Download 05 FX115 Ex Cmplx Num

Activity 5: Exploring Complex Numbers
CALCULATOR:
John Neral
Casio: fx-115ES
Teaching Notes and Solutions
Objectives: Students will demonstrate the ability to perform mathematical calculations
involving complex numbers using the Casio fx-115ES.
Getting Started:
Real numbers are defined as the set of rational and irrational numbers. As we continue our
exploration of mathematics, it is important to realize that there is a set of numbers, which
extends beyond the real number system. These numbers are referred to as complex numbers. A
complex number is defined as a number in the form of a + bi where a and b are both defined as
real numbers.
However, i denotes an imaginary number. So, in the form of a + bi, a is the real part of the
complex number while bi is the imaginary part. In some ways, one might look at a complex
number and make a connection with an algebraic binomial. In this sense, complex numbers can
operate just like a binomial providing those operations follow the sum, product, and equality of
complex numbers
Answers:
1.
2.
3.
4.
12 – 5i
13 + 14i
32 + 24i
18 + 8 i
97 97
5. When (3 + 2i) is squared, it equals (5 + 12i). When (3 – 2i) is squared, it equals (5 – 12i). When
we multiply (5 + 12i) by (5 – 12i) it equals 169.
6. Given the problem, (3 + 5i)(8 + 7i), Vicki obtained an answer of (24 + 12i) which is incorrect. It
appears as if Vicki multiplied the real part of this complex number and added the imaginary part.
However, to multiply these complex numbers correctly, she must FOIL them. Therefore, Vicki must
multiply (3 • 8), then (3 • 7i), then (5i • 8), and then (5i • 7i). Those products equal 24 + 21i + 40i +
32i2. Because i2 = -1, 32i2 = -32. Then, we can simply the product to be 24 + 21i + 40i – 32 to equal
-11 + 61i.
7. The conjugate of (2 + 3i) is (2 – 3i) and the product equals 13.
Activity 5: Exploring Complex Numbers
1
© Casio, Inc. • For Classroom Use Only
Name ___________________________________________________ Date ______________
Activity 5: Exploring Complex Numbers
CALCULATOR:
Casio: fx-115ES
Student Worksheet Activity 5
There is always a balance between understanding how any mathematical process or function works and
incorporating technology into problem solving. It is important for students to understand the properties
of complex numbers and the relationship between i, i2, i3, i4, and so on. However, once a student has
gained the necessary understanding of these relationships as well as how to operate with complex
numbers, the student can use technology to verify their computation and/or enhance their problem solving based upon the integration of such technology.
Calculator Notes:
To Solve Problems Involving Complex Numbers:
•
Turn the calculator ON .
•
Press MODE .
•
Press 2 for Complex Numbers (CMPLX)
•
To enter a complex number using i, press ALPHA followed by the ENG key.
The i is written in red above the ENG key on the right side.
Note: When entered the i, you must press the key twice in order to enter it onto the display.
Problems:
Simplify the following problems using the calculator.
1. (5 + 4i) + (7 – 9i)
2. (2 + i)(8 + 3i)
3. (6 + 2i)2
4.
2
(9 – 4i)
5. Explain why (3 + 2i)2(3 – 2i)2 = 169.
__________________________________________________________________________
6.
__________________________________________________________________________
Given the problem, (3 + 5i)(8 + 7i), Vicki obtained an answer of (24 + 12i). Is this correct? If it
is not correct, explain what steps Vicki did incorrectly and what is the correct answer?
__________________________________________________________________________
__________________________________________________________________________
7.
__________________________________________________________________________
What is the conjugate of (2 + 3i) and what is the product when you multiply (2 + 3i) by its
conjugate?
__________________________________________________________________________
Activity 5: Exploring Complex Numbers
2
© Casio, Inc. • For Classroom Use Only
Activity 6: Exploring Exponents
CALCULATOR:
John Neral
Casio: fx-115ES
Teaching Notes and Solutions
Objectives: Students will demonstrate the ability to explore patterns involving exponents,
solve problems involving exponents, and explore the properties of exponential functions.
