21.1 Algebra of Complex Numbers Problem Set
ALGEBRA OF COMPLEX NUMBERS | PRACTICE PROBLEMS Complete the following to reinforce your understanding of the concept covered in this module.
PROBLEM 1: Perform the following operation and write the result in the standard form of a complex number: (6 − 2𝑖)(2 − 3𝑖) A. 7 − 21𝑖 B. 3 + 25𝑖 C. 6 − 22𝑖 D. 6 + 22𝑖
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PROBLEM 2: Simplifying the complex number −18 results in: A. 3 2𝑖 B. 3 3𝑖 C. 2 5𝑖 D. 3 5𝑖
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PROBLEM 3: Write the following complex number in standard form: 3 9−𝑗
A.
/0
B.
22
C.
25
D.
1/ 34 60 /4 1/
+ + + +
0 1/ 3 1/ 4 1/ 2 1/
𝑗 𝑗 𝑗 𝑗
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PROBLEM 4: Multiply the following and write the answer in standard form: 2 − −100 1 + −36 A. 62 + 2𝑖 B. 72 + 3𝑖 C. 82 + 4𝑖 D. 92 + 5𝑖
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PROBLEM 5: Simplifying the following complex expression, (9 + 3𝑖)(6 + 8𝑖), results in: A. 30 + 90𝑖 B. 78 + 90𝑖 C. 24𝑖 / + 90𝑖 + 54 D. 54 + 90𝑖
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PROBLEM 6: Solve 𝑥 / + 4 = 0 A. 𝑥 = 2𝑖 B. 𝑥 = −2𝑖 C. 𝑥 = ±2𝑖 D. 𝑥 = 3
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PROBLEM 7: The value of 𝑖 /2 is most close to: A. −1 B. 1 C. −𝑖 D. 𝑖
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PROBLEM 8: The exponential form of the complex number 3 + 4𝑖 is best represented as: A. 𝑒 = 32.0° B. 5𝑒 = 32.0° C. 5𝑒 = 0/A.5° D. 7𝑒 = 32.0°
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PROBLEM 9: The product of the complex number 3 + 4𝑖 and 7 − 2𝑖 is most close to: A. 10 + 2𝑖 B. 13 + 22𝑖 C. 13 + 34𝑖 D. 29 + 22𝑖
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PROBLEM 10: The rationalized value of the stated complex quotient is best represented by: 6 + 2.5𝑖 3 + 4𝑖 A. −0.32 + 0.66𝑖 B. 0.32 − 0.66𝑖 C. 1.1 − 0.66𝑖 D. −1.7 + 1.1𝑖
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ALGEBRA OF COMPLEX NUMBERS | SOLUTIONS SOLUTION 1: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles. In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR. In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number
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• 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem statement, we are given: (6 − 2𝑖)(2 − 3𝑖) Carrying out the multiplication operation, we get: 12 − 18𝑖 − 4𝑖 + 6𝑖 / Recall that 𝑖 / = −1, which allows for us to simplify the result as: 12 − 22𝑖 − 6 We now need to work this result in to standard RECTANGULAR form. This can quickly be done by combining the constants and arranging the formula such that: 6 − 2𝑖 2 − 3𝑖 = 6 − 22𝑖 Therefore, the correct answer choice is C. 𝟔 − 𝟐𝟐𝒊
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SOLUTION 2: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. It is not possible to calculate a real number from the square root of a negative number. For example, −𝑛 does not represent a real number since there is no real number that can be squared to get the negative output. However, this general expression can be reduced in to two terms, rewritten as: 𝑛 −1 In our problem, we are given: −18 As we have illustrated, this can be broken in to two elements: 18 −1 Which can be further reduced to: 9 2 −1
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This original number is now in its reduced form. Recalling that −1 = 𝑖, we can rewrite the complex number as: 9 2 𝑖 Or: 3 2𝑖 Therefore, the correct answer choice is A. 𝟑 𝟐𝒊
SOLUTION 3: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles. In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR.
