10. 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 standard form. (6 − 2𝑖)(2 − 3𝑖) A. 7 − 21𝑖 B. 3 + 25𝑖 C. 6 − 22𝑖 D. 6 + 22𝑖

SOLUTION 1: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.



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Performing the multiplication operation: 6 − 2𝑖 2 − 3𝑖 = 12 − 18 − 4𝑖 + 6𝑖 / Recall that 𝑖 / = −1, which simplifies the result to: 12 − 22𝑖 − 6 Putting the final result into standard form we find: 6 − 2𝑖 2 − 3𝑖 = 6 − 22𝑖

Therefore, the correct answer choice is C. 𝟔 − 𝟐𝟐𝒊 PROBLEM 2: Simplify −18 A. 3 2𝑖 B. 3 3𝑖 C. 2 5𝑖 D. 3 5𝑖



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SOLUTION 2: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Recall that it is impossible to get a real number out of a square root of a negative number. We can however reduce and rewrite the square root: 18 −1 Which can be further reduced to: 9 2 −1 This original number is now in its reduced form. Recalling that −1 = 𝑖, we can rewrite the complex number as: 3 2𝑖

Therefore, the correct answer choice is A. 𝟑 𝟐𝒊



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PROBLEM 3: Write the following complex number in standard form: 3 9−𝑗

A.

/8

B.

::

C.

:=

D.

9/ ;< >8 /< 9/

+ + + +

8 9/ ; 9/ < 9/ : 9/

𝑗 𝑗 𝑗 𝑗

SOLUTION 3: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The standard form of a complex number is 𝑧 = 𝑎 + 𝑗𝑏. Therefore, we need to get rid of the fraction. To do that multiply by the conjugate of the denominator, such that:

3 (9 + 𝑗) 27 + 3𝑗 = / (9 − 𝑗) (9 + 𝑗) 9 +1



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Rearranging into the standard for: 27 3 + 𝑗 82 82

Therefore, the correct answer choice is D.

𝟐𝟕 𝟖𝟐

+

𝟑 𝟖𝟐

𝒋

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

SOLUTION 4: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.



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Before multiplying these numbers, the first step is to convert the square root of the negative numbers in their complex representations: −100 = 100 −1 = 10𝑖 −36 = 36 −1 = 6𝑖 We can now rewrite the complex expression as: (2 − 10𝑖)(1 + 6𝑖) This is much easier to deal with and multiplying gives the result: 2 − 10𝑖 1 + 6𝑖 = 2 + 2𝑖 − 60𝑖 / Recall that 𝑖 / = −1. Putting the results into the standard form gives: 2 + 2𝑖 − 60𝑖 / = 62 + 2𝑖

Therefore, the correct answer choice is A. 𝟔𝟐 + 𝟐𝒊



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PROBLEM 5: Simplify the following expression: (9 + 3𝑖)(6 + 8𝑖) A. 30 + 90𝑖 B. 78 + 90𝑖 C. 24𝑖 / + 90𝑖 + 54 D. 54 + 90𝑖

SOLUTION 5: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Performing the multiplication operation: 9 + 3𝑖 6 + 8𝑖 = 54 + 72𝑖 + 18𝑖 + 24𝑖 / Recall that 𝑖 / = −1, which simplifies the result to: 54 + 72𝑖 + 18𝑖 + 24𝑖 / = 54 + 90𝑖 − 24



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Putting the final result into standard form we find: 54 + 90𝑖 − 24 = 30 + 90𝑖

Therefore, the correct answer choice is A. 𝟑𝟎 + 𝟗𝟎𝒊 PROBLEM 6: Solve 𝑥 / + 4 = 0 A. 𝑥 = 2𝑖 B. 𝑥 = −2𝑖 C. 𝑥 = ±2𝑖 D. 𝑥 = 3

SOLUTION 6: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We write the given equation: 𝑥/ + 4 = 0



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We subtract 4 from each side: 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 As we know we can’t take the square root of a negative number we can simplify 𝑥 as: 𝑥 = ±2𝑖

Therefore, the correct answer choice is B. 𝒙 = ±𝟐𝒊 PROBLEM 7: Find the value of 𝑖 /: A. −1 B. 1 C. −𝑖 D. 𝑖



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SOLUTION 7: The TOPIC OF ALGEBRA COMPLEX NUMBERS can be referenced under the reference topic of MATHEMATICS on on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We know that: 𝑖8 = 1 𝑖 / = −1 𝑖 : = −𝑖 𝑖 > = +1 We can use these trends to predict any number as the exponent of the imaginary number 𝑖: 𝑖 >LM8 = 𝑖 𝑖 >LM/ = −1 𝑖 >LM: = −𝑖 𝑖 >LM> = +1 We can rewrite the provided imaginary number as: 𝑖 /: = 𝑖 (> N ;M:) = 𝑖

Therefore, the correct answer choice is C. – 𝒊

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PROBLEM 8: What is the exponential form of the complex number 3 + 4𝑖? A. 𝑒 R ;:.8 ° B. 5𝑒 R 8;:.8° C. 5𝑒 R 8/T.=° D. 7𝑒 R ;:.8°

SOLUTION 8: The FORMULAS FOR THE ALGEBRA OF COMPLEX NUMBERS can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

The standard form a complex number is represented by the formula 𝑧 = 𝑎 + 𝑗𝑏, Where: • 𝑎 is the real component • 𝑏 is the imaginary component • 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

R UVWXUY

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We are given: • The real component “𝑎” is equal to 𝑎 = 3 • The imaginary component “𝑏” is equal to 𝑏 = 4 Plugging these values of “𝑎” and “𝑏” into our formula we find:

3 + 4𝑖 =

\

3/ + 4/ 𝑒 R UVWXUY(])

We calculate the angle of the arctan (𝑡𝑎𝑛e8 ) as:

arctan

> :

= 53.1°

Therefore, the exponential form is represented as: 3 + 4𝑖 = 5𝑒 R ;:.8°

Therefore, the correct answer choice is B. 𝟓𝐞𝐢 𝟓𝟑.𝟏° .



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PROBLEM 9: What is the product of the complex number 3 + 4𝑖 and 7 − 2𝑖? A. 10 + 2𝑖 B. 13 + 22𝑖 C. 13 + 34𝑖 D. 29 + 22𝑖

SOLUTION 32: The FORMULAS FOR THE ALGEBRA OF COMPLEX NUMBERS can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We calculate the product as 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. 3 + 4𝑖 7 − 2𝑖 = 21 − 8𝑖 / + 28𝑖 − 6𝑖 3 + 4𝑖 7 − 2𝑖 = 21 + 8 + 28𝑖 − 6𝑖 3 + 4𝑖 7 − 2𝑖 = 29 + 22𝑖

Therefore, the correct answer choice is D. 𝟐𝟗 + 𝟐𝟐𝐢.



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PROBLEM 10: What is the rationalized value of the following complex quotient? 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

SOLUTION 10: The FORMULAS FOR THE ALGEBRA OF COMPLEX NUMBERS can be referenced under the topic 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, multiple the numerator and denominator by the complex conjugate of the denominator: 6 + 2.5𝑖 (6 + 2.5𝑖)(3 − 4𝑖) = 3 + 4𝑖 (3 + 4𝑖)(3 − 4𝑖) Using the F.O.I.L. method to expand and multiply the algebraic expressions we find: (6 + 2.5𝑖) 28 − 16.5𝑖 = = 1.1 − 0.66𝑖 (3 + 4𝑖) 25

Therefore, the correct answer choice is C. 𝟏. 𝟏 − 𝟎. 𝟔𝟔𝐢.

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