What Should the Nurse Do?
Marie, a 28-year-old pregnant female in her second trimester, presents at the community health center for her routine prenatal checkup. Marie expresses interest in the Lamaze childbirth education program to prepare for her upcoming delivery. She has a history of uncomplicated pregnancies and deliveries but is seeking a more informed and empowered experience this time. Marie reports no concerning symptoms, and her medical and psychiatric history is unremarkable. Vital signs are stable, and the physical examination reveals a healthy pregnancy progression. During the consultation, Marie emphasizes her desire for shared decision making and understanding the physiologic aspects of childbirth.
1 .
How has the history of Lamaze, starting with Dr. Fernand Lamaze’s breathing techniques, shaped its evolution into a comprehensive childbirth education program, and in what ways can Marie benefit from its transformation?
2 .
How do the six evidence-based healthy birth practices recommended by Lamaze, including spontaneous labor initiation and continuous labor support, align with the principles of physiologic birth, and why are these practices essential for birthing persons like Marie who are seeking an informed and empowered birth experience?
3 .
Why is shared decision making emphasized in Lamaze education, and how does it empower persons like Marie in their birthing experience?
4 .
Why does Lamaze education stress the significance of understanding the physiologic aspects of birth? Why is this knowledge valuable for birthing persons like Marie and their partners?
5 .
What does Lamaze education offer in terms of preparing birthing persons like Marie and their families for the postpartum period, and why is this information essential?
James, a 32-year-old expectant partner, attends a prenatal appointment at the community health clinic with his wife, Emma, who is 30 years old and in her second trimester of pregnancy. Emma is interested in exploring natural childbirth options and has specifically inquired about the Bradley Method. Both James and Emma are actively engaged in preparing for the birthing experience. Emma reports experiencing occasional headaches and mild back pain, common discomforts associated with pregnancy. Her medical and psychiatric history are otherwise unremarkable. Vital signs are within normal ranges. During the consultation, Emma emphasizes her interest in understanding the history of the Bradley Method, the significance of prenatal nutrition and exercise, and the importance of continuous labor support.
6 .
Considering James and Emma’s interest in the Bradley Method, how does the history of the Bradley Method, beginning with Dr. Robert A. Bradley’s focus on unmedicated, low-intervention births, reflect its evolution into a comprehensive childbirth education program?
7 .
Given Emma’s interest in the Bradley Method, how does the program incorporate prenatal nutrition and exercise recommendations, and why are these elements considered crucial for promoting a healthy pregnancy and uncomplicated birth?
8 .
In the context of James and Emma’s engagement with the Bradley Method, how does the program advocate for continuous labor support, and why is this considered vital for improving outcomes and experiences for birthing persons?
9 .
Given James and Emma’s interest in the Bradley Method, how does the program differentiate its teachings for the first and second stages of labor, and how does it prepare both the birthing person and the labor support person for each stage?
Sophie, a 28-year-old pregnant female, attends a prenatal appointment at the local birthing center with her husband, Michael, in their second trimester. Sophie is intrigued by HypnoBirthing, the Mongan Method of Childbirth Education, and expresses a desire to learn more about this approach for her upcoming delivery. She reports occasional feelings of anxiety and anticipatory stress related to childbirth. Sophie has a history of mild anxiety, which has been managed without medication, and no significant medical or psychiatric issues. She discloses that her anxiety tends to manifest as restlessness and occasional difficulty in sleeping. Despite these challenges, Sophie is motivated to explore natural methods to enhance her childbirth experience. During the consultation, Sophie’s vital signs are stable, and the physical examination is unremarkable. Additionally, Sophie mentions that she has been experiencing mild lower back pain, common in pregnancy, which she attributes to her job, involves prolonged periods of standing.
10 .
Considering Sophie’s interest in the HypnoBirthing Mongan Method, how does the history of this approach reflect a unique focus on the mental aspects of childbirth and how birth can be experienced with minimal discomfort?
11 .
In Sophie’s case, how does the Mongan Method aim to provide pain control, and what reported benefits might Sophie experience, as mentioned in the chapter?
12 .
How does the Mongan Method instill positive expectations in birthing persons, as observed in Sophie’s interest in exploring natural methods for enhancing her childbirth experience?
13 .
Considering Sophie’s occasional anxiety, how might the visualization and deepening process in the Mongan Method specifically assist Sophie during labor and birth?
14 .
In the context of Sophie’s pregnancy, how does the Mongan Method address and generalize the normal physical and emotional changes that birthing persons may undergo during childbirth?
Michelle, a 34-year-old pregnant female, attends a prenatal appointment at the local community center in her third trimester. She is intrigued by the potential benefits of the Alexander Technique for childbirth and expresses a desire to learn more about this approach. Michelle reports occasional lower back pain and discomfort related to her pregnancy, which has led her to seek alternative methods for pain management. She has a history of generalized anxiety disorder but is currently not on medication, managing her symptoms through mindfulness and relaxation techniques. Michelle’s vital signs are stable, and the physical examination reveals typical signs of a healthy pregnancy.
15 .
Where can Michelle find an instructor for the Alexander Technique?
16 .
How might the Alexander Technique benefit Michelle during labor and birth, and what evidence supports its use?
17 .
How would Michelle be educated about self-awareness and releasing muscle tension through the Alexander Technique?
18 .
Why might private classes in the Alexander Technique be advantageous to Michelle?
19 .
What positions does the Alexander Technique recommend for Michelle during pregnancy, labor, and postpartum?
Campbell, a 29-year-old pregnant female, visits her obstetrician at the local women’s health clinic for a routine prenatal appointment. Campbell is in her second trimester and has a history of mild hypertension, which is well controlled with medication. She reports occasional feelings of anxiety related to her upcoming delivery. Campbell’s vital signs are stable, and her physical examination shows typical signs of a healthy pregnancy. During the appointment, Campbell expresses an interest in creating a birth plan. She hopes to have a clear outline of her preferences and expectations for labor and delivery. Campbell mentions that she wants to be actively involved in decision making and feels that a birth plan will help her communicate her desires effectively to her health-care team.
20 .
How can a birth plan serve as a tool for effective communication between Campbell, her support team, and the health-care providers during labor and delivery?
21 .
In what ways can nurses contribute to the empowerment of birthing persons like Campbell through the support of their birth plans?
22 .
Considering Campbell’s case, what specific components or preferences might be important to include in her birth plan, given her history of mild hypertension and anxiety?


14.5: Polyprotic Acids
- Extend previously introduced equilibrium concepts to acids and bases that may donate or accept more than one proton
We can classify acids by the number of protons per molecule that they can give up in a reaction. Acids such as \(\ce{HCl}\), \(\ce{HNO3}\), and \(\ce{HCN}\) that contain one ionizable hydrogen atom in each molecule are called monoprotic acids . Their reactions with water are:
\[\ce{HCl}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{Cl-}(aq) \nonumber \]
\[\ce{HNO3}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{NO3-}(aq) \nonumber \]
\[\ce{HCN}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{CN-}(aq) \nonumber \]
Even though it contains four hydrogen atoms, acetic acid, \(\ce{CH3CO2H}\), is also monoprotic because only the hydrogen atom from the carboxyl group (\(\ce{-COOH}\)) reacts with bases:
Similarly, monoprotic bases are bases that will accept a single proton.
Diprotic Acids
Diprotic acids contain two ionizable hydrogen atoms per molecule; ionization of such acids occurs in two steps. The first ionization always takes place to a greater extent than the second ionization. For example, sulfuric acid, a strong acid, ionizes as follows:
- The first ionization is
\[ \ce{H2SO4}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HSO4-}(aq) \nonumber \]
with \(K_{\ce a1} > 10^2;\: {complete\: dissociation}\).
- The second ionization is
\[ \ce{HSO4-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{SO4^{2−}}(aq) \nonumber \]
with \( K_{\ce a2}=1.2×10^{−2}\).
This stepwise ioniza tion process occurs for all polyprotic acids. When we make a solution of a weak diprotic acid, we get a solution that contains a mixture of acids. Carbonic acid, \(\ce{H2CO3}\), is an example of a weak diprotic acid. The first ionization of carbonic acid yields hydronium ions and bicarbonate ions in small amounts.
- First Ionization
\[\ce{H2CO3}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO3-}(aq) \nonumber \]
with
\[K_{\ce{H2CO3}}=\ce{\dfrac{[H3O+][HCO3- ]}{[H2CO3]}}=4.3×10^{−7} \nonumber \]
The bicarbonate ion can also act as an acid. It ionizes and forms hydronium ions and carbonate ions in even smaller quantities.
- Second Ionization
\[\ce{HCO3-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CO3^2-}(aq) \nonumber \]
with
\[ K_{\ce{HCO3-}}=\ce{\dfrac{[H3O+][CO3^2- ]}{[HCO3- ]}}=4.7×10^{−11} \nonumber \]
\(K_{\ce{H2CO3}}\) is larger than \(K_{\ce{HCO3-}}\) by a factor of 10 4 , so H 2 CO 3 is the dominant producer of hydronium ion in the solution. This means that little of the \(\ce{HCO3-}\) formed by the ionization of H 2 CO 3 ionizes to give hydronium ions (and carbonate ions), and the concentrations of H 3 O + and \(\ce{HCO3-}\) are practically equal in a pure aqueous solution of H 2 CO 3 .
If the first ionization constant of a weak diprotic acid is larger than the second by a factor of at least 20, it is appropriate to treat the first ionization separately and calculate concentrations resulting from it before calculating concentrations of species resulting from subsequent ionization. This can simplify our work considerably because we can determine the concentration of H 3 O + and the conjugate base from the first ionization, then determine the concentration of the conjugate base of the second ionization in a solution with concentrations determined by the first ionization.