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In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number • 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem statement, we are given: 3 9−𝑗 The first call of duty is to get rid of the quotient. To do that, we will multiply by the conjugate of the denominator. The CONJUGATE of a complex number is the original complex number with the sign of the imagery part changed. It is generally useful to use the conjugate of the denominator when finding, or eliminating, the quotient of two complex numbers.
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The FORMULA for the COMPLEX CONJUGATE of a COMPLEX NUMBER can be referenced under the topic of ALGEBRA OF COMPLEX NUMBERS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The complex conjugate of a complex number 𝑧0 = (𝑎0 + 𝑗𝑏0 ) is defined as: 𝑧0∗ = (𝑎0 − 𝑗𝑏0 ) The COMPLEX NUMBER in the denominator is stated as: (9 − 𝑗) Which gives us the CONJUGATE of: (9 + 𝑗) Note that the only difference is the sign in front of the imaginary part of this COMPLEX NUMBER. Now multiplying both the numerator and denominator by the CONJUGATE, we get: 3 (9 + 𝑗) 27 + 3𝑗 = (9 − 𝑗) (9 + 𝑗) 9/ − 𝑗 / Recall that 𝑗 / = −1, which allows for us to simplify the result as:
27 + 3𝑗 81 + 1
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Or: 27 + 3𝑗 82 Rearranging this number so that it is represented in the standard RECTANGULAR FORM, we have: 27 3 + 𝑗 82 82 Therefore, the correct answer choice is D.
𝟐𝟕 𝟖𝟐
+
𝟑 𝟖𝟐
𝒋
SOLUTION 4: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles. In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR.
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In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number • 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem we are given: 2 − −100 1 + −36 We are asked to carry this operation out and reduce it in to standard RECTANGULAR FORM. Before multiplying these numbers, the first step is to rework the square root of the negative numbers in to their complex representations. Recall that it is not possible to calculate a real number from the square root of a negative number.
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For example, −𝑛 does not represent a real number since there is no real number that can be squared to get the negative output. However, this general expression can be reduced in to two terms, rewritten as: 𝑛 −1 In our problem, we are given: −100 As we have illustrated, this can be broken in to two elements: 100 −1 And ultimately: 10𝑖 We are also given: −36 As we have illustrated, this can be broken in to two elements: 36 −1
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And ultimately: 6𝑖 We can now take these new representations in to account and rewrite the complex expression as: (2 − 10𝑖)(1 + 6𝑖) This is much cleaner and easier to deal with. Multiplying out this expression gives the result: 2 + 2𝑖 − 60𝑖 / Recall that 𝑖 / = −1 which will allow us to put the result into the standard form, which is: 2 + 2𝑖 − 60 −1 Or: 62 + 2𝑖 Therefore, the correct answer choice is A. 𝟔𝟐 + 𝟐𝒊
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SOLUTION 5: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Performing the multiplication operation, we can expand our expression to: 9 + 3𝑖 6 + 8𝑖 = 54 + 72𝑖 + 18𝑖 + 24𝑖 / Combining the like terms, we get: 54 + 90𝑖 + 24𝑖 / Recall that 𝑖 / = −1, which simplifies the result to: 54 + 90𝑖 + 24(−1) Or: 54 + 90𝑖 − 24 Putting the final result into standard RECTANGULAR FORM, we get: 30 + 90𝑖 Therefore, the correct answer choice is A. 𝟑𝟎 + 𝟗𝟎𝒊
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SOLUTION 6: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given the equation: 𝑥/ + 4 = 0 We are after the solution for x that will make this statement true, therefore, we subtract 4 from each side and get: 𝑥 / = −4 We take the square root of each side of the equation: 𝑥 / = −4 We find that we can solve for 𝑥 as: 𝑥 = −4 However, it is not possible to calculate a real number from the square root of a negative number.