Getting Started:
Have students examine a table of values for an exponential function. Using the table function
of the fx-115ES, have them input the function f(x) = 2x. Set a start value of –5 and an ending
value of 5 with a step increment of 1. Ask: What does the shape of an exponential function
look like? When you plot the points on a coordinate plane it doesn’t possess the same properties as a linear function or absolute value function.
Students might say that it has some similar qualities of a quadratic function but it is not
symmetrical. Nevertheless, an exponential function is rooted in a solid understanding of how
exponents work. Explain that an exponent, by its definition, represents the number of times the
base is used as a factor. For example, if we were to write 34 as a product of factors, we would
write 4 x 4 x 4 x 4 and in simplified form, 34 = 81.
Ask: What happens when the exponent is a negative integer? When you evaluate a number in
standard form, what happens to the units digit as a positive exponent increases? Throughout
this activity, students will explore various properties of exponents and examine a table of values
to view its impact on an exponential function.
Answers:
The first eight powers of 3 are 3, 9, 27, 81, 243, 729, 2187, and 6561. The pattern in the units
digit is 3, 9, 7, 1. To calculate 3173, divide the exponent, 173, by 4 to obtain an answer of 43 R1.
The pattern repeats itself 43 times and the remainder 1 indicates that the units digit of 3173 is
in the first position of the pattern – the number 3.
1.
2.
3.
4.
5.
2-3 = 1/8, 2-2 = 1/4, 2-1 = 1/2, 20 = 1, 21 = 2, 22 = 4, 23 = 8.
Yes, 24 = 42 = 161. 24 = 16 as does 42.
101 = 10, 102 = 100, 103 = 1,000, 104 = 1,000, 105 = 100,000
(11 + 8)2 ≠ 112 + 82. The answer is false.
To calculate the amount of money received on the 30th day, you could evaluate the function
2(X-1) where x = 30. When you evaluate 2(30-1) it equals 536870912 pennies which is
$5,368,709.12. To calculate the total amount of money during the month you would need
to calculate:
30
(2(x-1)) = 1073741823 pennies which equals $10,737,418.23.
x =1
Activity 6: Exploring Exponents
1
© Casio, Inc. • For Classroom Use Only
Name ___________________________________________________ Date ______________
Activity 6: Exploring Exponents
CALCULATOR:
Casio: fx-115ES
Student Worksheet Activity 6
Examine the first eight powers of 3. Using the calculator create a table of values to determine a pattern
within the units digit. Then, use the pattern to determine the units digit for 3173.
Calculator Notes:
To Input a Function and Generate a Table of Values:
•
Turn the Calculator On .
•
Press MODE followed by 7 for Table.
•
Input the function after f(x)=.
•
Set the Start value and press = .
•
Set the End value and press = .
•
Set the Step value and press = .
•
Examine the table of values.
To Solve a Problem Involving Exponents:
•
Turn the calculator On .
•
Press MODE followed by 1 for COMP.
•
Press SHIFT SETUP followed by 1 for the MthIO (Math Input Output format).
•
Enter the base.
•
Press the exponent key to use the exponent template.
•
Enter the exponent.
Sample Problems:
1. Evaluate 2x where –3 ≤ x ≤ 3.
2. Does 24 = 42 = 161? Explain.
3. Evaluate 10x where 1 ≤ x ≤ 5. What pattern do you notice between the number in exponential
form and the number written in standard form?
4. Let x = 11 and y = 8. Examine the following question.
True or False: (x + y)2 = x2 + y2.
5. If you were given one penny on the first of the month and told that the amount of money would
double each day until the end of the month (let’s say that there are 30 days in the month) how
much money would you receive on the 30th day? What formula would you use to determine the
amount of money received on any particular day? How much money would you receive in total
for all 30 days of the month?
Activity 6: Exploring Exponents
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© Casio, Inc. • For Classroom Use Only