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Complex Analysis : Complex Numbers

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Example Questions

Example Question #1 : Complex Numbers

Evaluate: $\left| 1 - 3 i \right|$ $\displaystyle \left| 1 - 3 i \right|$

Possible Answers:

$\sqrt{10}$ $\displaystyle \sqrt{10}$

$- \sqrt{10}$ $\displaystyle - \sqrt{10}$

$\frac{\sqrt{10}}{2}$ $\displaystyle \frac{\sqrt{10}}{2}$

$\displaystyle 10$

Correct answer:

$\sqrt{10}$ $\displaystyle \sqrt{10}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| 1 - 3 i \right| = \sqrt{ 1 + 9 }$ $\displaystyle \left| 1 - 3 i \right| = \sqrt{ 1 + 9 }$

$= \sqrt{ 10 }$ $\displaystyle = \sqrt{ 10 }$

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Example Question #2 : Complex Numbers

Evaluate:

$\left| 9 - 9 i \right|$ $\displaystyle \left| 9 - 9 i \right|$

Possible Answers:

$- 9 \sqrt{2}$ $\displaystyle - 9 \sqrt{2}$

162 $\displaystyle 162$

$9 \sqrt{2}$ $\displaystyle 9 \sqrt{2}$

$\frac{9 \sqrt{2}}{2}$ $\displaystyle \frac{9 \sqrt{2}}{2}$

Correct answer:

$9 \sqrt{2}$ $\displaystyle 9 \sqrt{2}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| 9 - 9 i \right| = \sqrt{ 81 + 81 }$ $\displaystyle \left| 9 - 9 i \right| = \sqrt{ 81 + 81 }$

$= \sqrt{ 162 }$ $\displaystyle = \sqrt{ 162 }$

$= 9 \sqrt{2}$ $\displaystyle = 9 \sqrt{2}$

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Example Question #3 : Complex Numbers

Evaluate:

$\left| 6 - 3 i \right|$ $\displaystyle \left| 6 - 3 i \right|$

Possible Answers:

$3 \sqrt{5}$ $\displaystyle 3 \sqrt{5}$

$\frac{3 \sqrt{5}}{2}$ $\displaystyle \frac{3 \sqrt{5}}{2}$

$- 3 \sqrt{5}$ $\displaystyle - 3 \sqrt{5}$

$\displaystyle 45$

Correct answer:

$3 \sqrt{5}$ $\displaystyle 3 \sqrt{5}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| 6 - 3 i \right| = \sqrt{ 36 + 9 }$ $\displaystyle \left| 6 - 3 i \right| = \sqrt{ 36 + 9 }$

$= \sqrt{ 45 }$ $\displaystyle = \sqrt{ 45 }$

$= 3 \sqrt{5}$ $\displaystyle = 3 \sqrt{5}$

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Example Question #4 : Complex Numbers

Evaluate:

$\left| ( 9 + 7 i ) ( -5 + 6 i ) \right|$ $\displaystyle \left| ( 9 + 7 i ) ( -5 + 6 i ) \right|$

Possible Answers:

$\sqrt{7930}$ $\displaystyle \sqrt{7930}$

$- \sqrt{7930}$ $\displaystyle - \sqrt{7930}$

$\frac{\sqrt{7930}}{2}$ $\displaystyle \frac{\sqrt{7930}}{2}$

7930 $\displaystyle 7930$

Correct answer:

$\sqrt{7930}$ $\displaystyle \sqrt{7930}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 9 + 7 i ) ( -5 + 6 i ) \right| = \sqrt{ 81 + 49 }\cdot \sqrt{ 25 + 36 }$ $\displaystyle \left| ( 9 + 7 i ) ( -5 + 6 i ) \right| = \sqrt{ 81 + 49 }\cdot \sqrt{ 25 + 36 }$

$= \sqrt{130} \cdot \sqrt{61}$ $\displaystyle = \sqrt{130} \cdot \sqrt{61}$

$= \sqrt{7930}$ $\displaystyle = \sqrt{7930}$

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Example Question #5 : Complex Numbers