When we buy soda water (carbonated water), we are buying a solution of carbon dioxide in water. The solution is acidic because CO 2 reacts with water to form carbonic acid, H 2 CO 3 . What are \(\ce{[H3O+]}\), \(\ce{[HCO3- ]}\), and \(\ce{[CO3^2- ]}\) in a saturated solution of CO 2 with an initial [H 2 CO 3 ] = 0.033 M ?
\[\ce{H2CO3}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO3-}(aq) \hspace{20px} K_{\ce a1}=4.3×10^{−7} \label{step1} \tag{equilibrium step 1} \]
\[\ce{HCO3-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CO3^2-}(aq) \hspace{20px} K_{\ce a2}=4.7×10^{−11} \label{step2} \tag{equilibrium step 2} \]
Solution
As indicated by the ionization constants, H 2 CO 3 is a much stronger acid than \(\ce{HCO3-}\), so \(\ce{H2CO3}\) is the dominant producer of hydronium ion in solution. Thus there are two parts in the solution of this problem:
- Using the customary four steps, we determine the concentration of H 3 O + and \(\ce{HCO3-}\) produced by ionization of H 2 CO 3 .
- Then we determine the concentration of \(\ce{CO3^2-}\) in a solution with the concentration of H 3 O + and \(\ce{HCO3-}\) determined in (1).
To summarize:
1. First Ionization: Determine the concentrations of \(\ce{H3O+}\) and \(\ce{HCO3-}\).
Since \ref{step1} is has a much bigger \(K_{a1}=4.3×10^{−7}\) than \(K_{a2}=4.7×10^{−11}\) for \ref{step2}, we can safely ignore the second ionization step and focus only on the first step (but address it in next part of problem).
\[\ce{H2CO3}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO3-}(aq) \hspace{20px} K_{\ce a1}=4.3×10^{−7} \nonumber \]
As for the ionization of any other weak acid:
An abbreviated table of changes and concentrations shows:
| ICE Table | \(\ce{H2CO3}(aq)\) | \(\ce{H2O}(l)\) | \( \ce{H3O+}(aq) \) | \(\ce{HCO3-}(aq)\) |
|---|---|---|---|---|
| I nitial (M) | \(0.033 \:M\) | - | \(0\) | \(0\) |
| C hange (M) | \(- x\) | - | \(+x\) | \(+x\) |
| E quilibrium (M) | \(0.033 \:M - x\) | - | \(x\) | \( x\) |
Substituting the equilibrium concentrations into the equilibrium constant gives us:
\[K_{\ce{H2CO3}}=\ce{\dfrac{[H3O+][HCO3- ]}{[H2CO3]}}=\dfrac{(x)(x)}{0.033−x}=4.3×10^{−7} \nonumber \]
Solving the preceding equation making our standard assumptions gives:
\[x=1.2×10^{−4} \nonumber \]
Thus:
\[\ce{[H2CO3]}=0.033\:M \nonumber \]
\[\ce{[H3O+]}=\ce{[HCO3- ]}=1.2×10^{−4}\:M \nonumber \]
2. Second Ionization: Determine the concentration of \(CO_3^{2-}\) in a solution at equilibrium.
Since the \ref{step1} is has a much bigger \(K_a\) than \ref{step2}, we can the equilibrium conditions calculated from first part of example as the initial conditions for an ICER Table for the \ref{step2}:
\[\ce{HCO3-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CO3^2-}(aq) \nonumber \]
| ICE Table | \(\ce{HCO3-}(aq)\) | \(\ce{H2O}(l)\) | \( \ce{H3O+}(aq) \) | \(\ce{CO3^2-}(aq)\) |
|---|---|---|---|---|
| I nitial (M) | \(1.2×10^{−4}\:M\) | - | \(1.2×10^{−4}\:M\) | \(0\) |
| C hange (M) | \(- y\) | - | \(+y\) | \(+y\) |
| E quilibrium (M) | \(1.2×10^{−4}\:M - y\) | - | \(1.2×10^{−4}\:M + y\) | \( y\) |
\[ \begin{align*} K_{\ce{HCO3-}}&=\ce{\dfrac{[H3O+][CO3^2- ]}{[HCO3- ]}} \\[4pt] &=\dfrac{(1.2×10^{−4}\:M + y) (y)}{(1.2×10^{−4}\:M - y)} \end{align*} \nonumber \]
To avoid solving a quadratic equation, we can assume \(y \ll 1.2×10^{−4}\:M \) so
\[K_{\ce{HCO3-}} = 4.7×10^{−11} \approx \dfrac{(1.2×10^{−4}\:M ) (y)}{(1.2×10^{−4}\:M)} \nonumber \]
Rearranging to solve for \(y\)
\[y \approx \dfrac{ (4.7×10^{−11})(1.2×10^{−4}\:M )}{ 1.2×10^{−4}\:M} \nonumber \]
\[[\ce{CO3^2-}]=y \approx 4.7×10^{−11} \nonumber \]
To summarize :
In part 1 of this example, we found that the \(\ce{H2CO3}\) in a 0.033- M solution ionizes slightly and at equilibrium \([\ce{H2CO3}] = 0.033\, M\), \([\ce{H3O^{+}}] = 1.2 × 10^{−4}\), and \(\ce{[HCO3- ]}=1.2×10^{−4}\:M\). In part 2, we determined that \(\ce{[CO3^2- ]}=5.6×10^{−11}\:M\).
The concentration of \(H_2S\) in a saturated aqueous solution at room temperature is approximately 0.1 M . Calculate \(\ce{[H3O+]}\), \(\ce{[HS^{−}]}\), and \(\ce{[S^{2−}]}\) in the solution:
\[\ce{H2S}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HS-}(aq) \hspace{20px} K_{\ce a1}=8.9×10^{−8} \nonumber \]
\[\ce{HS-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{S^2-}(aq) \hspace{20px} K_{\ce a2}=1.0×10^{−19} \nonumber \]
- Answer
-
\([\ce{H2S}] = 0.1 M\), \(\ce{[H3O+]} = [HS^{−}] = 0.0001\, M\), \([S^{2−}] = 1 × 10^{−19}\, M\)
We note that the concentration of the sulfide ion is the same as K a2 . This is due to the fact that each subsequent dissociation occurs to a lesser degree (as acid gets weaker).
Triprotic Acids
A triprotic acid is an acid that has three dissociable protons that undergo stepwise ionization: Phosphoric acid is a typical example:
- The first ionization is
\[\ce{H3PO4}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{H2PO4-}(aq) \nonumber \]
with \(K_{\ce a1}=7.5×10^{−3} \).
- The second ionization is
\[\ce{H2PO4-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HPO4^2-}(aq) \nonumber \]
with \( K_{\ce a2}=6.2×10^{−8} \).
- The third ionization is
\[\ce{HPO4^2-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{PO4^3-}(aq) \nonumber \]
with \( K_{\ce a3}=4.2×10^{−13} \).
As with the diprotic acids, the differences in the ionization constants of these reactions tell us that in each successive step the degree of ionization is significantly weaker. This is a general characteristic of polyprotic acids and successive ionization constants often differ by a factor of about 10 5 to 10 6 . This set of three dissociation reactions may appear to make calculations of equilibrium concentrations in a solution of H 3 PO 4 complicated. However, because the successive ionization constants differ by a factor of 10 5 to 10 6 , the calculations can be broken down into a series of parts similar to those for diprotic acids.
Polyprotic bases can accept more than one hydrogen ion in solution. The carbonate ion is an example of a diprotic base , since it can accept up to two protons. Solutions of alkali metal carbonates are quite alkaline, due to the reactions:
\[\ce{H2O}(l)+\ce{CO3^2-}(aq)⇌\ce{HCO3-}(aq)+\ce{OH-}(aq) \nonumber \]
and
\[\ce{H2O}(l)+\ce{HCO3-}(aq)⇌\ce{H2CO3}(aq)+\ce{OH-}(aq) \nonumber \]
Summary
An acid that contains more than one ionizable proton is a polyprotic acid. The protons of these acids ionize in steps. The differences in the acid ionization constants for the successive ionizations of the protons in a polyprotic acid usually vary by roughly five orders of magnitude. As long as the difference between the successive values of K a of the acid is greater than about a factor of 20, it is appropriate to break down the calculations of the concentrations of the ions in solution into a series of steps.
Glossary
- diprotic acid
- acid containing two ionizable hydrogen atoms per molecule. A diprotic acid ionizes in two steps
- diprotic base
- base capable of accepting two protons. The protons are accepted in two steps
- monoprotic acid
- acid containing one ionizable hydrogen atom per molecule
- stepwise ionization
- process in which an acid is ionized by losing protons sequentially
- triprotic acid
- acid that contains three ionizable hydrogen atoms per molecule; ionization of triprotic acids occurs in three steps


4.6: Graduation Speeches- Speeches You Give in Pointy Hats
Gather the Details
Brainstorm with Friends
Organizational Format
Manuscript Format
Pick a Theme
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Graduation is a big day for graduates, their families, and teachers. If you are called to give a graduation speech, you want to make it special. I want to share with you what makes a good graduation speech and give you tips on how to write one that will make an impact.
As we begin, you need to wrap your mind around two main things:
There are two main types of graduation speakers, the student speaker, and the headline speaker. At one college at our university, there is a speech contest to be the graduation speaker and at another college, it is someone who has been nominated by a faculty member. How you get there varies from place to place At the local high school, the valedictorian is often the speaker. I recently went to high school graduation and they had seven valedictorians so they had seven speakers–yes, it was as long as you can imagine.
