate pairs are always opposite, by definition,
their sum is 0. This eliminates the imaginary part of the complex number, leaving only the sum of the
real parts. The sum of the real parts will always be double the real part in the original complex
numbers.
Multiplying: Using the distributive property, multiplying complex conjugates will always result in a
number of the form a2  abi  abi  b2i 2 . The sum of the two imaginary terms will always be 0,
eliminating the imaginary part of the number.
Subtracting: The real parts of complex conjugates are the same number that when subtracted results
in a value of 0, leaving only the imaginary part.
Adapted from:
Texas Instruments: Complex Numbers
Page 3 of 3