Evaluate:

$\left| ( 9 + 2 i ) ( -9 + 4 i ) \right|$ $\displaystyle \left| ( 9 + 2 i ) ( -9 + 4 i ) \right|$

Possible Answers:

8245 $\displaystyle 8245$

$\frac{\sqrt{8245}}{2}$ $\displaystyle \frac{\sqrt{8245}}{2}$

$\sqrt{8245}$ $\displaystyle \sqrt{8245}$

$- \sqrt{8245}$ $\displaystyle - \sqrt{8245}$

Correct answer:

$\sqrt{8245}$ $\displaystyle \sqrt{8245}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 9 + 2 i ) ( -9 + 4 i ) \right| = \sqrt{ 81 + 4 }\cdot \sqrt{ 81 + 16 }$ $\displaystyle \left| ( 9 + 2 i ) ( -9 + 4 i ) \right| = \sqrt{ 81 + 4 }\cdot \sqrt{ 81 + 16 }$

$= \sqrt{85} \cdot \sqrt{97}$ $\displaystyle = \sqrt{85} \cdot \sqrt{97}$

$= \sqrt{8245}$ $\displaystyle = \sqrt{8245}$

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Example Question #6 : Complex Numbers

Evaluate:

$\left| ( 7 + 6 i ) ( -8 + 6 i ) \right|$ $\displaystyle \left| ( 7 + 6 i ) ( -8 + 6 i ) \right|$

Possible Answers:

$- 10 \sqrt{85}$ $\displaystyle - 10 \sqrt{85}$

8500 $\displaystyle 8500$

$10 \sqrt{85}$ $\displaystyle 10 \sqrt{85}$

$5 \sqrt{85}$ $\displaystyle 5 \sqrt{85}$

Correct answer:

$10 \sqrt{85}$ $\displaystyle 10 \sqrt{85}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 7 + 6 i ) ( -8 + 6 i ) \right| = \sqrt{ 49 + 36 }\cdot \sqrt{ 64 + 36 }$ $\displaystyle \left| ( 7 + 6 i ) ( -8 + 6 i ) \right| = \sqrt{ 49 + 36 }\cdot \sqrt{ 64 + 36 }$

$= \sqrt{85} \cdot 10$ $\displaystyle = \sqrt{85} \cdot 10$

$= 10 \sqrt{85}$ $\displaystyle = 10 \sqrt{85}$

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Example Question #7 : Complex Numbers

Evaluate:

$\left| ( 5 + 6 i ) ( -8 + 5 i ) \right|$ $\displaystyle \left| ( 5 + 6 i ) ( -8 + 5 i ) \right|$

Possible Answers:

5429 $\displaystyle 5429$

$- \sqrt{5429}$ $\displaystyle - \sqrt{5429}$

$\sqrt{5429}$ $\displaystyle \sqrt{5429}$

$\frac{\sqrt{5429}}{2}$ $\displaystyle \frac{\sqrt{5429}}{2}$

Correct answer:

$\sqrt{5429}$ $\displaystyle \sqrt{5429}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 5 + 6 i ) ( -8 + 5 i ) \right| = \sqrt{ 25 + 36 }\cdot \sqrt{ 64 + 25 }$ $\displaystyle \left| ( 5 + 6 i ) ( -8 + 5 i ) \right| = \sqrt{ 25 + 36 }\cdot \sqrt{ 64 + 25 }$

$= \sqrt{61} \cdot \sqrt{89}$ $\displaystyle = \sqrt{61} \cdot \sqrt{89}$

$= \sqrt{5429}$ $\displaystyle = \sqrt{5429}$

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Example Question #8 : Complex Numbers

Evaluate:

$\left| ( 10 + 3 i ) ( -5 + 4 i ) \right|$ $\displaystyle \left| ( 10 + 3 i ) ( -5 + 4 i ) \right|$

Possible Answers:

4469 $\displaystyle 4469$

$\frac{\sqrt{4469}}{2}$ $\displaystyle \frac{\sqrt{4469}}{2}$

$\sqrt{4469}$ $\displaystyle \sqrt{4469}$

$- \sqrt{4469}$ $\displaystyle - \sqrt{4469}$

Correct answer:

