1.6: Chapter Review
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SECTION 1.6 PROBLEM SET: CHAPTER REVIEW
1) Find an equation of the x-axis.
2) Find the slope of the line whose equation is 2x + 3y = 6.
3) Find the slope of the line whose equation is y = - 3x + 5.
4) Find both the x and y intercepts of the line 3x - 2y = 12.
5) Find an equation of the line whose slope is 3 and y-intercept 5.
6) Find an equation of the line whose x-intercept is 2 and y-intercept 3.
7) Find an equation of the line that has slope 3 and passes through the point (2, 15).
8) Find an equation of the line that has slope -3/2 and passes through the point (4, 3).
9) Find an equation of the line that passes through the points (0, 32) and (100, 212).
10) Find an equation of the line that passes through the point (2, 5) and is parallel to the line y = 3x + 4.
11) Find the point of intersection of the lines 2x - 3y = 9 and 3x + 4y = 5.
12) Is the point (3, - 2) on the line 5x - 2y = 11?
13) Find two points on the line given by the parametric equations, x = 2 + 3t, y = 1 - 2t.
14) Find two points on the line 2x - 6 = 0.
15) Graph the line 2x - 3y + 6 = 0.
16) Graph the line y = - 2x + 3.
17) A female college student who is 60 inches tall weighs 100 pounds. Another female student who is 66 inches tall weighs 124 pounds. Assume the relationship between the female students' weights and heights is linear. Find an equation for weight as a function of height. Use this relationship to predict the weight of a female student who is 70 inches tall.
18) In deep-sea diving, the pressure exerted by water plays a great role in designing underwater equipment. If at a depth of 10 feet there is a pressure of 21 lb/in 2 , and at a depth of 50 ft there is a pressure of 75 lb/in 2 , write a linear equation giving a relationship between depth and pressure. Use this relationship to predict pressure at a depth of 100 ft.
19) The variable cost to manufacture an item is $30 per item; the fixed costs are $2750. Find the cost function.
20) The variable cost to manufacture an item is $10 per item, and it costs $2,500 to produce 100 items. Write the cost function, and use this function to estimate the cost of manufacturing 300 items.
21) It costs $2,700 to manufacture 100 items of a product, and $4,200 to manufacture 200 items.
x= the number of items; y= cost. Find the cost function; use it to predict the cost to produce 1000 items.
22) In 1990, the average house in Emerald City cost $280,000 and in 2007 the same house cost $365,000. Assuming a linear relationship, write an equation that will give the price of the house in any year, and use this equation to predict the price of a similar house in the year 2020.
23) The population of Mexico in 1995 was 95.4 million and in 2010 it was 117.9 million. Assuming a linear relationship, write an equation that will give the population of Mexico in any year, and use this equation to predict the population of Mexico in the year 2025.
SECTION 1.6 PROBLEM SET: CHAPTER REVIEW
24) At Nuts for Soup Lunch Bar, they sell 150 bowls of soup if the high temperature for the day is 40 ºF. For every 5 ºF increase in high temperature for the day, they sell 10 fewer bowls of soup.
- Assuming a linear relationship, write an equation that will give y = the number of bowls of soup sold as a function of x = the daily high temperature.
- How many bowls of soup are sold when the temperature is 75 ºF?
- What is the temperature when 100 bowls of soup are sold?
25) Two-hundred items are demanded at a price of $5, and 300 items are demanded at a price of $3. If x represents the price, and y the number of items, write the demand function.
26) A supply curve for a product is the number of items of the product that can be made available at different prices. A doll manufacturer can supply 2000 dolls if the dolls are sold for $30 each, but he can supply only 400 dolls if the dolls are sold for $10 each. If x represents the price of dolls and y the number of items, write an equation for the supply curve.
27) Suppose you are trying to decide on a price for your latest creation - a coffee mug that never tips. Through a survey, you have determined that at a price of $2, you can sell 2100 mugs, but at a price of $12 you can only sell 100 mugs.  What price should you charge so that the supply equals demand, and at that price how many coffee mugs will you be able to sell?
28) A car rental company offers two plans. Plan I charges $16 a day and 25 cents a mile, while Plan II charges $45 a day but no charge for miles. If you were to drive 200 miles in a day, which plan is better? For what mileage are both rates the same?
29) The supply curve for a product is y = 250x - 1000. The demand curve for the same product is
y = - 350x + 8,000, where x is the price and y the number of items produced. Find the following.
- At a price of $10, how many items will be in demand?
- At what price will 4,000 items be supplied?
- What is the equilibrium price for this product?
- How many items will be manufactured at the equilibrium price?
30) The supply curve for a product is y = 625x - 600 and the demand curve for the same product is
y = - 125x + 8,400, where x is the price and y the number of items produced.
Find the equilibrium price and determine the number of items that will be produced at that price.
31) Both Jenny and Masur work in the sales department for Sports Supply. Jenny gets paid $120 per day plus 4% commission on the sales. Masur gets paid $132 per day plus 8% commission on the sales in excess of $1,000. For what sales amount would they both earn the same daily amounts?