When thinking about giving a graduation speech, you have to ask, “What does the audience need from me?” They need you to reflect on the past, celebrate the present, and focus on the future. This chapter will walk you through the essentials of giving a graduation speech and then give you several example speeches as samples of key elements.
This is the fun part. Sit down with friends and make a list of all the things that come to mind about the college experience. When brainstorming, write down everything you think of and don’t try to judge whether it should be included, just go with it. There is an entire chapter on how to brainstorm here.
Most all student graduation speeches include the past, present, and future format.
Most student graduation speeches are in manuscript format. That helps you from getting overwhelmed at the moment and that also gives the school a chance to censor– I mean to approve of–your content. There is an entire chapter on writing a manuscript that you can refer to here.
Many graduation speeches use a theme. Here are some of the most common graduation themes.
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It can be helpful to pick a theme and connect a metaphor to your theme. There is an entire chapter on how to do that here.
“There is no such thing as failure.
Failure is just life trying to push you in another direction.”
Oprah Winfrey, Harvard University Commencement Speech
Most introductions acknowledge the occasion, offer thanks, and lead into the main idea. Shutterfly suggests these as openings.
I have written a chapter on each component of ceremonial speaking and you can reference those you need:
Notice how Jaclyn Marston reflects on specific classes and memories. (Watch starting at 54 seconds).
Watch how Lin Manuel Miranda references the familiar and the obscure in his address to the University of Pennsylvania (start watching at 1:12).
Notice how she uses the theme–“What do you want to be” when you grow up and alters it to “What do you want to do?” She opens with this and wraps back around to this same idea at the end.
Notice how this speaker admits his shortcomings. We feel like he is honest and vulnerable so we hang on his everyword.
Headline speakers are usually someone famous or notable. Speeches by those individuals almost always include stories and challenges. I have included several here. Pick two of them to analyze.
Today I want to tell you three stories from my life. That’s it. No big deal. Just three stories. Steve Jobs
These highlights of Lou Holtz’s graduation speech is full of great challenges and life lessons.
Tim Minchin
“One: Be micro-ambitious.  You never know where you might end up.
Two: Don’t seek happiness. Keep busy and aim to make someone else happy and you might find you get some as a side effect.
Three: Understanding that you can’t truly take credit for your successes nor truly blame others for their failures will humble you and make you more compassionate.
Four: Exercise. Take care of your body: you’re going to need it.
Five: Be hard on your opinions. Be intellectually rigorous. Identify your biases, your prejudices, your privileges.
Six: Even if you’re not a teacher, be a teacher. Share your ideas. Don’t take for granted your education.
Seven: Define yourself by what you love. Be demonstrative and generous in your praise of those you admire. Send thank you cards and give standing ovations. Be pro stuff not just anti stuff.

Nine: Finally, don’t rush.
As you can see, graduation speeches can be serious or lighthearted; they can be personal, motivational, and informative. The key thing is that the speech should be authentic. It should be as unique as the speaker.
Remember This!
Remember This!
Jaclyn Marson describes the process of how she wrote her Graduation Speech.
Dunham, A. (2019). Valedictorian comes out as autistic during speech. [Video] YouTube. https://youtu.be/GtPGrLoU5Uk Standard YouTube License
Holtz, L. (2017). Lou Holtz’s inspirational speech. Commencement speech.[Video] YouTube. https://www.youtube.com/watch?v=M3LOo_Ccyws Standard YouTube License
Jobs, S. (2008). Steve Jobs’ 2005 Stanford Commencement Address. [Video] YouTube. https://www.youtube.com/watch?v=UF8uR6Z6KLc Standard YouTube License.
Jostens, (n.d.). Celebrate high school memories. Inspire your grad community. https://www.jostens.com/resources/students-and-parents/graduation-guides/how-to-write-a-grad-speech
Marson, J. (2020). How to write an amazing graduation speech–Jaclyn Marson podcast Ep 1. [Video] YouTube. https://www.youtube.com/watch?v=t5CUSzp9SrM Standard YouTube License.
Marston, J. (2016). Beautiful and moving graduation speech 2016. [Video] YouTube. https://www.youtube.com/watch?v=_F3K3Z_5CEE Standard YouTube License.
Minchin, T. (2013). 9 life lessons-Time Minchin UWA Address. [Video] YouTube. https://www.youtube.com/watch?v=yoEezZD71sc Standard YouTube License.
Rosen, L. (2019). Leah Rosen: “The power of this place,” Duke University 2019 commencement student speaker. https://www.youtube.com/watch?v=p4N Standard YouTube License.
Shutterfly. (n.d) How to start a graduation speech. https://www.shutterfly.com/ideas/graduation-speech/
Stewart, M. (2020). Student speaker. Commencement 2020. University of Utah. [Video] YouTube. h ttps://www.youtube.com/watch?v=AZFJnZvuQIo Standard YouTube License.
University of Pennsylvania. (2016). Penn’s 2016 commencement ceremony- Commencement speaker Lin-Manuel Miranda. [Video] YouTube. https://www.youtube.com/watch?v=ewHcsFlolz4&t=0s Standard YouTube License.


9.1: Gasses and Atmospheric Pressure
Chapter 2 , we learned about the three principle states of matter; solids, liquids and gasses. We explained the properties of the states of matter using the kinetic molecular theory (KMT). Substances in the gaseous state, according to the KMT , have enough kinetic energy to break all of the attractive forces between the individual gas particles and are therefore free to separate and rapidly move throughout the entire volume of their container. Because there is so much space between the particles in a gas, a gas is highly compressible. High compressibility and the ability of gases to take on the shape and volume of its container are two of the important physical properties of gasses.
InThe gas that we are all most familiar with is the mixture of elements and compounds that we call the “atmosphere”. The air that we breath is mostly nitrogen and oxygen, with much smaller amounts of water vapor, carbon dioxide, noble gasses and the organic compound, methane (Table 9.1).
Table 9.1. Approximate Composition of the Atmosphere
| Gas | Concentration, Parts per Billion | Percentage |
|---|---|---|
| N 2 | 7.8 × 10 8 | 78% |
| O 2 | 2.0 × 10 8 | 20% |
| H 2 O | About 10 6 – 10 7 | < 1% |
| Ar | 9.3 × 10 6 | < 1% |
| CO 2 | 3.5 × 10 5 | < 0.05% |
| Ne | 1.8 × 10 4 | trace |
| He | 5.2 × 10 3 | trace |
| CH 4 | 1.6 × 10 3 | trace |
A gas that is enclosed in a container exerts a pressure on the inner walls of that container. This pressure is the result of the countless collisions of the gas particles with the container wall. As each collision occurs, a small amount of energy is transferred, generating a net pressure. Although we are generally unaware of it, the gasses in the atmosphere generate a tremendous pressure on all of us. At sea level, atmospheric pressure is equal to 14.7 pounds per square inch. Putting this in perspective, for a person of average height and build, the total pressure from the atmosphere pressing on their body is about 45,000 pounds! Why aren’t we squashed? Remember, we also have air inside our bodies and the pressure from the inside balances the pressure outside, keeping us nice and firm, not squishy!
The proper SI unit for pressure is the Pascal (Pa), where 1 Pa = 1 kg m -1 s -2 . In chemistry, however, it is more common to measure pressure in terms of atmospheres (atm) where 1 atm is atmospheric pressure at sea level, or 1 atm = 14.7 pounds per square inch (1 atm = 101,325 Pa). Atmospheric pressure is typically measured using a device called a barometer . A simple mercury barometer (also called a Torricelli barometer, after its inventor) consists of a glass column, about 30 inches high, closed at one end and filled with mercury. The column is inverted and placed in an open, mercury-filled reservoir. The weight of the mercury in the tube causes the column to drop to the point that the mass of the mercury column matches the atmospheric pressure exerted on the mercury in the reservoir. The atmospheric pressure is then read as the height of the mercury column. Again, working at sea level, 1 atmosphere is exactly equal to a column height of 760 mm of mercury. The units for the conversion are 1 atm = 760 mm Hg, and this is an exact relationship with regard to significant figures. The unit torr (after Torricelli) is sometimes used in place of mm Hg.


7.1: Pressure
Pressure is related to the vertical distance from the surface of water to a reference point somewhere below the water surface typically expressed in feet. For example, a water storage tank has a certain water level inside of it. The reference point below could be a water faucet a certain distance in feet below the water level. This distance in feet is the head pressure (in feet). The other common unit for pressure is POUNDS PER SQUARE INCH, (psi). You can convert back and forth between feet and pounds per square inch using one of the following conversion factors.
\[1 \text { foot }=0.433 \mathrm{psi} \quad \text { or } \quad 1 \mathrm{psi}=2.31 \text { feet } \nonumber \]
To stay consistent with UDA, view the following conversion factors as
\[\dfrac{1 \text { foot }}{0.433 \text { psi }} \quad \text { or } \quad \dfrac{1 \text { psi }}{2.31 \text { feet }} \nonumber \]
Remember, as with all conversion factors, these can be written as the inverse.
\[\dfrac{0.433 \mathrm{psi}}{1 \mathrm{foot}} \quad \text { or } \quad \dfrac{2.31 \text { feet }}{1 \mathrm{psi}} \nonumber \]
Example \(\PageIndex{1}\)
- Assume both cylinders above are filled with water, which one has a greater pressure at the base of it?
- What is the pressure at the base of each cylinder?
Solution
- The answer is neither! The pressure at the base of each cylinder is the same because the height of the water is the same.
- \(\dfrac{30 \text { feet }}{1} \times \dfrac{1 \mathrm{psi}}{2.31 \mathrm{feet}}=12.98 \mathrm{psi}\) or (if you round) \(13 \mathrm{psi}\)
Exercise 7.1
Calculate the following pressure problems
- What is the pressure in psi at the bottom of a 30 foot tall tank if it is full?
