5.1.2: Homework

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Reading Questions

What is the definition of the imaginary unit $i$? What is the value of $i^2$?
How do you express the square root of a negative number, like $\sqrt{-49}$, in terms of $i$?
What are the two main components of a complex number when written in standard form $a+bi$?
Describe the complex plane. What does the horizontal axis represent, and what does the vertical axis represent?
How do you add two complex numbers, for example, $(3-2i) + (1+5i)$?
How do you subtract two complex numbers, for example, $(5+i) - (2-3i)$?
Explain the process of multiplying two complex numbers, such as $(2+3i)(1-i)$. What crucial substitution is made involving $i$?
What is the complex conjugate of a complex number $a+bi$? How is it found?
What is the result when a complex number is multiplied by its complex conjugate? (e.g., $(a+bi)(a-bi)$)
Why is the complex conjugate used when dividing complex numbers? What is the goal?
Outline the steps to divide $\frac{2+5i}{4-i}$, as shown in Example $ \PageIndex{ 7 } $.
What are the first four powers of $i$ (i.e., $i^1, i^2, i^3, i^4$)?
How can you simplify a higher power of $i$, such as $i^{35}$, using the cyclic nature of its powers? (Refer to Example $ \PageIndex{ 10 } $)
If a complex number is $0+3i$, what is its real part and what is its imaginary part? Is it considered a pure imaginary number?
Explain how to add complex numbers.
What is the basic principle in multiplication of complex numbers?
Give an example to show the product of two imaginary numbers is not always imaginary.
What is a characteristic of the plot of a real number in the complex plane?

Homework

Algebraic

For the following exercises, evaluate the algebraic expressions.

If $f(x)= x^2 +x−4$, evaluate $f(2i)$.
If $f(x)= x^3 −2$, evaluate $f(i)$.
If $f(x)= x^2 +3x+5$, evaluate $f(2+i)$.
If $f(x)=2 x^2 +x−3$, evaluate $f(2−3i)$.
If $f(x)= \frac{x+1}{2−x} $, evaluate $f(5i)$.
If $f(x)= \frac{1+2x}{x+3} $, evaluate $f(4i)$.

Graphical

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

$Graph of a parabola intersecting the real axis.$
$Graph of a parabola not intersecting the real axis.$

For the following exercises, plot the complex numbers on the complex plane.

$1−2i$
$−2+3i$
$i$
$−3−4i$

Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

$( 3+2i )+(5−3i)$
$( −2−4i )+( 1+6i )$
$( −5+3i )−(6−i)$
$( 2−3i )−(3+2i)$
$(−4+4i)−(−6+9i)$
$( 2+3i )(4i)$
$( 5−2i )(3i)$
$( 6−2i )(5)$
$( −2+4i )( 8 )$
$( 2+3i )(4−i)$
$( −1+2i )(−2+3i)$
$( 4−2i )(4+2i)$
$( 3+4i )( 3−4i )$
$\frac{3+4i}{2}$
$\frac{6−2i}{3}$
$\frac{−5+3i}{2i}$
$\frac{6+4i}{i}$
$\frac{2−3i}{4+3i}$
$\frac{3+4i}{2−i}$
$\frac{2+3i}{2−3i}$
$\sqrt{ −9 } +3\sqrt{ −16 }$
$−\sqrt{ −4 } −4\sqrt{ −25 }$
$\frac{2+\sqrt{ −12 }}{2}$
$\frac{4+\sqrt{ −20 }}{2}$
$i^8$
$i^{15}$
$i^{22}$

Technology

For the following exercises, use a calculator to help answer the questions.

Evaluate $(1+i)^k$ for $k = 4$, $8$, and $12$. Predict the value if $k=16$.
Evaluate $(1−i)^k$ for $k = 2$, $6$, and $10$. Predict the value if $k=14$.
Evaluate $(1+i)^k − (1−i)^k$ for $k = 4$, $8$, and $12$. Predict the value if $k=16$.
Show that a solution of $x^6 +1=0$ is $\frac{\sqrt{ 3 }}{2} + \frac{1}{2} i$.
Show that a solution of $x^8 −1=0$ is $\frac{\sqrt{ 2 }}{2} + \frac{\sqrt{ 2 }}{2} i$.

Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

$\frac{1}{i} + \frac{4}{i^3}$
$\frac{1}{i^{11}} − \frac{1}{i^{21}}$
$i^7 ( 1+ i^2 )$
$i^{−3} +5 i^7$
$\frac{( 2+i )( 4−2i )}{(1+i)}$
$\frac{( 1+3i )( 2−4i )}{(1+2i)}$
$\frac{( 3+i )^2}{( 1+2i )^2}$
$\frac{3+2i}{2+i} +( 4+3i )$
$\frac{4+i}{i} + \frac{3−4i}{1−i}$
$ \frac{3+2i}{1+2i} − \frac{2−3i}{3+i}$

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