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Precalculus : Polar Coordinates and Complex Numbers

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Example Questions

Example Question #1 : Find The Quotient Of Complex Numbers

Write

$\small \frac{1+2i}{1+i}$

in the form $\small a+bi$ for some real numbers $\small a$ and $\small b$ .

Possible Answers:

$\small \small \frac{3}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$

$\small \frac{3}{2}+\frac{1}{2}i$

$\small 1+2i$

$\small \small \frac{3}{2}-\frac{1}{2}i$

Correct answer:

$\small \frac{3}{2}+\frac{1}{2}i$

Explanation:

The correct answer is

$\small \frac{3}{2}+\frac{1}{2}i$

Using simple algebra and multiplying the expression by the complex conjugate of the denominator, we get:

$\small \small \frac{1+2i}{1+i}=\frac{1+2i}{1+i}\cdot\frac{1-i}{1-i}= (1-i)\frac{1+2i}{2}$

$\small =\frac{1-i+2i-2i^2}{2}=\frac{1+i-(-2)}{2}=\frac{3+i}{2}=\frac{3}{2}+\frac{1}{2}i$

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Example Question #2 : Find The Quotient Of Complex Numbers

Divide:

$\frac{4+5i}{5-i}$

Possible Answers:

$\frac{5}{8}+\frac{29i}{24}$

$\frac{15}{26}-\frac{29i}{26}$

$\frac{29}{26}+\frac{15i}{26}$

$\frac{15}{26}+\frac{29i}{26}$

Correct answer:

$\frac{15}{26}+\frac{29i}{26}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 5+i .

$\left(\frac{4+5i}{5-i}\right)\left(\frac{5+i}{5+i}\right)$

Now, distribute and simplify.

$\left(\frac{4+5i}{5-i}\right)\left(\frac{5+i}{5+i}\right)=\frac{20+29i+5i^2}{25-i^2}$

Recall that i^2=-1

$\frac{20+29i+5(-1)}{25-(-1)}=\frac{15+29i}{26}=\frac{15}{26}+\frac{29i}{26}$

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Example Question #5 : Find The Quotient Of Complex Numbers

Divide:

$\frac{7+i}{2-i}$

Possible Answers:

$\frac{13}{5}+\frac{9i}{5}$

$\frac{13}{5}-\frac{9i}{5}$

$\frac{15}{2}+3i$

$\frac{5}{13}+\frac{5}{9i}$

Correct answer:

$\frac{13}{5}+\frac{9i}{5}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 2+i .

$\left(\frac{7+i}{2-i}\right)\left(\frac{2+i}{2+i}\right)$

Now, distribute and simplify.

$\left(\frac{7+i}{2-i}\right)\left(\frac{2+i}{2+i}\right)=\frac{14+9i+i^2}{4-i^2}$

Recall that i^2=-1

$\frac{14+9i+i^2}{4-i^2}=\frac{14+9i+(-1)}{4-(-1)}=\frac{13+9i}{5}=\frac{13}{5}+\frac{9i}{5}$

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Example Question #1 : Find The Quotient Of Complex Numbers

Divide.

$\frac{2-9i}{3+i}$

Possible Answers:

$\frac{3}{4}+\frac{29i}{4}$

$-\frac{3}{10}-\frac{29i}{10}$

$-\frac{10}{3}-\frac{10}{29i}$

$\frac{3}{10}+\frac{29i}{10}$

Correct answer:

$-\frac{3}{10}-\frac{29i}{10}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3-i .

$\left(\frac{2-9i}{3+i}\right)\left(\frac{3-i}{3-i}\right)$

Now, distribute and simplify.

$\left(\frac{2-9i}{3+i}\right)\left(\frac{3-i}{3-i}\right)=\frac{6-29i+9i^2}{9-i^2}$

Recall that i^2=-1

$\frac{6-29i+9i^2}{9-i^2}=\frac{6-29i+9(-1)}{9-(-1)}=\frac{-3-29i}{10}=-\frac{3}{10}-\frac{29i}{10}$

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Example Question #7 : Find The Quotient Of Complex Numbers

Divide.

$\frac{4+3i}{6-i}$

Possible Answers:

$\frac{9}{7}+\frac{27}{22i}$

$\frac{21}{37}+\frac{22i}{37}$

$\frac{7}{9}-\frac{22i}{27}$

$\frac{2}{3}+\frac{i}{9}$

Correct answer:

$\frac{21}{37}+\frac{22i}{37}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 6+i .

$\left(\frac{4+3i}{6-i}\right)\left(\frac{6+i}{6+i}\right)$

Now, distribute and simplify.