- What is the pressure in feet if the psi is 130?
- A storage tank is 45 feet tall and half full. What is the pressure in psi?
- A water tank sits on a 50 foot hill and is 25 feet tall. Assuming the tank is full, what is the pressure in psi at the bottom of the hill?
- A fire hydrant was hit and water is spraying up approximately 75 feet. What is the approximate pressure in psi?


1.9: Finance Style
Students very often will come to their first finance class well-read in an impressive host of subjects, including history, literature, and more, but not necessarily in economics or finance.
Finance, as certain other “scientific” subjects, is written in a markedly different style; it is notably terse and succinct. Words are not parsed, and expansive, colorful prose is consistently eschewed. Sparse, precise language is the rule.
Unlike literature, for instance, one may find that s/he has to read the same sentence several times until s/he gets it. You just can’t put your feet up, and slice through many pages in relatively short order.
Do not get frustrated when this happens. Kick around the notions discussed in your head, until you get it. Then kick it over again; you may find that you can see the same thing from different angles with increasing thought. That should invigorate you. Start thinking in Abstract Terms – like a Financial Economist.
Spend extra time on each paragraph and page. You may find that you have to substantially slow down the pace of your reading.
As formulae are presented, carefully check the calculations to be sure you agree. You will find this extremely helpful in increasing your understanding and insight. Always keep your calculator handy as you read.
Be methodical and take your time. You will find that you will adjust to the new style, and you will find enjoyment in your increasing mastery! Think deliberately. Don’t think too fast!


8.2: The Republic
Roman Sculpture under the Republic
During the Roman Republic, members of all social classes used a variety of sculptural techniques to promote their distinguished social statuses.
Describe the defining characteristics of sculpture during the Roman Republic
Key Points
- Continuous narration arose during the Republic as a means of telling a story from beginning to end in a visual form . Art patrons used it on public monuments to celebrate their status in society.
- The patronage of public sculpture extended to freedmen, as seen in the massive and uniquely decorated tomb of Marcus Vergilius Eurysaces the Baker.
- Veristic portraiture depicts the subject in hyper- realism , wrinkles and individual facial characteristics are carefully formed in these images.
- Portraits of Pompey the Great and Julius Caesar set precedents for future imperial portraiture by quoting iconographic characteristics and introducing a likeness of the living imperator on coins.
- Death masks provided a means for a family to remember their ancestors as well as to remind others of the illustriousness of their ancestry.
Key Terms
- verism : The style of portraiture favored during the Roman Republic in which the subjects were portrayed in a super-realistic likeness; from the Latin word for true.
- continuous narration : A visual art technique that illustrates multiple scenes of a story within a single frame and without any separation between scenes and actions.
- Imperator : A Latin word meaning general or ruler; a title given to triumphant generals during the Republic.
Roman Art in the Republic
Early Roman art was influenced by the art of Greece and that of the neighboring Etruscans, themselves greatly influenced by their Greek trading partners. As the expanding Roman Republic began to conquer Greek territory, its official sculpture became largely an extension of the Hellenistic style , with its departure from the idealized body and flair for the dramatic. This is partly due to the large number of Greek sculptors working within Roman territory.
However, Roman sculpture during the Republic departed from the Greek traditions in several ways.
- It was the first to feature a new technique called continuous narration.
- Commoners, including freedmen, could commission public art and use it to cast their professions in a positive light.
- Portraiture throughout the Republic celebrated old age with its verism .
- In the closing decades of the Republic, Julius Caesar counteracted traditional propriety by becoming the first living person to place his own portrait on a coin.
In the examples that follow, the patrons use these techniques to promote their status in society.
The Altar of Domitius Ahenobarbus
Despite its most common title, the Altar of Domitius Ahenobarbus (late second century BCE) was more likely a base intended to support cult statues in the cella of a Temple of Neptune (Poseidon) located in Rome on the Field of Mars. The frieze is the second oldest Roman bas- relief currently known.
Domitius Ahenobarbus, a naval general, likely commissioned the altar and the temple in gratitude of a naval victory between 129 and 128 BCE. The reliefs combine mythology and contemporary civic life.
One panel of the altar depicts the census, a uniquely Roman event of contemporary civic life. It is one of the earliest reliefs sculpted in continuous narration, in which the viewer reads from left to right the recording of the census, the purification of the army before the altar of Mars, and the levy of the soldiers.
The other three panels depict the mythological wedding of Neptune and Amphitrite. At the center of his scene, Neptune and Amphitrite are seated in a chariot drawn by two Tritons (messengers of the sea) who dance to music. They are accompanied by a multitude of fantastic creatures, Tritons, and Nereides (sea nymphs) who form a retinue for the wedding couple, which, like the census scene, can be read from left to right.
At the left, a Nereid riding on a sea-bull carries a present. Next, Amphitrite’s mother Doris advances towards the couple, mounted on a hippocampus (literally, a sea horse) and holding wedding torches in each hand to light the procession’s way. Eros hovers behind her. Behind the wedding couple, a Nereid riding a hippocampus carries another present.
Tomb of Marcus Vergilius Eurysaces the Baker
The patronage of public sculpture was not limited to the ruling classes during the Republic. The tomb of Marcus Vergilius Eurysaces the Baker (c. 50–20 BCE) is one of the largest and best-preserved freedman funerary monuments in Rome. Its sculpted frieze is a classic example of the plebeian style in Roman sculpture.
The deceased built the tomb for himself and perhaps his wife Atistia in the final decades of the Republic. While the tomb’s inscription lacks an L to denote the status of a freedman, the tripartite name of the deceased follows the pattern of names given to and adopted by former slaves.
The tomb, approximately 33 feet tall, commemorates the deceased and his profession. It three main components are a frieze at the top and the cylindrical niches (probably symbolic of a kneading machine or grain measuring vessels) below it.
The surviving text of the inscription translates as “This is the monument of Marcus Vergilius Eurysaces, baker, contractor, public servant.” The frieze represents various stages in the baking of bread in continuous narration.
Although time-worn, the naturalistic depiction of human and animal bodies in a variety of poses is still evident. This record of each stage in a mundane process demonstrates the sense of pride the deceased must have had in his profession. Because the wearing of togas was not conducive to manual labor, the simple clothing on the figures marks them as plebeians, or commoners.
Portraiture
Roman portraiture during the Republic is identified by its considerable realism, known as veristic portraiture. Verism refers to a hyper-realistic portrayal of the subject’s facial characteristics. The style originated from Hellenistic Greece; however, its use in the Roman Republic is due to Roman values , customs, and political life.
As with other forms of Roman art, portraiture borrowed certain details from Greek art but adapted these to their own needs. Veristic images often show their male subjects with receding hairlines, deep winkles, and even with warts. While the faces of the portraits often display incredible detail and likeness, the subjects’ bodies are idealized and do not correspond to the age shown in the face.
The popularity and usefulness of verism appears to derive from the need to have a recognizable image. Veristic portrait busts provided a means of reminding people of distinguished ancestors or of displaying one’s power, wisdom, experience, and authority. Statues were often erected of generals and elected officials in public forums—and a veristic image ensured that a passerby would recognize the person when they actually saw them.
The Late Republic
The use of veristic portraiture began to diminish in the first century BCE. During this time, civil wars threatened the empire, and individual men began to gain more power. The portraits of Pompey the Great and Julius Caesar, two political rivals who were also the most powerful generals in the Republic, began to change the style of the portraits and their use.
The portraits of Pompey are not fully idealized, nor were they created in the same veristic style of Republican senators. Pompey borrowed a specific parting and curl of his hair from Alexander the Great . This similarity served to link Pompey visually with the likeness of Alexander and to remind people that he possessed similar characteristics and qualities.
The portraits of Julius Caesar are more veristic than those of Pompey. Despite staying closer to stylistic convention, Caesar was the first man to mint coins with his own likeness printed on them. In the decades prior to this, it had become increasingly common to place an illustrious ancestor on a coin, but putting a living person—especially oneself—on a coin departed from Roman propriety. By circulating coins issued with his image, Caesar directly showed the people that they were indebted to him for their own prosperity and therefore should support his political pursuits.
Death Masks
The creation and use of death masks demonstrate Romans’ veneration of their ancestors. These masks were created from molds taken of a person at the time of his or her death. Made of wax, bronze , marble, and terra cotta , death masks were kept by families and displayed in the atrium of their homes.
Visitors and clients who entered the home would have been reminded of the family’s ancestry and the honorable qualities of their ancestors. Such displays served to bolster the reputation and credibility of the family.
Death masks were also worn and paraded through the streets during funeral procession. Again, this served not only a memorial for the dead, but also to link the living members of a family to their illustrious ancestors in the eyes of the spectator.
Roman Architecture under the Republic
Roman architecture relies heavily on the use of concrete and the arch to create unique interior spaces and architectural forms.
Explain the importance of both concrete and the arch in Roman architecture.
Key Points
- Roman architecture began as an imitation of the Classical Greek style but eventually grew into its own style with technological advances and modifications on traditional Greek elements. Roman temple design is based on a mixed use of Etruscan and Greek models. They are typically strictly frontal, on a high podium with a flight of stairs, and have a deep, colonnaded portico in front of the cella .
- Concrete is an essential building material in Roman architecture. It is lightweight, strong, and durable and could even be used underwater.
- Most Roman buildings were constructed with concrete and brick and then faced in stucco , expensive stone, or marble.
- The arch is a highly significant architectural shape in Roman architecture, often employed to allow for wider openings in structures. Arches can be used together to create vaults (barrel and groin) and domes , as well as to create unique interior spaces .
