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Algebra II : Imaginary Numbers

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Example Questions

Example Question #1 : Imaginary Numbers

$\frac{-5+10i}{3+4i}=?$ $\displaystyle \frac{-5+10i}{3+4i}=?$

Possible Answers:

-1+2i $\displaystyle -1+2i$

1+2i $\displaystyle 1+2i$

1-2i $\displaystyle 1-2i$

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

$-\frac{5}{3}+\frac{5}{2}i$ $\displaystyle -\frac{5}{3}+\frac{5}{2}i$

Correct answer:

1+2i $\displaystyle 1+2i$

Explanation:

$\frac{-5+10i}{3+4i}$ $\displaystyle \frac{-5+10i}{3+4i}$

$=\frac{(-5+10i)(3-4i)}{(3+4i)(3-4i)}$ $\displaystyle =\frac{(-5+10i)(3-4i)}{(3+4i)(3-4i)}$

$=\frac{-15+20i+30i-40i^{2}}{9-16i^{2}}$ $\displaystyle =\frac{-15+20i+30i-40i^{2}}{9-16i^{2}}$

$=\frac{-15+50i+40}{9+16}$ $\displaystyle =\frac{-15+50i+40}{9+16}$

$=\frac{25+50i}{25}$ $\displaystyle =\frac{25+50i}{25}$

=1+2i $\displaystyle =1+2i$

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Example Question #1 : Irrational Numbers

Multiply:

$\displaystyle (7 + 3i) (1 - 2i)$

Possible Answers:

7 - 17i $\displaystyle 7 - 17i$

7 - 5i $\displaystyle 7 - 5i$

$\displaystyle 13 - 11i$

1 - 11i $\displaystyle 1 - 11i$

$\displaystyle 13 - 17i$

Correct answer:

$\displaystyle 13 - 11i$

Explanation:

Use the FOIL technique:

$\displaystyle (7 + 3i) (1 - 2i)$

$= 7 \cdot 1 - 7 \cdot 2i + 3i \cdot 1 - 3i \cdot 2i$ $\displaystyle = 7 \cdot 1 - 7 \cdot 2i + 3i \cdot 1 - 3i \cdot 2i$

$= 7 - 14i + 3i - 6i ^{2}$ $\displaystyle = 7 - 14i + 3i - 6i ^{2}$

$\displaystyle = 7 - 14i + 3i - 6(-1)$

$\displaystyle = 7 - 14i + 3i +6$

$\displaystyle = 13 - 11i$

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Example Question #1 : Complex Imaginary Numbers

Evaluate $\small \frac{-6+2i}{10-3i}$ $\displaystyle \small \frac{-6+2i}{10-3i}$

Possible Answers:

$\small \frac{-66+2i}{109}$ $\displaystyle \small \frac{-66+2i}{109}$

$\small \small \frac{-54+38i}{91-60i}$ $\displaystyle \small \small \frac{-54+38i}{91-60i}$

$\small \frac{-54+38i}{109}$ $\displaystyle \small \frac{-54+38i}{109}$

$\small \small \frac{-66+2i}{91-60i}$ $\displaystyle \small \small \frac{-66+2i}{91-60i}$

You cannot divide by complex numbers

Correct answer:

$\small \frac{-66+2i}{109}$ $\displaystyle \small \frac{-66+2i}{109}$

Explanation:

To divide by a complex number, we must transform the expression by multiplying it by the complex conjugate of the denominator over itself. In the problem, $\small 10-3i$ $\displaystyle \small 10-3i$ is our denominator, so we will multiply the expression by $\small \frac{10+3i}{10+3i}$ $\displaystyle \small \frac{10+3i}{10+3i}$ to obtain:

$\small \frac{-6+2i}{10-3i}\frac{10+3i}{10+3i}=\frac{-60+20i-18i+6i^2}{100-30i+30i-9i^2}$ $\displaystyle \small \frac{-6+2i}{10-3i}\frac{10+3i}{10+3i}=\frac{-60+20i-18i+6i^2}{100-30i+30i-9i^2}$.

We can then combine like terms and rewrite all $\small i^2$ $\displaystyle \small i^2$ terms as $\small -1$ $\displaystyle \small -1$. Therefore, the expression becomes:

$\small \frac{-60+2i+6i^2}{100-9i^2}=\frac{-66+2i}{109}$ $\displaystyle \small \frac{-60+2i+6i^2}{100-9i^2}=\frac{-66+2i}{109}$

Our final answer is therefore $\small \frac{-66+2i}{109}$ $\displaystyle \small \frac{-66+2i}{109}$

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Example Question #2 : Imaginary Numbers

Simplify the following product:

$\displaystyle {} (5+3i)(-2+i)$

Possible Answers:

$\displaystyle {} -13+11i$

$\displaystyle {} -13-i$

{}-7-i $\displaystyle {}-7-i$

$\displaystyle {} -10+3i$

Correct answer:

$\displaystyle {} -13-i$

Explanation:

Multiply these complex numbers out in the typical way:

${}(5+3i)(-2+i) = -10+5i-6i+3i^2$ $\displaystyle {}(5+3i)(-2+i) = -10+5i-6i+3i^2$

and recall that i^2=-1 $\displaystyle i^2=-1$ by definition. Then, grouping like terms we get

$\displaystyle {} (-10-3)+(5i-6i) = -13-i$

which is our final answer.