32) A company's revenue and cost in dollars are given by R = 25x and C = 10x + 9,000, where x represents the number of items. Find the number of items that must be produced to break-even.
33) A firm producing a certain type of CFL lightbulb has fixed costs of $6,800, and a variable cost of $2.30 per bulb. The bulbs sell for $4 each. How many bulbs must be produced to break-even?
34) A company producing tire pressure gauges has fixed costs of $7,500, and variable cost of $1.50 cents per item. If the gauges sell for $4.50, how many must be produced to break-even?
35) A company is introducing a new cordless travel shaver before the Christmas holidays. It hopes to sell 15,000 of these shavers in December. The variable cost is $11 per item and the fixed costs $100,000. If the shavers sell for $19 each, how many must be produced and sold to break-even?


2.2: Pain Assessment
“Pain is whatever the experiencing person says it is, existing whenever the experiencing person says it does” (McCaffery, 1968, cited in Rosdahl & Kowalski, 2007, p. 704). Pain is a subjective experience, and self-report of pain is the most reliable indicator of a patient’s experience. Determining pain is an important component of a physical assessment, and pain is sometimes referred to as the “fifth vital sign.”
Pain assessment is an ongoing process rather than a single event (see Figure 2.1). A more comprehensive and focused assessment should be performed when someone’s pain changes notably from previous findings, because sudden changes may indicate an underlying pathological process (Jarvis, Browne, MacDonald-Jenkins, & Luctkar-Flude, 2014).
Always assess pain at the beginning of a physical health assessment to determine the patient’s comfort level and potential need for pain comfort measures. At any other time you think your patient is in pain, you can use the mnemonic LOTTAARP (location, onset, timing, type, associated symptoms, alleviating factors, radiation, precipitating event) to help you remember what questions to ask your patient. See Checklist 14 for the questions to ask and steps to take to assess pain.
Checklist 14: Pain AssessmentDisclaimer: Always review and follow your hospital policy regarding this specific skill. |
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Steps |
Additional Information |
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| 1. Start your assessments by asking patients to rate their pain on a scale from 0 to 10, with 10 being the worst possible pain and 0 being no pain. | |||
| L: Location | Where are you feeling pain? | ||
| O: Onset |
When did the pain start? How long have you been in pain? |
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| T: Timing |
Is the pain constant or intermittent?
Has the intensity changed over time? |
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| T: Type | What does the pain feel like? | ||
| A: Associated symptoms | Do you have any associated symptoms such as nausea, vomiting, fever, etc.? | ||
| A: Alleviating factors |
What makes the pain feel better?
Do you take any medications for this pain? If so, are they effective? |
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| R: Radiation | Does the pain move anywhere else? | ||
| P: Precipitating event |
What was happening when the pain started? What has caused the pain to occur? Has this happened before? |
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| 2. Provide analgesia as prescribed and other comfort measures, such as distraction, massage, and the application of warmth or cold, as appropriate. | |||
| 3. Report and document assessment findings and related health problems according to agency policy. | |||
| Data source: Assessment Skill Checklists, 2014 |
Critical Thinking Exercises
- You are caring for a patient who has just returned from a surgical procedure. The patient has a history of chronic pain. Would the patient’s assessment provide the same data as an assessment of a person who does not have a history of chronic pain?
- What is more important: the subjective or the objective data in a pain assessment?


22.5D: Serosa
Serosa consists of a secretory epithelial layer and a thin connective tissue layer that reduce the friction from muscle movements.
- Diagram the structure of the serosa
Key Points
- The serous layer provides a partition between the internal organs and the abdominal cavity.
- Cells of the serous layer secrete a serous fluid that provides lubrication to reduce friction.
- The connective tissue layer provides blood vessels and nerves.
- The three serous cavities within the human body are the pericardial cavity (surrounding the heart ), the pleural cavity (surrounding the lungs), and peritoneal cavity (surrounding most organs of the abdomen).
- The serous membrane covers the heart; it has an inner layer (the parietal pericardium ) and an outer layer (the visceral pericardium).
- The serosa of the uterus is called the perimetrium.
Key Terms
- pleural cavity : The body cavity that surrounds the lungs and is enclosed by the pleura.
- coelom : A fluid-filled cavity within the body of an animal. The digestive system is suspended within the cavity that is lined by a tissue called the peritoneum.
- serosa : A membrane that lines an internal cavity to protect the contents and secretes serum.
The Serous Membrane
In anatomy, the serous membrane (or serosa) is a smooth membrane that consists of a thin connective tissue layer and a thin layer of cells that secrete serous fluid. Serous membranes line and enclose several body cavities, known as serous cavities, where they secrete a lubricating fluid to reduce friction from muscle movements.
Serosa is not to be confused with adventitia, a connective tissue layer that binds together structures rather than reduces friction between them.
Layers of stomach lining : The serosa is labeled at far right, and is colored yellow.