Key Terms
- pozzolana : A type of volcanic ash used for mortar or for cement that sets under water.
- psuedoperipteral : A building with free standing columns in the front and engaged columns along the peripheral walls; this style is often used on Roman temples.
- barrel vault : A simple roof having a curved, often semicircular, cross section used to span large distances in railway stations, churches, etc., and usually supported on columns.
- exedra : A semicircular recess, with stone benches, used as a place for discussion.
- cella : The inner chamber of a temple where the cult image or statue is kept.
Roman Architecture
Roman architecture began as an imitation of Classical Greek architecture but eventually evolved into a new style. Unfortunately, almost no early Republican buildings remain intact. The earliest substantial remains date to approximately 100 BCE.
Innovations such as improvements to the round arch and barrel vault , as well as the inventions of concrete and the true hemispherical dome, allowed Roman architecture to become more versatile than its Greek predecessors. While the Romans were reluctant to abandon classical motifs , they modified their temple designs by abandoning pedimental sculptures, altering the traditional Greek peripteral colonnades , and opting for central exterior stairways.
Likewise, although Roman architects did not abandon traditional column orders, they did modify them with the Tuscan, Roman Ionic, and Composite orders. This diagram shows the Greek orders on the left and their Roman modifications on the right.
Roman Temples
Most Roman temples derived from Etruscan prototypes. Like Etruscan temples, Roman temples are frontal with stairs that lead up to a podium, and a deep portico filled with columns. They are also usually rectilinear , and the interiors consist of at least one cella that contained a cult statue.
If multiple gods were worshiped in one temple, each god would have its own cella and cult image. For example, Capitolia—the temples dedicated to the Capitoline Triad—would always be built with three cellae, one for each god of the triad: Jupiter, Juno, and Minerva.
Roman temples were typically made of brick and concrete and then faced in either marble or stucco. Engaged columns (columns that protrude from walls like reliefs) adorn the exteriors of the temples. This creates an effect of columns completely surrounding a cella, an effect known as psuedoperipteral . The altar, used for sacrifices and offerings , always stood outside in front of the temple.
While most Roman temples followed this typical plan, some were dramatically different. At times, the Romans erected round temples that imitated the Greek tholos . Examples can be found in the Temple of Hercules Victor (late second century BCE), in the Forum Boarium in Rome . The temple consists a circular cella within a concentric ring of 20 Corinthian columns. Like its Etruscan predecessors, the temple rests on a tufa foundation. Its original roof and architrave are now lost.
Concrete
The Romans perfected the recipe for concrete during the third century BCE by mixing together water, lime, and pozzolana , volcanic ash mined from the countryside surrounding Mt. Vesuvius. Concrete became the primary building material for the Romans, and it is largely the reason that they were such successful builders.
Most Roman buildings were built with concrete and brick that was then covered in façade of stucco, expensive stone, or marble. Concrete was a cheaper and lighter material than most other stones used for construction. This helped the Romans build structures that were taller, more complicated, and quicker to build than any previous ones.
Once dried, concrete was also extremely strong, yet flexible enough to remain standing during moderate seismic activity. The Romans were even able to use concrete underwater, allowing them build harbors and breakers for their ports. The ruins of a tomb on the Via Appia (the most famous thoroughfare through ancient Rome) expose the stones and aggregate that the Romans used to mix concrete.
Arches, Vaults, and Domes
The Romans effectively combined concrete and the structural shape of the arch. These two elements became the foundations for most Roman structures. Arches can bear immense weight, as they are designed to redistribute weight from the top, to its sides, and down into the ground . While the Romans did not invent the arch, they were the first culture to manipulate it and rely on its shape.
An arch is a pure compression form . It can span a large area by resolving forces into compressive stresses (pushing downward) that, in turn, eliminate tensile stresses (pushing outward). As the forces in the arch are carried to the ground, the arch will push outward at the base (called thrust). As the height of the arch decreases, the outward thrust increases. In order to maintain arch action and prevent the arch from collapsing, the thrust needs to be restrained, either with internal ties or external bracing, such as abutments (labeled 8 on the diagram below).
The arch is a shape that can be manipulated into a variety of forms that create unique architectural spaces. Multiple arches can be used together to create a vault. The simplest type is known as a barrel vault.
Barrel vaults consist of a line of arches in a row that create the shape of a tunnel. When two barrel vaults intersect at right angles, they create a groin vault . These are easily identified by the x-shape they create in the ceiling of the vault. Furthermore, because of the direction, the thrust is concentrated along this x-shape, so only the corners of a groin vault need to be grounded. This allows an architect or engineer to manipulate the space below the groin vault in a variety of ways.
Arches and vaults can be stacked and intersected with each other in a multitude of ways. One of the most important forms that they can create is the dome. This is essentially an arch that is rotated around a single point to create a large hemispherical vault. The largest dome constructed during the Republic was on the Temple of Echo at Baiae, named for its remarkable acoustic properties.
Arches and concrete are found in many iconic Roman structures. The Sanctuary of Fortuna Primigenia (c. 120 BCE) at Palestrina, Italy is a massive temple structure built into the hillside in a series of terraces, exedras , and porticoes. Concrete was used as the primary building material and barrel vaults provide structural support, both as a terracing method for the hill and in creating interesting architectural spaces for the sanctuary.
Roman aqueducts are another iconic use of the arch. The arches that make up an aqueduct provided support without requiring the amount of building material necessary for arches supported by solid walls. The Aqua Marcia (144–140 BCE) was the longest of the eleven aqueducts that served the city of Rome during the Republic. It supplied water to the Viminal Hill in the north of Rome, and from there to the Caelian, Aventine, Palatine, and Capitoline Hills. Where the Aqua Marcia had contact with water, it was coated with a waterproof mortar.
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3.1: Complex Numbers
Learning Objectives
- Express square roots of negative numbers as multiples of \(i\).
- Plot complex numbers on the complex plane.
- Add and subtract complex numbers.
- Multiply and divide complex numbers.
The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as
\[x^2+4=0\]
Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
Expressing Square Roots of Negative Numbers as Multiples of i
We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number i is defined as the square root of negative 1.
\[\sqrt{-1}=i\]
So, using properties of radicals,
\[i^2=(\sqrt{-1})^2=−1\]
We can write the square root of any negative number as a multiple of i. Consider the square root of –25.
\[\begin{align} \sqrt{-25}&=\sqrt{25 {\cdot} (-1)}\\ &=\sqrt{25}\sqrt{-1} \\ &= 5i \end{align}\]
We use 5 i and not −5 i because the principal root of 25 is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written \(a+bi\) where \(a\) is the real part and \(bi\) is the imaginary part. For example, \(5+2i\) is a complex number. So, too, is \(3+4\sqrt{3}i\).
Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.
Imaginary and Complex Numbers
A complex number is a number of the form \(a+bi\) where
- \(a\) is the real part of the complex number.
- \(bi\) is the imaginary part of the complex number.
If \(b=0\), then \(a+bi\) is a real number. If \(a=0\) and \(b\) is not equal to 0, the complex number is called an imaginary number . An imaginary number is an even root of a negative number.
Standard Form
Given an imaginary number, express it in standard form.
- Write \(\sqrt{−a}\) as \(\sqrt{a}\sqrt{−1}\).
- Express \(\sqrt{−1}\) as \(i\) .
- Write \(\sqrt{a}{\cdot}i\) in simplest form.
Example \(\PageIndex{1}\): Expressing an Imaginary Number in Standard Form
Express \(\sqrt{−9}\) in standard form.
Solution
\[\sqrt{−9}=\sqrt{9}\sqrt{−1}=3i \nonumber\]
In standard form, this is \(0+3i\).
Exercise \(\PageIndex{1}\)
Express \(\sqrt{−24}\) in standard form.
- Answer
-
\(\sqrt{−24}=0+2i\sqrt{6}\)
Plotting a Complex Number on the Complex Plane
We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs \((a,b)\), where \(a\) represents the coordinate for the horizontal axis and \(b\) represents the coordinate for the vertical axis.
Let’s consider the number \(−2+3i\). The real part of the complex number is−2 and the imaginary part is \(3i\). We plot the ordered pair \((−2,3)\) to represent the complex number \(−2+3i\) as shown in Figure \(\PageIndex{2}\)
Complex Plane
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in Figure \(\PageIndex{3}\).
How To ...
Given a complex number, represent its components on the complex plane.
- Determine the real part and the imaginary part of the complex number.
- Move along the horizontal axis to show the real part of the number.
- Move parallel to the vertical axis to show the imaginary part of the number.
- Plot the point.
Example \(\PageIndex{2}\): Plotting a Complex Number on the Complex Plane
Plot the complex number \(3−4i\) on the complex plane.
Solution
The real part of the complex number is 3, and the imaginary part is \(−4i\). We plot the ordered pair \((3,−4)\) as shown in Figure \(\PageIndex{4}\).
Exercise \(\PageIndex{1}\)
Plot the complex number \(−4−i\) on the complex plane.
- Answer
Adding and Subtracting Complex Numbers
Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.
Complex Numbers: Addition and Subtraction
Adding complex numbers:
\[(a+bi)+(c+di)=(a+c)+(b+d)i\]
Subtracting complex numbers:
\[(a+bi)−(c+di)=(a−c)+(b−d)i\]
How To...
Given two complex numbers, find the sum or difference.
- Identify the real and imaginary parts of each number.
- Add or subtract the real parts.
- Add or subtract the imaginary parts.
Example \(\PageIndex{3}\): Adding Complex Numbers
Add \(3−4i\) and \(2+5i\).