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For example, −𝑛 does not represent a real number since there is no real number that can be squared to get the negative output. However, this general expression can be reduced in to two terms, rewritten as: 𝑛 −1 Simplifying −4 we get: 4 −1 And we know that 𝑖 = −1 so: 4𝑖 Or: ±2𝑖 Therefore: 𝑥 = ±2𝑖 Therefore, the correct answer choice is C. 𝒙 = ±𝟐𝒊
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SOLUTION 7: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. An imaginary number has a value that changes based on the value of the exponent. Just as a negative number squared becomes positive, an imaginary number can go from positive to negative, or negative to positive depending on its exponent. For example: 𝑖0 = 1 𝑖 / = −1 𝑖 2 = −𝑖 𝑖 6 = +1 Using these established trends allows us to predict any number as the exponent of the imaginary number “𝑖”, for example: 𝑖 6RS0 = 𝑖 𝑖 6RS/ = −1 𝑖 6RS2 = −𝑖 𝑖 6RS6 = +1
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Given the IMAGINARY NUMBER: 𝑖 /2 We can rewrite the provided imaginary number as: 𝑖 /2 = 𝑖 (6 T 3S2) Therefore, referring back to our trends, we can say that: 𝑖 /2 = −𝑖 Therefore, the correct answer choice is C. – 𝒊
SOLUTION 8: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The standard form of a complex number is represented by the formula 𝑧 = 𝑎 + 𝑗𝑏. Where: • 𝑎 is the real component • 𝑏 is the imaginary component
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• 𝑗 = −1 (some disciplines use 𝑖 = −1) We can convert the standard form into an equivalent exponential form using the expression:
𝑎 + 𝑏𝑖 =
(𝑎/ + 𝑏 / )𝑒
= VWXYVZ
[ \
We are given in our problem statement that: • The real component “𝑎” is equal to 3 • The imaginary component “𝑏” is equal to 4 Plugging these values of “𝑎” and “𝑏” into our formula we get:
3 + 4𝑖 =
3/
+
]
4/ 𝑒 = VWXYVZ(^)
Calculating the angle of the arctan (we may be more accustomed to seeing stated as the tanb0 ), we get:
arctan
6 2
= 53.1°
Therefore, the exponential form is represented as: 3 + 4𝑖 = 5𝑒 = 32.0°
Therefore, the correct answer choice is B. 𝟓𝐞𝐢 𝟓𝟑.𝟏° Made with by Prepineer | Prepineer.com
SOLUTION 9: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We calculate the product by multiplying the algebraic expressions just as we would in basic algebra using the F.O.I.L. technique. This technique has you multiple the first, the outer, the inner, and the last terms in the expression. Laying out both complex numbers in to a proper expression, we have: 3 + 4𝑖 7 − 2𝑖 Carrying out the operation, we have: 21 − 8𝑖 / + 28𝑖 − 6𝑖 Combining like terms we get: 21 − 8𝑖 / + 22𝑖
Recall that 𝑖 / = −1, which simplifies the result to:
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21 − 8(−1) + 22𝑖 Or: 29 + 22𝑖 Which means that the products of our complex set of numbers is: 3 + 4𝑖 7 − 2𝑖 = 29 + 22𝑖 Therefore, the correct answer choice is D. 𝟐𝟗 + 𝟐𝟐𝐢
SOLUTION 10: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In order to rationalize a complex number, we must work what we have in to standard form. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles.
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In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR. In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number • 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem statement, we are given: 6 + 2.5𝑖 3 + 4𝑖 The first call of duty is to get rid of the quotient. To do that, we will multiply by the conjugate of the denominator. The CONJUGATE of a complex number is the original complex number with the sign of the imagery part changed.