$\left(\frac{4+3i}{6-i}\right)\left(\frac{6+i}{6+i}\right)=\frac{24+22i+3i^2}{36-i^2}$

Recall that i^2=-1

$\frac{24+22i+3i^2}{36-i^2}=\frac{24+22i+3(-1)}{36-(-1)}=\frac{21+22i}{37}$

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Example Question #1 : Find The Quotient Of Complex Numbers

Divide.

$\frac{5-2i}{3-8i}$

Possible Answers:

$\frac{73}{31}-\frac{73}{64i}$

$\frac{31}{73}-\frac{34i}{73}$

$\frac{31}{73}+\frac{34i}{73}$

$\frac{21}{29}+\frac{6i}{29}$

Correct answer:

$\frac{31}{73}+\frac{34i}{73}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3+8i .

$\left(\frac{5-2i}{3-8i}\right)\left(\frac{3+8i}{3+8i}\right)$

Now, distribute and simplify.

$\left(\frac{5-2i}{3-8i}\right)\left(\frac{3+8i}{3+8i}\right)=\frac{15+34i-16i^2}{9-64i^2}$

Recall that i^2=-1

$\frac{15+34i-16i^2}{9-64i^2}=\frac{15+34i-16(-1)}{9-64(-1)}=\frac{31+34i}{73}=\frac{31}{73}+\frac{34i}{73}$

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Example Question #2 : Find The Quotient Of Complex Numbers

Divide.

$\frac{5+i}{3+i}$

Possible Answers:

$\frac{8}{5}-\frac{i}{5}$

$\frac{5}{8}+\frac{i}{5}$

$\frac{8}{5}+\frac{i}{5}$

$\frac{3}{8}-\frac{i}{4}$

Correct answer:

$\frac{8}{5}-\frac{i}{5}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3-i .

$\left(\frac{5+i}{3+i}\right)\left(\frac{3-i}{3-i}\right)$

Now, distribute and simplify.

$\left(\frac{5+i}{3+i}\right)\left(\frac{3-i}{3-i}\right)=\frac{15-2i-i^2}{9-i^2}$

Recall that i^2=-1

$\frac{15-2i-i^2}{9-i^2}=\frac{15-2i-(-1)}{9-(-1)}=\frac{16-2i}{10}=\frac{8}{5}-\frac{i}{5}$

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Example Question #10 : Find The Quotient Of Complex Numbers

Divide.

$\frac{-9+i}{1-i}$

Possible Answers:

-5-4i

-2+3i

-4-5i

5+4i

Correct answer:

-5-4i

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 1-i .

$\left(\frac{-9+i}{1-i}\right)\left(\frac{1+i}{1+i}\right)$

Now, distribute and simplify.

$\left(\frac{-9+i}{1-i}\right)\left(\frac{1+i}{1+i}\right)=\frac{-9-8i+i^2}{1-i^2}$

Recall that i^2=-1

$\frac{-9-8i+i^2}{1-i^2}=\frac{-9-8i+(-1)}{1-(-1)}=\frac{-10-8i}{2}=-5-4i$

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Example Question #461 : Pre Calculus

Divide.

$\frac{6-2i}{3-i}$

Possible Answers:

2-3i

3-3i

1+3i

Correct answer:

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3+i .

$\left(\frac{6-2i}{3-i}\right)\left(\frac{3+i}{3+i}\right)$

Now, distribute and simplify.

$\left(\frac{6-2i}{3-i}\right)\left(\frac{3+i}{3+i}\right)=\frac{18-6i+6i-2i^2}{3^2-i^2}$

Recall that i^2=-1

$\frac{18-2i^2}{3^2-i^2}=\frac{18-2(-1)}{9-(-1)}=\frac{20}{10}=2$

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Example Question #462 : Pre Calculus

Divide.

$\frac{5+i}{i}$

Possible Answers:

1-5i

5-i

1+5i

5+i

Correct answer:

1-5i

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{5+i}{i}\right)\left(\frac{-i}{-i}\right)$

Now, distribute and simplify.

$\left(\frac{5+i}{i}\right)\left(\frac{-i}{-i}\right)=\frac{-5i-i^2}{-i^2}$

Recall that i^2=-1

$\frac{-5i-i^2}{-i^2}=\frac{-5i-(-1)}{-(-1)}=\frac{1-5i}{1}=1-5i$

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Precalculus : Polar Coordinates and Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #2 : Find The Quotient Of Complex Numbers

Example Question #5 : Find The Quotient Of Complex Numbers

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #7 : Find The Quotient Of Complex Numbers

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #2 : Find The Quotient Of Complex Numbers

Example Question #10 : Find The Quotient Of Complex Numbers

Example Question #461 : Pre Calculus

Example Question #462 : Pre Calculus

All Precalculus Resources

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