Solution
We add the real parts and add the imaginary parts.
\[\begin{align*} (a+bi)+(c+di)&=(a+c)+(b+d)i \\ (3−4i)+(2+5i)&=(3+2)+(−4+5)i \\ &=5+i \end{align*}\]
Exercise \(\PageIndex{3}\)
Subtract \(2+5i\) from \(3–4i\).
- Answer
-
\((3−4i)−(2+5i)=1−9i\)
Multiplying Complex Numbers
Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.
Multiplying a Complex Numbers by a Real Number
Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,
How To...
Given a complex number and a real number, multiply to find the product.
- Use the distributive property.
- Simplify.
Example \(\PageIndex{4}\): Multiplying a Complex Number by a Real Number
Find the product \(4(2+5i).\)
Solution
Distribute the 4.
\[\begin{align*} 4(2+5i)&=(4⋅2)+(4⋅5i) \\ &=8+20i \end{align*}\]
Exercise \(\PageIndex{4}\)
Find the product \(−4(2+6i)\).
- Answer
-
\(−8−24i\)
Multiplying Complex Numbers Together
Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get
\[(a+bi)(c+di)=ac+adi+bci+bdi^2 \nonumber\]
Because \(i^2=−1\), we have
\[(a+bi)(c+di)=ac+adi+bci−bd \nonumber\]
To simplify, we combine the real parts, and we combine the imaginary parts.
\[(a+bi)(c+di)=(ac−bd)+(ad+bc)i \nonumber\]
How To...
Given two complex numbers, multiply to find the product.
- Use the distributive property or the FOIL method.
- Simplify.
Example \(\PageIndex{5}\): Multiplying a Complex Number by a Complex Number
Multiply \((4+3i)(2−5i)\).
Solution
Use \((a+bi)(c+di)=(ac−bd)+(ad+bc)i\)
\[\begin{align*} (4+3i)(2−5i)&=(4⋅2−3⋅(−5))+(4⋅(−5)+3⋅2)i \\ &=(8+15)+(−20+6)i \\ &=23−14i \end{align*}\]
Exercise \(\PageIndex{5}\)
Multiply \((3−4i)(2+3i)\).
- Answer
-
\(18+i\)
Dividing Complex Numbers
Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of \(a+bi\) is \(a−bi\).
Note that complex conjugates have a reciprocal relationship: The complex conjugate of \(a+bi\) is \(a−bi\), and the complex conjugate of \(a−bi\) is \(a+bi\). Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide \(c+di\) by \(a+bi\), where neither a nor \(b\) equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
\[\dfrac{c+di}{a+bi} \, \text{ where $a{\neq}0$ and $b{\neq}0$} \nonumber\]
Multiply the numerator and denominator by the complex conjugate of the denominator.
\[\dfrac{(c+di)}{(a+bi)}{\cdot}\dfrac{(a−bi)}{(a−bi)}=\dfrac{(c+di)(a−bi)}{(a+bi)(a−bi)} \nonumber\]
Apply the distributive property.
\[=\dfrac{ca−cbi+adi−bdi^2}{a^2−abi+abi−b^2i^2} \nonumber\]
Simplify, remembering that \(i^2=−1\).
\[=\dfrac{ca−cbi+adi−bd(−1)}{a^2−abi+abi−b^2(−1)} \\ =\dfrac{(ca+bd)+(ad−cb)i}{a^2+b^2} \nonumber\]
Definition: The Complex Conjugate
The complex conjugate of a complex number \(a+bi\) is \(a−bi\). It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.
- When a complex number is multiplied by its complex conjugate, the result is a real number.
- When a complex number is added to its complex conjugate, the result is a real number.
Example \(\PageIndex{6}\): Finding Complex Conjugates
Find the complex conjugate of each number.
- \(2+i\sqrt{5}\)
- \(−\frac{1}{2}i\)
Solution
a. The number is already in the form \(a+bi\). The complex conjugate is \(a−bi\), or \(2−i\sqrt{5}\).
b. We can rewrite this number in the form \(a+bi\) as \(0−\frac{1}{2}i\). The complex conjugate is \(a−bi\), or \(0+\frac{1}{2}i\). This can be written simply as \(\frac{1}{2}i\).
Analysis
Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by \(i\).
How To...
Given two complex numbers, divide one by the other.
- Write the division problem as a fraction.
- Determine the complex conjugate of the denominator.
- Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
- Simplify.
Example \(\PageIndex{7}\): Dividing Complex Numbers
Divide \((2+5i)\) by \((4−i)\).
Solution
We begin by writing the problem as a fraction.
\[\dfrac{(2+5i)}{(4−i)} \nonumber\]
Then we multiply the numerator and denominator by the complex conjugate of the denominator.
\[\dfrac{(2+5i)}{(4−i)}{\cdot}\dfrac{(4+i)}{(4+i)} \nonumber\]
To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).
\[\begin{align} \dfrac{(2+5i)}{(4−i)}{\cdot}\dfrac{(4+i)}{(4+i)}&=\dfrac{8+2i+20i+5i^2}{16+4i−4i−i^2} \\ &=\dfrac{8+2i+20i+5(−1)}{16+4i−4i−(−1)} &\text{ because $i^2=-1$} \\ &=\dfrac{3+22i}{17} \\ &=\dfrac{3}{17}+\dfrac{22}{17}i & \text{Separate real and imaginary parts.} \end{align}\]
Note that this expresses the quotient in standard form.
Example \(\PageIndex{8}\): Substituting a Complex Number into a Polynomial Function
Let \(f(x)=x^2−5x+2\). Evaluate \(f(3+i)\).
Solution
Substitute \(x=3+i\) into the function \(f(x)=x^2−5x+2\) and simplify.
\[\begin{align*} f(3+i) &= (3+i)^2 - 5(3+i) + 2\; \qquad \text{Substitute 3+i for x}\\ &= (3+6i+i^2) - (15+5i) + 2\; \qquad \text{Multiply}\\ &= 9+6i+(-1)-15-5i+2\; \qquad \text{Subsitute -1 for }i^2\\ &= -5+i\; \qquad \text{Combine like terms} \end{align*}\]
Analysis
We write \(f(3+i)=−5+i\). Notice that the input is \(3+i\) and the output is \(−5+i\).
Exercise \(\PageIndex{8}\)
Let \(f(x)=2x^2−3x\). Evaluate \(f(8−i)\).
- Answer
-
\(102−29i\)
Example \(\PageIndex{9}\): Substituting an Imaginary Number in a Rational Function
Let \(f(x)=\frac{2+x}{x+3}\). Evaluate \(f(10i)\).
Solution
Substitute \(x=10i\) and simplify.
\[\begin{align*} &\dfrac{2+10i}{10i+3} &\text{Substitute $10i$ for $x$.} \\ &\dfrac{2+10i}{3+10i} &\text{Rewrite the denominator in standard form.} \\ &\dfrac{2+10i}{3+10i}{\cdot}\dfrac{3–10i}{3–10i} &\text{Multiply the numerator and denominator by the complex conjugate of the denominator.}\\ &\dfrac{6–20i+30i–100i^2}{9–30i+30i–100i^2} &\text{Multiply using the distributive property or the FOIL method.} \\ &\dfrac{6–20i+30i–100(–1)}{9–30i+30i–100(–1)} &\text{Substitute –1 for $i^2$.}\\ &\dfrac{106+10i}{109} &\text{Simplify.}\\ &\dfrac{106}{109}+\dfrac{10}{109} &\text{Separate the real and imaginary parts.} \end{align*}\]
Exercise \(\PageIndex{9}\)
Let \(f(x)=\frac{x+1}{x−4}\). Evaluate \(f(−i)\).
- Answer
-
\(−\frac{3}{17}+\frac{5i}{17}\)
Simplifying Powers of \(i\)
The powers of \(i\) are cyclic. Let’s look at what happens when we raise i to increasing powers.
\[\begin{align*} i^1&=i \\[4pt] i^2&=−1 \\[4pt] i^3&=i^2⋅i=−1⋅i=−i \\[4pt] i^4&=i^3⋅i=−i⋅i=−i^2=−(−1)=1 \\[4pt] i^5&=i^4⋅i=1⋅i=i \end{align*}\]
We can see that when we get to the fifth power of \(i\), it is equal to the first power. As we continue to multiply \(i\) by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of \(i\).
\[\begin{align*} i^6&=i^5⋅i=i⋅i=i^2=−1 \\[4pt] i^7&=i^6⋅i=i^2⋅i=i^3=−i \\[4pt] i^8&=i^7⋅i=i^3⋅i=i^4=1 \\[4pt] i^9&=i^8⋅i=i^4⋅i=i^5=i \end{align*}\]
Example \(\PageIndex{10}\): Simplifying Powers of \(i\)
Evaluate \(i^{35}\).
Solution
Since \(i^4=1\), we can simplify the problem by factoring out as many factors of i^4 as possible. To do so, first determine how many times 4 goes into 35: \(35=4⋅8+3\).
\[i^{35}=i^{4⋅8+3}=i^{4⋅8}⋅i^{3}=(i^4)^8⋅i^3=1^8⋅i^3=i^3=−i \nonumber \]
Q & A
Can we write \(i^35\) in other helpful ways?
As we saw in Example \(\PageIndex{10}\), we reduced \(i^{35}\) to \(i^3\) by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of \(i^{35}\) may be more useful. Table \(\PageIndex{1}\) shows some other possible factorizations.
| Factorization of \(i^{35}\) | \(i^{34}{\cdot}i\) | \(i^{33}{\cdot}i^2\) | \(i^{31}{\cdot}i^4\) | \(i^{19}{\cdot}i^{16}\) |
|---|---|---|---|---|
| Reduced form | \(\big(i^2\big)^{17}{\cdot}i\) | \(i^{33}{\cdot}(-1)\) | \(i^{31}{\cdot}1\) | \(i^{19}{\cdot}\big(i^4\big)^4\) |
| Simplified form | \((-1)^{17}{\cdot}i\) | \(-i^{33}\) | \(i^{31}\) | \(i^{19}\) |
Key Concepts
- The square root of any negative number can be written as a multiple of \(i\).
