1.
Convert the polar point \(\left(4,\frac{7\pi}{6}\right)_p\) to rectangular form. Leave your components in exact form.
Section 8.3 Complex Numbers
In this section, we will learn about complex numbers. We will learn how to plot complex numbers in the complex plane, convert complex numbers between polar, rectangular, and Euler forms, and how to arithmetic with complex numbers.
Subsection Textbook Reference
This relates to content in Β§10.5 and Β§2.4 of Algebra and Trigonometry 2e.
Exercises Preparation Exercises
Answer the following. These questions are intended to check the prerequisite skills needed to complete the rest of the lab.
2.
Convert the rectangular point \(\left(-5,2\right)_r\) to polar form. Use your calculator to find an angle in radians rounded to two digits behind the decimal place.
3.
For the right triangle shown in FigureΒ 8.3.1, find the missing length of \(r\) and the missing angle \(\theta\) using SOHCAHTOA. You can ue a calculator to get an approximation, accurate to 4 digits behind the decimal place.
Exercises Practice Exercises
1.
Plot the complex numbers on the complex plane, with real units on the \(x\)-axis and imaginary units on the \(y\)-axis.
(a)
\(2+3i\)
(b)
\(-1+2i\)
(c)
\(-3i\)
(d)
\(3-i\)
2.
Find the absolute value and argument of the complex numbers.
(a)
\(2+3i\)
(b)
\(-1+2i\)
3.
(a)
Convert the standard form complex number \(5-12i\) into polar form. Use radians and round your argument to two digits behind the decimal place.
(b)
Convert the polar form complex number \(7\left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right)\) into standard form. Keep your values exact using your memorized values on the unit circle.
4.
Simplify the expressions completely.
(a)
\((2i)^4\)
(b)
\((2i+3)\cdot(4i-1)\)
(c)
\((2i+3)-(4i-1)\)
5.
(a)
Convert the standard form complex number \(5-12i\) into Eulerβs form. Use radians and round your argument to two digits behind the decimal place.
(b)
Convert the Eulerβs form complex number \(7e^{\frac{4\pi i}{3}}\) into standard form. Keep your values exact using your memorized values on the unit circle.
Subsection Definitions
Definition 8.3.2. The Imaginary Unit \(i\).
The imaginary unit \(i\) is defined to be \(i=\sqrt{-1}\text{.}\) Since there is no real number that is multiplied by itself to create a negative number, \(i\) must not be a real number. It fits into a new category of numbers called imaginary numbers, which are multiples of \(i\text{.}\) Itβs called a "unit" because its absolute value is \(1\)
Example 8.3.3.
Definition 8.3.4. Standard Form of a Complex Number.
A complex number is any real number plus any imaginary number. Any complex number can be written in standard form (also called rectangular form), which is the form \(a+bi\text{,}\) where both \(a\) and \(b\) are real numbers.
Example 8.3.5.
Definition 8.3.6. The Complex Plane.
The complex plane is the two dimensional axes system to pot complex numbers. The horizontal axis, often just labeled \(x\text{,}\) measures the real part of a complex number. The vertical axis, often labeled \(y\text{,}\) measures the imaginary part of a complex number.
Example 8.3.7.
The number \(-3+4i\) would be plotted \(3\) units to the left and \(4\) units up from the origin at the same place that \((-3,4)\) would have been plotted in the regular two-dimensional real axes.
Definition 8.3.8. Absolute Value of a Complex Number.
The absolute value of the complex number is the distance from the origin to the number in the complex plane. The absolute value is written as usual, \(|z|\text{.}\) To find the absolute value of \(z= a+ bi\text{,}\) use the Pythagorean Theorem: \(|z|= \sqrt{a^2 + b^2}\text{.}\)
Note: the \(i\) is dropped during this calculation!
Note: the absolute value of a complex number is sometimes also called the magnitude or modulus.
Example 8.3.9.
To find the absolute value of \(z = -3+4i\text{,}\) we would write that \(|z| = \sqrt{(-3)^2 + 4^2} = 5\text{.}\)
Definition 8.3.10. Argument of a Complex Number.
The argument of the complex number, \(z =a+ bi\text{,}\) is an angle, \(\theta\text{,}\) measured from the standard position to the number in the complex plane. To find this angle, ΞΈ, use the formula \(tan(\theta) = \frac{b}{a}\) solve for \(\theta\) that is in the correct quadrant.
Note: the \(i\) is dropped during this calculation!
Example 8.3.11.
To find the argument of \(z =β3 + 4i\text{,}\) we would first note that \(z\) is in the second quadrant, and we would write \(tan(\theta) = \frac{4}{β3}\text{.}\) Next, we would take the inverse-tangent on both sides, however, inverse-tangent does not give us angles in quadrant II, so we know that to find the correct argument of \(z\) we should add \(\pi\text{.}\) So the argument of \(z\) is \(\theta = \tan^{-1}\left(\frac{4}{-3}\right) + \pi \approx 2.21\text{,}\) as shown in FigureΒ 8.3.12
A graph on the complex plane. The point \(-3+4i\) is plotted with a dotted line connecting it to the origin. The dotted line is labeled \(|z| = 5\text{,}\) showing the absolute value, and the angle from the positive \(x\)-axis is labeled \(\theta \approx 2.21\text{,}\) showing the argument.
Definition 8.3.13. Polar Form of a Complex Number.
A complex number can be written in polar form, which is the form \(r\left(\cos(\theta)+i\sin(\theta)\right)\text{,}\) where \(r=|z|\) and \(\theta\) is the argument of the complex number.
Example 8.3.14.
Definition 8.3.15. Eulerβs Form of a Complex Number.
Eulerβs form of a complex number, \(z\text{,}\) is \(z=re^{i\theta}\text{,}\) where \(r=|z|\) and \(\theta\) is the argument of \(z\text{.}\)
Example 8.3.16.
Exercises Exit Exercises
1.
Describe the difference between the point \((5,8)\) in the two-dimensional real coordinate plane and the complex number \(5+8i\) in the complex plane.
2.
(a)
Plot the complex number \(-6-4i\) onto the complex plane.
(b)
Find the absolute value of the complex number \(-6-4i\text{.}\) Simplify the radical completely and leave your answer in exact form. Illustrate the value on your plot from exercise a.
(c)
Find the argument of the complex number \(-6-4i\text{.}\) Use a calculator to round your answer to three digits behind the decimal place. Illustrate the value on your plot from exercise a.
(d)
Write the complex number \(-6-4i\) into polar form.
(e)
Write the complex number \(-6-4i\) into Eulerβs form.
Reflection Reflection
1.
On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