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It is generally useful to use the conjugate of the denominator when finding, or eliminating, the quotient of two complex numbers. The FORMULA for the COMPLEX CONJUGATE of a COMPLEX NUMBER can be referenced under the topic of ALGEBRA OF COMPLEX NUMBERS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The complex conjugate of a complex number 𝑧0 = (𝑎0 + 𝑗𝑏0 ) is defined as: 𝑧0∗ = (𝑎0 − 𝑗𝑏0 ) The COMPLEX NUMBER in the denominator is stated as: 3 + 4𝑖 Which gives us the CONJUGATE of: 3 − 4𝑖 Note that the only difference is the sign in front of the imaginary part of this COMPLEX NUMBER.
Now multiplying both the numerator and denominator by the CONJUGATE, we get:
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(6 + 2.5𝑖) (3 − 4𝑖) (3 + 4𝑖) (3 − 4𝑖) Using the F.O.I.L. method, we can expand both the numerator and denominator, such that: (6 + 2.5𝑖) (3 − 4𝑖) 18 − 24𝑖 + 7.5𝑖 − 10𝑖 / = (3 + 4𝑖) (3 − 4𝑖) 9 − 12𝑖 + 12𝑖 − 16𝑖 / Combing like terms we have: 18 − 16.5𝑖 − 10𝑖 / 9 − 16𝑖 / Recall that 𝑖 / = −1 which allows us to simplify further, giving us: 18 − 16.5𝑖 − 10(−1) 9 − 16(−1) Or: 18 − 16.5𝑖 + 10 9 + 16
Condensing further we have:
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28 − 16.5𝑖 25 The last move is to put this expression in to RECTANGULAR FORM, which is: 28 − 16.5𝑖 = 1.1 − 0.66𝑖 25 Therefore, the correct answer choice is C. 𝟏. 𝟏 − 𝟎. 𝟔𝟔𝐢
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PROBLEM 1: Perform the following operation and write the result in the standard form of a complex number: (6 − 2𝑖)(2 − 3𝑖) A. 7 − 21𝑖 B. 3 + 25𝑖 C. 6 − 22𝑖 D. 6 + 22𝑖
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PROBLEM 2: Simplifying the complex number −18 results in: A. 3 2𝑖 B. 3 3𝑖 C. 2 5𝑖 D. 3 5𝑖
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PROBLEM 3: Write the following complex number in standard form: 3 9−𝑗
A.
/0
B.
22
C.
25
D.
1/ 34 60 /4 1/
+ + + +
0 1/ 3 1/ 4 1/ 2 1/
𝑗 𝑗 𝑗 𝑗
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PROBLEM 4: Multiply the following and write the answer in standard form: 2 − −100 1 + −36 A. 62 + 2𝑖 B. 72 + 3𝑖 C. 82 + 4𝑖 D. 92 + 5𝑖
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PROBLEM 5: Simplifying the following complex expression, (9 + 3𝑖)(6 + 8𝑖), results in: A. 30 + 90𝑖 B. 78 + 90𝑖 C. 24𝑖 / + 90𝑖 + 54 D. 54 + 90𝑖
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PROBLEM 6: Solve 𝑥 / + 4 = 0 A. 𝑥 = 2𝑖 B. 𝑥 = −2𝑖 C. 𝑥 = ±2𝑖 D. 𝑥 = 3
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PROBLEM 7: The value of 𝑖 /2 is most close to: A. −1 B. 1 C. −𝑖 D. 𝑖
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PROBLEM 8: The exponential form of the complex number 3 + 4𝑖 is best represented as: A. 𝑒 = 32.0° B. 5𝑒 = 32.0° C. 5𝑒 = 0/A.5° D. 7𝑒 = 32.0°
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PROBLEM 9: The product of the complex number 3 + 4𝑖 and 7 − 2𝑖 is most close to: A. 10 + 2𝑖 B. 13 + 22𝑖 C. 13 + 34𝑖 D. 29 + 22𝑖
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PROBLEM 10: The rationalized value of the stated complex quotient is best represented by: 6 + 2.5𝑖 3 + 4𝑖 A. −0.32 + 0.66𝑖 B. 0.32 − 0.66𝑖 C. 1.1 − 0.66𝑖 D. −1.7 + 1.1𝑖
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ALGEBRA OF COMPLEX NUMBERS | SOLUTIONS SOLUTION 1: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles. In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR. In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number