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Example Question #2 : Complex Imaginary Numbers

Identify the real part of 1-3i $\displaystyle 1-3i$

Possible Answers:

$\displaystyle -3$

$\displaystyle -2$

$\displaystyle 3$

none of the above.

$\displaystyle 1$

Correct answer:

$\displaystyle 1$

Explanation:

A complex number in its standard form is of the form: a+bi $\displaystyle a+bi$, where $\displaystyle a$ stands for the real part and $\displaystyle b$ stands for the imaginary part. The symbol $\displaystyle i$ stands for $\sqrt{-1}$ $\displaystyle \sqrt{-1}$.

The real part in this problem is 1.

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Example Question #1961 : Mathematical Relationships And Basic Graphs

Simplify:

$\displaystyle (4-6i)+(7+2i)$

Possible Answers:

11-4i $\displaystyle 11-4i$

5+2i $\displaystyle 5+2i$

3-4i $\displaystyle 3-4i$

-3+4i $\displaystyle -3+4i$

Correct answer:

11-4i $\displaystyle 11-4i$

Explanation:

To add complex numbers, find the sum of the real terms, then find the sum of the imaginary terms.

$\displaystyle (4-6i)+(7+2i)=(4+7)+(-6+2)i=11-4i$

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Example Question #1 : Imaginary Numbers

Simplify:

$\displaystyle (-5+5i)+(2-9i)$

Possible Answers:

-3i $\displaystyle -3i$

3-4i $\displaystyle 3-4i$

-3-4i $\displaystyle -3-4i$

-3+4i $\displaystyle -3+4i$

Correct answer:

-3-4i $\displaystyle -3-4i$

Explanation:

To add complex numbers, find the sum of the real terms, then find the sum of the imaginary terms.

$\displaystyle (-5+5i)+(2-9i)=(-5+2)+(5-9)i=-3-4i$

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Example Question #1 : Imaginary Numbers

Simplify:

$\displaystyle (-3+2i)+(5+8i)$

Possible Answers:

12i $\displaystyle 12i$

2+10i $\displaystyle 2+10i$

15-16i $\displaystyle 15-16i$

10+2i $\displaystyle 10+2i$

Correct answer:

2+10i $\displaystyle 2+10i$

Explanation:

To add complex numbers, find the sum of the real terms, then find the sum of the imaginary terms.

$\displaystyle (-3+2i)+(5+8i)=(-3+5)+(2+8)i=2+10i$

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Example Question #2 : Imaginary Numbers

Simplify:

$\displaystyle (2+5i)-(5+6i)$

Possible Answers:

3+11i $\displaystyle 3+11i$

-4i $\displaystyle -4i$

3+i $\displaystyle 3+i$

-3-i $\displaystyle -3-i$

Correct answer:

-3-i $\displaystyle -3-i$

Explanation:

To subtract complex numbers, subtract the real terms together, then subtract the imaginary terms.

$\displaystyle (2+5i)-(5+6i)=(2-5)+(5-6)i=-3-i$

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Example Question #3 : Complex Imaginary Numbers

Simplify:

$\displaystyle (12+8i)-(20+16i)$

Possible Answers:

8+8i $\displaystyle 8+8i$

-8-8i $\displaystyle -8-8i$

32+24i $\displaystyle 32+24i$

-16i $\displaystyle -16i$

Correct answer:

-8-8i $\displaystyle -8-8i$

Explanation:

To subtract complex numbers, subtract the real terms, then subtract the imaginary terms.

$\displaystyle (12+8i)-(20+16i)=(12-20)+(8-16)i=-8-8i$

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Algebra II : Imaginary Numbers

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Imaginary Numbers

Example Question #1 : Irrational Numbers

Example Question #1 : Complex Imaginary Numbers

Example Question #2 : Imaginary Numbers

Example Question #2 : Complex Imaginary Numbers

Example Question #1961 : Mathematical Relationships And Basic Graphs

Example Question #1 : Imaginary Numbers

Example Question #1 : Imaginary Numbers

Example Question #2 : Imaginary Numbers

Example Question #3 : Complex Imaginary Numbers

All Algebra II Resources

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