- To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.
- Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.
- Complex numbers can be multiplied and divided.
- To multiply complex numbers, distribute just as with polynomials.
- To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.
- The powers of \(i\) are cyclic, repeating every fourth one.
Glossary
complex conjugate
the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number
complex number
the sum of a real number and an imaginary number, written in the standard form \(a+bi\), where \(a\) is the real part, and \(bi\) is the imaginary part
complex plane
a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number
imaginary number
a number in the form bi where \(i=\sqrt{−1}\)


8: Microsoft Word
Learning Objectives
If you do well in this unit, you should be able to:
- Indicate the names and functions of the Word interface components.
- Create, edit, save, and print documents to include documents with lists and tables.
- Format text and to use styles.
- Add a header and footer to a document.
- Add a footnote to a document.
- Add a graphic to a document.
- Use the Spelling and Grammar Checker as well as Microsoft Help.
- Manipulate documents using functions such as find and replace; cut, copy, replace.
CC licensed content, Original
- Introduction to Computer Applications and Concepts Word Learning Objectives. Authored by : Jim Shannon. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : http://www.nvcc.edu/eli/index.html . License : CC BY: Attribution


13.14: Assignment- Social Diversity in the Workplace
You are one of four co-founders of a new start-up company producing a new line of meal-replacement drinks. You and your co-founders have had two promising years of producing and selling your product, and now you’re looking to expand the team and grow the business.
One of the things you’re most looking forward to with expanding the company is increasing the diversity of the team. You began the company with some friends who had similar interests who have hired like-minded friends, making the company feel very homogeneous. As you’ve grown as a company, you’re started to worry about the product becoming too niche without the input of other points of view. In casual conversation, another co-founder brought up the idea that the like-mindedness of the team is essential for the company’s continued success. You refuted the idea and mentioned the importance of having diverse opinions, but you didn’t feel like you had enough information or data to make your point credible at the time.
Your task as a leader is to effectively communicate the importance of social diversity in the workplace to the whole company. You may choose one of the following methods to inform the team about the planned company expansion by emphasizing the importance of hiring diverse employees. For this assignment, you can choose one of the three methods:
- Write an email to be sent to the full company.
- Create a presentation to show first to your co-founders, then to the rest of the company.
- Create an informal report showing the positive impact of social diversity in other companies.
In your short rationale (no more than 150 words) explain why you chose the method you did.
Please note: There is no “correct” way to present this message; it will depend on which method you think would be best for your colleagues.
Grading Rubric
| Criteria | Not Evident | Developing | Exemplary | Points |
|---|---|---|---|---|
| Discuss the plan to expand the team |
0 pts
No presentation completed |
5 pts
Expansion is not discussed |
10 pts
Expansion is discussed |
10 pts |
| Discuss the importance of social diversity |
0 pts
No presentation completed |
10 pts
Discussion shows a lack of understanding of diversity |
20 pts
Discussion shows an understanding of diversity and respect of individuals with different qualities and abilities |
20 pts |
| Provide support for the benefits of social diversity |
0 pts
No slide deck submitted |
5 pts
No support is provided |
10 pts
Support is provided |
10 pts |
| Provide a short rationale |
0 pts
No response to peers |
5
pts
Rationale does not meet required words criteria |
10 pts
Rationale is under 150 words |
10 pts |
| Total: | 50 pts |
Contributors and Attributions
- Assignment: Social Diversity in the Workplace. Provided by : Lumen Learning. License : CC BY: Attribution


3: Unit III- Wiring Connections
Learning Objectives
When you have completed the Learning Tasks in this Competency, you will be able to:
- define the terms used and explain the principles of soldering
- describe the methods for making properly soldered connections
- maintain soldering equipment
- use wireless connectors
It is important for you to be familiar with techniques for soldering electrical connections and how to use wireless connectors. For example, the ends of the finely stranded wires used for power supply cords on most portable power tools are soldered to permit a long-lasting, troublefree connection. Solder also produces secure, durable electrical connections for switches, plugs, and tools. Wireless connectors are commonly used in many electrical applications because they are quick and easy to use.


2.9: Formatting Business Writing
- Discuss common guidelines to create a professionally formatted document
In order to properly format a document, we must first understand the primary components of that document. These skills—including spacing, emphasis, and more—are the same skills used in emails, memos, business reports, and employability documents.
Two of the most widely used examples of word processing software are Microsoft Word and Google Docs. Both Word and Google Docs provide the business writer with the formatting tools needed to create professional documents.
Specifications of the basic letter, memo, or email format:
- Paragraphs are not indented
- Line spacing is 1.0 (or 1.15) in the paragraph
- Line spacing is 2 between paragraphs
- Left margin is justified/aligned, which means it looks like a straight line
- Right margin is ragged (left aligned paragraph) or straight line (block paragraph)
- Font size is 10 or 12 depending on font type (Arial and Times New Roman are common fonts because they are universal across PC and Mac platforms)
Business Letter Format
There are numerous online sources describing the format of a business letter. Many businesses have specific formats for communications coming from their organizations, and of course, those formats take precedence. There is likely pre-printed letterhead with the company logo and address available. Today, it is commonplace to include company email and website addresses in the address block or heading.
As you begin to use both Word and Google Docs, you will become familiar with templates that help with many of the spacing issues.This section comes from a widely recognized source, Purdue OWL (Online Writing Lab) . There are several common styles: Block, Modified Block, and Semi-Block. These instructions address the most common parts of a business letter using block format.
- Sender’s Address
- Date
- Inside Address
- Salutation
- Body
- Closing
- Enclosures or attachments
Business Memo and Email Format
With the increasing use of email, memos are less frequently found in the workplace. For business writers, to think of memo and email formats as the same is not too far a reach. Pay careful attention to the subject line in both emails and memos. As discussed in the previous sections, the subject line plays a critical role in providing guidance for the reader regarding urgency, topic, and actions to take. In an email or memo, use bold paragraph headings for scanability. Doing this helps you stay organized, and readers love it because it helps them pick out the most important information easily.
For more information on how to format a memo take a look at this document on standard memo formatting or Module 3: Written Communication’s page on memos.
Style
When formatting any written communication, it’s important to end up with a document that’s pleasing to the eye. Here are a few quick tips on making your document look good. For more tips, take a look at Format: Make Your Message Inviting.
White Space
White space between words, white space between paragraphs. White space between sentences.
When you see a block of text, break it apart into shorter paragraphs, sentences, words or bullet points.
Sculpting as you go.
You dear writer, shape the meaning of your content with the words, but shape the presentation by formatting.
By the white space.
—Demian Farnworth [1]
For business writers, the choice of white space is not as free-form as sculpture, yet there are standard conventions to apply and business writing choices that aid the reader. Standard conventions appear in the final section of this module.
Notice how the quote from Farnworth has short lines and uses the blanks space showing through to break up the text. The rhythm of the words is represented visually. In business writing, paragraphs, just like sentences, are kept relatively concise. There is no prescribed length, but writers tend to avoid one sentence paragraphs. Writers also tend to avoid paragraphs that result in long, uninterrupted blocks of text. Regardless of length, each paragraph will have have a topic sentence with just as much support as it needs—no more, no less.
Lists
Lists are excellent tools for two reasons: They create white space and they create a pattern that is easy for a reader to understand and recall. There are two types of lists: numbered (enumerated) and bulleted.
Use numbered lists when order or importance matters:
| Avoid | Try |
|---|---|
| To make toast, first plug in the toaster. Second, slide the bread in to the slot. Third, push down on the toasting lever. |
Steps to make toast:
|
In this example, the order of steps to make toast matters. Pushing down on the toasting lever provides no value without the bread loaded into the slot. Also notice how much easier the “Try” column was to understand and retain.
| Avoid | Try |
|---|---|
| The reasons the store is staying open later are that first, s tudies show per store profits increase by 3%. Second, c ustomers like a pharmacy open in the evening for emergencies. Third, t he Associate Manager lives close by, so a bit of a commute is not a worry. |
The reasons the store is staying open later:
|
Notice that the Associate Manager’s preference likely was not as important to the additional store hours as sales and customer preference were.
Use bulleted lists when the items are of equal value.
| Vegetables that may be mixed in store-made salads are l ettuce, t omatoes, c arrots, or s pinach |
Vegetables that may be mixed in store-made salads:
|
In this example, notice that any of the vegetables may be used. One item is not more important than the other. It does not matter the order in which the items are selected.
List items may be one word long, a phrase long, or a sentence long. They each must be constructed with parallel organization (see previous section on Parallel Construction).
Lists items may end with punctuation or without as long as they are all the same. Notice the consistency in the lists above.
Headings
Headings provide significant assistance in longer messages or reports as a way of guiding the reader and providing white space to separate ideas and messages. Most software programs have heading levels built in to a style that the writer may select. Using this pre-formatted style ensures consistency throughout a message.
Serif vs. Sans Serif Font
When choosing a font type, the first thing to decide is if you need a serif or sans serif font. What is a serif?
While there is some debate on the topic, the general recommendations are as follows:
- When writing for print media, use a serif font for your body text and a sans serif font for headings.
- When writing for digital media, use a sans serif font for your body text, and you can use a serif or sans serif font for headings.
Font type impacts the selection of font size. For most business writing, the optimal fonts size is 10 or 12.
- Farnworth, Demian. "How To Dramatically Improve Your Writing With White Space." nd. Web. 13 June 2018.