$\sqrt{4469}$ $\displaystyle \sqrt{4469}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 10 + 3 i ) ( -5 + 4 i ) \right| = \sqrt{ 100 + 9 }\cdot \sqrt{ 25 + 16 }$ $\displaystyle \left| ( 10 + 3 i ) ( -5 + 4 i ) \right| = \sqrt{ 100 + 9 }\cdot \sqrt{ 25 + 16 }$

$= \sqrt{109} \cdot \sqrt{41}$ $\displaystyle = \sqrt{109} \cdot \sqrt{41}$

$= \sqrt{4469}$ $\displaystyle = \sqrt{4469}$

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Example Question #9 : Complex Numbers

Evaluate:

$\left| ( 9 + i ) ( -9 + 6 i ) \right|$ $\displaystyle \left| ( 9 + i ) ( -9 + 6 i ) \right|$

Possible Answers:

$3 \sqrt{1066}$ $\displaystyle 3 \sqrt{1066}$

9594 $\displaystyle 9594$

$- 3 \sqrt{1066}$ $\displaystyle - 3 \sqrt{1066}$

$\frac{3 \sqrt{1066}}{2}$ $\displaystyle \frac{3 \sqrt{1066}}{2}$

Correct answer:

$3 \sqrt{1066}$ $\displaystyle 3 \sqrt{1066}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 9 + i ) ( -9 + 6 i ) \right| = \sqrt{ 81 + 1 }\cdot \sqrt{ 81 + 36 }$ $\displaystyle \left| ( 9 + i ) ( -9 + 6 i ) \right| = \sqrt{ 81 + 1 }\cdot \sqrt{ 81 + 36 }$

$= \sqrt{82} \cdot 3 \sqrt{13}$ $\displaystyle = \sqrt{82} \cdot 3 \sqrt{13}$

$= 3 \sqrt{1066}$ $\displaystyle = 3 \sqrt{1066}$

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Example Question #10 : Complex Numbers

Evaluate:

$\left| ( 8 + 5 i ) ( -5 + 5 i ) \right|$ $\displaystyle \left| ( 8 + 5 i ) ( -5 + 5 i ) \right|$

Possible Answers:

$\frac{5 \sqrt{178}}{2}$ $\displaystyle \frac{5 \sqrt{178}}{2}$

4450 $\displaystyle 4450$

$- 5 \sqrt{178}$ $\displaystyle - 5 \sqrt{178}$

$5 \sqrt{178}$ $\displaystyle 5 \sqrt{178}$

Correct answer:

$5 \sqrt{178}$ $\displaystyle 5 \sqrt{178}$

Explanation:

The general formula to figure out the modulus is

$\left|a+bi\right|=\sqrt{a^2+b^2}$ $\displaystyle \left|a+bi\right|=\sqrt{a^2+b^2}$

We apply this to get

$\left| ( 8 + 5 i ) ( -5 + 5 i ) \right| = \sqrt{ 64 + 25 }\cdot \sqrt{ 25 + 25 }$ $\displaystyle \left| ( 8 + 5 i ) ( -5 + 5 i ) \right| = \sqrt{ 64 + 25 }\cdot \sqrt{ 25 + 25 }$

$= \sqrt{89} \cdot 5 \sqrt{2}$ $\displaystyle = \sqrt{89} \cdot 5 \sqrt{2}$

$=5 \sqrt{178}$ $\displaystyle =5 \sqrt{178}$

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All Complex Analysis Resources

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Complex Analysis : Complex Numbers

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #1 : Complex Numbers

Example Question #2 : Complex Numbers

Example Question #3 : Complex Numbers

Example Question #4 : Complex Numbers

Example Question #5 : Complex Numbers

Example Question #6 : Complex Numbers

Example Question #7 : Complex Numbers

Example Question #8 : Complex Numbers

Example Question #9 : Complex Numbers

Example Question #10 : Complex Numbers

All Complex Analysis Resources

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