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• 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem statement, we are given: (6 − 2𝑖)(2 − 3𝑖) Carrying out the multiplication operation, we get: 12 − 18𝑖 − 4𝑖 + 6𝑖 / Recall that 𝑖 / = −1, which allows for us to simplify the result as: 12 − 22𝑖 − 6 We now need to work this result in to standard RECTANGULAR form. This can quickly be done by combining the constants and arranging the formula such that: 6 − 2𝑖 2 − 3𝑖 = 6 − 22𝑖 Therefore, the correct answer choice is C. 𝟔 − 𝟐𝟐𝒊
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SOLUTION 2: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. It is not possible to calculate a real number from the square root of a negative number. For example, −𝑛 does not represent a real number since there is no real number that can be squared to get the negative output. However, this general expression can be reduced in to two terms, rewritten as: 𝑛 −1 In our problem, we are given: −18 As we have illustrated, this can be broken in to two elements: 18 −1 Which can be further reduced to: 9 2 −1
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This original number is now in its reduced form. Recalling that −1 = 𝑖, we can rewrite the complex number as: 9 2 𝑖 Or: 3 2𝑖 Therefore, the correct answer choice is A. 𝟑 𝟐𝒊
SOLUTION 3: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles. In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR.
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In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number • 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem statement, we are given: 3 9−𝑗 The first call of duty is to get rid of the quotient. To do that, we will multiply by the conjugate of the denominator. The CONJUGATE of a complex number is the original complex number with the sign of the imagery part changed. It is generally useful to use the conjugate of the denominator when finding, or eliminating, the quotient of two complex numbers.
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The FORMULA for the COMPLEX CONJUGATE of a COMPLEX NUMBER can be referenced under the topic of ALGEBRA OF COMPLEX NUMBERS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The complex conjugate of a complex number 𝑧0 = (𝑎0 + 𝑗𝑏0 ) is defined as: 𝑧0∗ = (𝑎0 − 𝑗𝑏0 ) The COMPLEX NUMBER in the denominator is stated as: (9 − 𝑗) Which gives us the CONJUGATE of: (9 + 𝑗) Note that the only difference is the sign in front of the imaginary part of this COMPLEX NUMBER. Now multiplying both the numerator and denominator by the CONJUGATE, we get: 3 (9 + 𝑗) 27 + 3𝑗 = (9 − 𝑗) (9 + 𝑗) 9/ − 𝑗 / Recall that 𝑗 / = −1, which allows for us to simplify the result as:
27 + 3𝑗 81 + 1
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Or: 27 + 3𝑗 82 Rearranging this number so that it is represented in the standard RECTANGULAR FORM, we have: 27 3 + 𝑗 82 82 Therefore, the correct answer choice is D.
𝟐𝟕 𝟖𝟐
+
𝟑 𝟖𝟐
𝒋
SOLUTION 4: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles. In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR.
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In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number • 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem we are given: 2 − −100 1 + −36 We are asked to carry this operation out and reduce it in to standard RECTANGULAR FORM. Before multiplying these numbers, the first step is to rework the square root of the negative numbers in to their complex representations. Recall that it is not possible to calculate a real number from the square root of a negative number.