Contributors and Attributions
- Formatting Business Writing. Authored by : Susan Kendall. Provided by : Lumen Learning. License : CC BY: Attribution


8.6: Rise of Islam
Legend traces the Arabs back to Isma‘il, the son of Abraham and his Egyptian maid, Haga, a link that would later help to legitimize Islam by connecting it to the Hebrew tradition. In reality, Arabs inhabited the pre-Islamic Arabian Peninsula and shared socio-linguistic commonalities with such other Semitic-speaking peoples in the area as the Hebrews, Assyrians, Arameans, and even the Amhara of Ethiopia. Most of the population of Arabia prior to the rise of Islam resided in the south of the peninsula, in modern day Yemen, where they practiced terraced agriculture and herded ruminants in a relatively small area.
Farther to the north of Yemen, along the highland spine of western Arabia and up against the littoral of the Red Sea, was the Hijaz, a prominent cultural and economic region. Situated in this remote fastness was the dusty city of Mecca, the holiest place in the peninsula and the location of the Ka‘ba , or “cube,” which contained many of the traditional Arabian religious images, including many Christian icons. So important was the Ka‘ba to the religion of polytheistic tribes of Arabia that they negotiated a truce lasting one month every year that allowed for safe pilgrimage to the shrine.
The Hijaz was the most arable part of the Arabian Peninsula north of Yemen and distinguished by irrigated agriculture that supported fruit trees and essential grains. Local traders exported a range of Hijazi agricultural products to Syria in the north in return for imperative imports like textiles and olive oil so the region benefited from robust trade. Regional commerce depended on the security of trade, and piracy on the Red Sea threatened to disrupt business. Under these conditions, merchants diverted their trade overland. Many goods journeyed up the Red Sea Rift from Yemen on their way to the eastern Mediterranean. Caravans of camels carried these goods, as well as Hijazi exports, to the Levant. Most of the caravans stopped in Mecca, the halfway point up the spine of the peninsula, thus their commerce brought much needed wealth and tax revenue to the city.
The Arabs first domesticated the camel, probably sometime between 3000 and 1000 BCE. Caravan operators eventually availed themselves of these useful dromedaries because they were so adept at crossing the region’s massive deserts. Capable of drinking 100 liters of water in mere minutes, they could endure days of travel without needing to replenish themselves again. Moreover, camels instinctively remembered the locations of important, life-sustaining oases. So important were these beasts of burden that the tribes that controlled the camels controlled the trade. And the Quraysh Tribe of Mecca commanded many of the camels in the Hijaz region; therefore, they commanded much of the trade.
Life in the Arabian Peninsula centered around the tribe, which usually consisted of a group of relatives who claimed a shared ancestry. Tribal traditions found meaning in the poetic concept of muruwah , which represented the notion of the ideal tribal man. This uniquely Arabian brand of chivalry focused on bravery, patience, persistence in revenge, generosity, hospitality, and protection of the poor and weak. In the absence of formal government, tribes offered physical security to its individual members. Tribes mitigated violence and theft through the shared understanding that retribution for such acts would follow swiftly. Tribes also organized to compete over increasingly scarce resources, as they had a responsibility to provide for the economic needs of their individual members. Nevertheless, tribal traditions had been breaking down prior to the rise of Islam; no longer were the dominant members of society adhering to the principles set forth in muruwah .
Into this evolving cultural milieu Muhammad (c.570 – 632) was born in the city of Mecca. Muhammad’s father, ‘Abdallah, was a member of the Hashimite Clan, a less prosperous branch of the Quraysh Tribe. ‘Abdullah died just prior to his son’s birth, and Muhammad’s mother passed away when he was just six years old. Orphaned at such a young age, his tribe intervened to ensure Muhammad’s survival. His uncle, Abu Thalib, the leader of the Hashimite Clan and an important member of the Quraysh Tribe, eventually took custody of the young boy. These early privations influenced Muhammad’s later desire to take care of those who could not care for themselves.
In his youth, Muhammad found employment in the regional caravan trade as a dependable herder and driver of camels. During this period, he cultivated a reputation of an empathetic and honest man, one who earned the respect of many Meccans. His upright character soon attracted the attention of a wealthy merchant known as Khadija who hired Muhammad to manage her caravans. Once Muhammad proved his reliability, Khadija, who was fifteen years older than Muhammad, proposed to him, and they married. This marriage afforded Muhammad a financial security that allowed him to begin meditating on religion in the abstract.
Muhammad had been concerned about the direction society had recently been taking and that the concepts defined by muruwah were no longer being upheld. He believed that some of the most influential members of society, namely the merchant elite of the Quraysh Tribe, were no longer respecting their traditional responsibilities to the weaker members of society because of their own greed. He thought that the People of the Book , specifically, Christians and Jews, might have a better answer for the ills afflicting Meccan society. Muhammad had contact with the Christians and Jews of the peninsula and even traveled to Christian Syria while working in the caravan trade. In this context, the Angel Gabriel appeared to Muhammad at a cave nearby to Mecca in 610, during the holy month of Ramadan. The Angel Gabriel instructed him to “recite,” and then he spoke the divine word of God. His revelations became the Quran .
At first, Muhammad displayed the very human reactions of fear and distrust to the apparition of the Angel Gabriel. He also expressed embarrassment because he did not want to be associated with the pagan diviners of the region. Fortunately, his wife Khadija had a cousin who was a hanif , someone who was neither a Christian nor a Jew, but who believed in a vague concept of a monotheistic god. Her cousin trusted the veracity of Muhammad’s revelations. So with trepidation, Muhammad eventually accepted his role as God’s vehicle. His wife became the first convert to Islam, with Abu Thalib’s son ‘Ali converting soon afterwards.
The Religion of Islam
As a religion of the Abrahamic faith, Islam holds much in common with Judaism and Christianity. Islam grew out of the Judeo-Christian tradition, a link which helped to legitimize the new religion. In fact, Muslims believe in the same God, or Allah in Arabic, as the Jewish and Christian God. Although Muslims trust that the People of the Book had received the word of God, they believe that it had become distorted over time, so God sent the Angel Gabriel to deliver His word to Muhammad, the Seal of the Prophets, or Khatam an-Nabiyyin, for Muslims believe that he represented God’s final word to man. Muhammad never claimed to be founding a new religion, rather he served as the last in a long line of God’s messengers, beginning with the Hebrew prophets, and including Jesus. His revelations, therefore, represent the pure, unadulterated version of God’s message. The Prophet’s followers memorized the revelations and ultimately recorded them in a book called the Quran. Together with the Quran, the Hadith , traditions of Muhammad used to illustrate a concept, and the Sunna , the teachings of the Prophet not found in the Quran, helped guide and inform Muslims on proper behavior. And with that knowledge came great responsibility, as God held His people to a high standard of behavior, based on their obedience, or submission to His will. In fact, the word Islam means submission in Arabic, and a Muslim is one who submits (to God).
Derived from a Hadith, the Five Pillars of Islam are essential, obligatory actions that serve as the foundation of the faith. The first pillar, known as the witness, or shahada , is a profession of faith, in which believers declare that “There is no God but God, and Muhammad is His messenger.” Prayer, also called salat , is the second pillar of Islam. Islam expects faithful Muslims to pray five times a day, kneeling towards Mecca, at dawn, noon, afternoon, sunset, and evening. One should perform ritual ablutions prior to their prayers in order to approach God as being symbolically clean and pure. The third pillar is almsgiving, or zakat in Arabic. Islam requires Muslims to contribute a proportion of their wealth to the upkeep of the Islamic community. This proportion, or tithe, accorded with the size of one’s wealth; therefore, the rich should expect to contribute more than the poor. Fasting, or sawm , is the fourth pillar of Islam and takes place during Ramadan , the ninth month of the Islamic lunar calendar. For the duration of Ramadan, believers consume neither food nor drink from dawn to dusk. This practice is meant to remind them of what it is like to be poor and go hungry. The fifth and final pillar of Islam is pilgrimage, or hajj . Islam expects all able-bodied Muslims to make a journey to Mecca at least once in their lifetime. All five pillars combine to unite the Islamic community.


24.1: Introduction
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Learning Objectives
Teaching effectiveness describes instructors’ ability to affect student success. It is usually defined according to several factors, such as how well instructors organize courses, how well they know the course material, how clearly they communicate with students, how frequently they provide timely feedback, and other criteria. In the classroom, effectiveness sometimes depends on the instructor’s enthusiasm or disposition. During fully online and blended learning courses, students often need more structure and support to succeed because their course activities usually require them to take greater responsibility for their own learning success. Therefore, many of the criteria take on even more importance when evaluating online teaching effectiveness.
Online teaching is often held to higher standards than classroom teaching, and sometimes these standards have nothing to do with the teacher’s ability. For example, a technological breakdown can have a negative impact on students’ evaluation of an instructor’s work, though the instructor is rarely responsible for the technical failure.
To succeed, you should find some allies to help. If you are new to online teaching and learning, let your students know. They will usually give you a lot of leeway. Some of the students may offer to help you set up or facilitate technology-based activities or at least respond positively to your requests for technological help. Overall, you will find it well worth the effort to evaluate and improve your online teaching effectiveness.
There are many ways to evaluate teaching effectiveness in either the physical or virtual environments. Getting pointers and advice before the term begins can save you from making revisions later. Formative feedback, collected during an ongoing course, improves that specific course. Summative feedback, collected after a course ends, improves the next iterations. Feedback that applies to the instructor’s process can also improve other courses.
In this chapter, I discuss seeking feedback from peer instructors, distance education practitioners, students, and even yourself. If you teach fully online, you should be able to convert any face-to-face feedback strategy to the online environment, even if I do not specifically point out how to do it.