Algebra II : Complex Imaginary Numbers

Study concepts, example questions & explanations for Algebra II

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Irrational Numbers

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

Possible Answers:

\displaystyle -1+2i

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

\displaystyle 1+2i

\displaystyle 1-2i

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

Correct answer:

\displaystyle 1+2i

Explanation:

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

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

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

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

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

\displaystyle =1+2i

Example Question #2 : Imaginary Numbers

Multiply:

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

Possible Answers:

\displaystyle 7 - 17i

\displaystyle 7 - 5i

\displaystyle 13 - 11i

\displaystyle 13 - 17i

\displaystyle 1 - 11i

Correct answer:

\displaystyle 13 - 11i

Explanation:

Use the FOIL technique:

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

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

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

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

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

\displaystyle = 13 - 11i

Example Question #1 : Irrational Numbers

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

Possible Answers:

You cannot divide by complex numbers

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

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

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

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

Correct answer:

\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, \displaystyle \small 10-3i is our denominator, so we will multiply the expression by \displaystyle \small \frac{10+3i}{10+3i} to obtain:

\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 \displaystyle \small i^2 terms as \displaystyle \small -1. Therefore, the expression becomes:

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

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

Example Question #1 : Imaginary Numbers

Simplify the following product:

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

Possible Answers:

\displaystyle {} -13+11i

\displaystyle {} -10+3i

\displaystyle {} -13-i

\displaystyle {}-7-i

Correct answer:

\displaystyle {} -13-i

Explanation:

Multiply these complex numbers out in the typical way:

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

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

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

which is our final answer.

Example Question #2 : Complex Imaginary Numbers

Identify the real part of \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: \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 \displaystyle \sqrt{-1}.

The real part in this problem is 1.

Example Question #1961 : Mathematical Relationships And Basic Graphs

Simplify:

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

Possible Answers:

\displaystyle 11-4i

\displaystyle 5+2i

\displaystyle 3-4i

\displaystyle -3+4i

Correct answer:

\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

Example Question #1 : Complex Imaginary Numbers

Simplify:

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

Possible Answers:

\displaystyle -3i

\displaystyle -3-4i

\displaystyle 3-4i

\displaystyle -3+4i

Correct answer:

\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

Example Question #3 : Imaginary Numbers

Simplify:

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

Possible Answers:

\displaystyle 10+2i

\displaystyle 12i

\displaystyle 15-16i

\displaystyle 2+10i

Correct answer:

\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

Example Question #4 : Imaginary Numbers

Simplify:

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

Possible Answers:

\displaystyle -4i

\displaystyle -3-i

\displaystyle 3+11i

\displaystyle 3+i

Correct answer:

\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

Example Question #5 : Imaginary Numbers

Simplify:

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

Possible Answers:

\displaystyle 32+24i

\displaystyle -8-8i

\displaystyle 8+8i

\displaystyle -16i

Correct answer:

\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

Learning Tools by Varsity Tutors