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For example, −𝑛 does not represent a real number since there is no real number that can be squared to get the negative output. However, this general expression can be reduced in to two terms, rewritten as: 𝑛 −1 In our problem, we are given: −100 As we have illustrated, this can be broken in to two elements: 100 −1 And ultimately: 10𝑖 We are also given: −36 As we have illustrated, this can be broken in to two elements: 36 −1
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And ultimately: 6𝑖 We can now take these new representations in to account and rewrite the complex expression as: (2 − 10𝑖)(1 + 6𝑖) This is much cleaner and easier to deal with. Multiplying out this expression gives the result: 2 + 2𝑖 − 60𝑖 / Recall that 𝑖 / = −1 which will allow us to put the result into the standard form, which is: 2 + 2𝑖 − 60 −1 Or: 62 + 2𝑖 Therefore, the correct answer choice is A. 𝟔𝟐 + 𝟐𝒊
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SOLUTION 5: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Performing the multiplication operation, we can expand our expression to: 9 + 3𝑖 6 + 8𝑖 = 54 + 72𝑖 + 18𝑖 + 24𝑖 / Combining the like terms, we get: 54 + 90𝑖 + 24𝑖 / Recall that 𝑖 / = −1, which simplifies the result to: 54 + 90𝑖 + 24(−1) Or: 54 + 90𝑖 − 24 Putting the final result into standard RECTANGULAR FORM, we get: 30 + 90𝑖 Therefore, the correct answer choice is A. 𝟑𝟎 + 𝟗𝟎𝒊
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SOLUTION 6: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given the equation: 𝑥/ + 4 = 0 We are after the solution for x that will make this statement true, therefore, we subtract 4 from each side and get: 𝑥 / = −4 We take the square root of each side of the equation: 𝑥 / = −4 We find that we can solve for 𝑥 as: 𝑥 = −4 However, it is not possible to calculate a real number from the square root of a negative number.
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For example, −𝑛 does not represent a real number since there is no real number that can be squared to get the negative output. However, this general expression can be reduced in to two terms, rewritten as: 𝑛 −1 Simplifying −4 we get: 4 −1 And we know that 𝑖 = −1 so: 4𝑖 Or: ±2𝑖 Therefore: 𝑥 = ±2𝑖 Therefore, the correct answer choice is C. 𝒙 = ±𝟐𝒊
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SOLUTION 7: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. An imaginary number has a value that changes based on the value of the exponent. Just as a negative number squared becomes positive, an imaginary number can go from positive to negative, or negative to positive depending on its exponent. For example: 𝑖0 = 1 𝑖 / = −1 𝑖 2 = −𝑖 𝑖 6 = +1 Using these established trends allows us to predict any number as the exponent of the imaginary number “𝑖”, for example: 𝑖 6RS0 = 𝑖 𝑖 6RS/ = −1 𝑖 6RS2 = −𝑖 𝑖 6RS6 = +1
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Given the IMAGINARY NUMBER: 𝑖 /2 We can rewrite the provided imaginary number as: 𝑖 /2 = 𝑖 (6 T 3S2) Therefore, referring back to our trends, we can say that: 𝑖 /2 = −𝑖 Therefore, the correct answer choice is C. – 𝒊
SOLUTION 8: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The standard form of a complex number is represented by the formula 𝑧 = 𝑎 + 𝑗𝑏. Where: • 𝑎 is the real component • 𝑏 is the imaginary component
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• 𝑗 = −1 (some disciplines use 𝑖 = −1) We can convert the standard form into an equivalent exponential form using the expression:
𝑎 + 𝑏𝑖 =
(𝑎/ + 𝑏 / )𝑒
= VWXYVZ
[ \
We are given in our problem statement that: • The real component “𝑎” is equal to 3 • The imaginary component “𝑏” is equal to 4 Plugging these values of “𝑎” and “𝑏” into our formula we get:
3 + 4𝑖 =
3/
+
]
4/ 𝑒 = VWXYVZ(^)
Calculating the angle of the arctan (we may be more accustomed to seeing stated as the tanb0 ), we get:
arctan
6 2
= 53.1°
Therefore, the exponential form is represented as: 3 + 4𝑖 = 5𝑒 = 32.0°
Therefore, the correct answer choice is B. 𝟓𝐞𝐢 𝟓𝟑.𝟏° Made with by Prepineer | Prepineer.com
SOLUTION 9: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We calculate the product by multiplying the algebraic expressions just as we would in basic algebra using the F.O.I.L. technique. This technique has you multiple the first, the outer, the inner, and the last terms in the expression. Laying out both complex numbers in to a proper expression, we have: 3 + 4𝑖 7 − 2𝑖 Carrying out the operation, we have: 21 − 8𝑖 / + 28𝑖 − 6𝑖 Combining like terms we get: 21 − 8𝑖 / + 22𝑖
Recall that 𝑖 / = −1, which simplifies the result to:
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21 − 8(−1) + 22𝑖 Or: 29 + 22𝑖 Which means that the products of our complex set of numbers is: 3 + 4𝑖 7 − 2𝑖 = 29 + 22𝑖 Therefore, the correct answer choice is D. 𝟐𝟗 + 𝟐𝟐𝐢
SOLUTION 10: The TOPIC of ALGEBRA COMPLEX NUMBERS can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In order to rationalize a complex number, we must work what we have in to standard form. When working with complex numbers, there are two forms in which we will encounter; RECTANGULAR and POLAR. If we were to be working with the POLAR form of a COMPLEX NUMBER, the biggest giveaway is the fact that we would be working with angles.
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In this instance, no angles are given or can be inferred, therefore, our standard form will be RECTANGULAR. In RECTANGULAR FORM, a complex number is written in terms of its real and imaginary components: 𝑧 = 𝑎 + 𝑗𝑏 Where: • 𝑎 is the real component of the complex number • 𝑏 is the imaginary component of the complex number • 𝑗 = −1 is the imaginary unit (𝑖 is also commonly used to represent an imaginary unit in the form of 𝑖 = −1) Back to our problem statement, we are given: 6 + 2.5𝑖 3 + 4𝑖 The first call of duty is to get rid of the quotient. To do that, we will multiply by the conjugate of the denominator. The CONJUGATE of a complex number is the original complex number with the sign of the imagery part changed.
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It is generally useful to use the conjugate of the denominator when finding, or eliminating, the quotient of two complex numbers. The FORMULA for the COMPLEX CONJUGATE of a COMPLEX NUMBER can be referenced under the topic of ALGEBRA OF COMPLEX NUMBERS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The complex conjugate of a complex number 𝑧0 = (𝑎0 + 𝑗𝑏0 ) is defined as: 𝑧0∗ = (𝑎0 − 𝑗𝑏0 ) The COMPLEX NUMBER in the denominator is stated as: 3 + 4𝑖 Which gives us the CONJUGATE of: 3 − 4𝑖 Note that the only difference is the sign in front of the imaginary part of this COMPLEX NUMBER.
Now multiplying both the numerator and denominator by the CONJUGATE, we get:
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(6 + 2.5𝑖) (3 − 4𝑖) (3 + 4𝑖) (3 − 4𝑖) Using the F.O.I.L. method, we can expand both the numerator and denominator, such that: (6 + 2.5𝑖) (3 − 4𝑖) 18 − 24𝑖 + 7.5𝑖 − 10𝑖 / = (3 + 4𝑖) (3 − 4𝑖) 9 − 12𝑖 + 12𝑖 − 16𝑖 / Combing like terms we have: 18 − 16.5𝑖 − 10𝑖 / 9 − 16𝑖 / Recall that 𝑖 / = −1 which allows us to simplify further, giving us: 18 − 16.5𝑖 − 10(−1) 9 − 16(−1) Or: 18 − 16.5𝑖 + 10 9 + 16
Condensing further we have:
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28 − 16.5𝑖 25 The last move is to put this expression in to RECTANGULAR FORM, which is: 28 − 16.5𝑖 = 1.1 − 0.66𝑖 25 Therefore, the correct answer choice is C. 𝟏. 𝟏 − 𝟎. 𝟔𝟔𝐢
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