Each serous membrane is composed of a secretory epithelial layer and a connective tissue layer underneath. The epithelial layer, known as mesothelium, consists of a single layer of avascular flat nucleated cells (simple squamous epithelium) that produce the lubricating serous fluid. This fluid has a consistency similar to thin mucus.
These cells are bound tightly to the underlying connective tissue. The connective tissue layer provides the blood vessels and nerves for the overlying secretory cells, and also serves as the binding layer that allows the whole serous membrane to adhere to organs and other structures.
For the heart, the surrounding serous membranes include: the outer, inner, parietal pericardium, and visceral pericardium (epicardium). Other parts of the body may also have specific names for these structures. For example, the serosa of the uterus is called the perimetrium.
The pericardial cavity (surrounding the heart), pleural cavity (surrounding the lungs) and peritoneal cavity (surrounding most organs of the abdomen) are the three serous cavities within the human body. While serous membranes have a lubricative role to play in all three cavities, in the pleural cavity it has a greater role to play in the function of breathing.
The serous cavities are formed from the intraembryonic coelom and are basically an empty space within the body surrounded by a serous membrane. Early in embryonic life, visceral organs develop adjacent to a cavity and invaginate into the bag-like coelom.
Therefore each organ becomes surrounded by a serous membrane—they do not lie within the serous cavity. The layer in contact with the organ is known as the visceral layer, while the parietal layer is in contact with the body wall.
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6.14D: High Pressure
Under very high hydrostatic pressure(HHP) of up to 700 MPa, water inactivates pathogens such as E. coli and Salmonella .
LEARNING OBJECTIVES
Explain high pressure as a antimicrobial control
Key Takeaways
Under very high hydrostatic pressure of up to 700 MPa (100,000 psi), water inactivates pathogens such as Listeria, E. coli and Salmonella .
High pressure processing (HPP), pascalization or bridgmanization, is a method of preserving and sterilizing food, in which a product is processed under very high pressure, leading to the inactivation of certain microorganisms and enzymes in the food. The technique was named after Blaise Pascal, whose work included detailing the effects of pressure on fluids. Pascalization is preferred over heat treatment in the food industry as it eliminates changes in the quality of foods due to thermal degradation, resulting in fresher taste, texture, appearance and nutrition. Processing conveniently takes place at ambient or refrigeration temperatures.
Experiments into the effects of pressure on microorganisms were first recorded in the late nineteenth century. The frist reports showed that bacterial spores were not always inactivated by pressure, while vegetative bacteria were usually killed. Around 1970, researchers renewed their efforts in studying bacterial spores after it was discovered that using moderate pressures was more effective than using higher pressures. These spores, which caused a lack of preservation in the earlier experiments, were inactivated faster by moderate pressure, but in a manner different from what occurred with vegetative microbes. When subjected to moderate pressures, bacterial spores germinate, and the resulting spores are easily killed using pressure, heat, or ionizing radiation.
Research into the effects of high pressures on microorganisms was largely focused on deep-sea organisms until the 1980s, when advancements in ceramic processing were made. This resulted in the production of machinery that allowed for processing foods at high pressures at a large scale, and generated some interest in the technique, especially in Japan. Although commercial products preserved by pascalization first emerged in 1990, the technology behind pascalization is still being perfected for widespread use.
In pascalization, food products are sealed and placed into a steel compartment containing a liquid, often water, and pumps are used to create pressure. The pumps may apply pressure constantly or intermittently. During pascalization, more than 50,000 pounds per square inch (340 MPa) may be applied for around fifteen minutes, leading to the inactivation of yeast, mold, and bacteria. In the process, the food’s proteins are denatured, hydrogen bonds are fortified, and noncovalent bonds in the food are disrupted, while the product’s main structure remains intact. Because pascalization is not heat-based, covalent bonds are not affected, causing no change in the food’s taste.
Experiments were also performed with anthrax, typhoid, and tuberculosis, which was a potential health risk for the researchers. Indeed, before the process was improved, one employee of the Experimental Station became ill with typhoid fever.
Key Points
- High pressure processing (HPP), pascalization or bridgmanization, is a method of preserving and sterilizing food, in which a product is processed under very high pressure, leading to the inactivation of certain microorganisms and enzymes in the food.
- The frist reports showed that bacterial spores were not always inactivated by pressure, while vegetative bacteria were usually killed. Later it was discovered that using moderate pressures was more effective in eliminating spores than using higher pressures.
- Experiments were also performed with anthrax, typhoid, and tuberculosis, which was a potential health risk for the researchers.
Key Terms
- bridgmanization : Pascalization is also known as bridgmanization, named for physicist Percy Williams Bridgman.


12.3: Scatter Plots
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Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables x and y . The most common and easiest way is a scatter plot . The following example illustrates a scatter plot.
Example \(\PageIndex{1}\)
In Europe and Asia, m-commerce is popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from a machine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was there a relationship between the year and the number of m-commerce users? Construct a scatter plot. Let \(x =\) the year and let \(y =\) the number of m-commerce users, in millions.
| \(x\) (year) | \(y\) (# of users) |
|---|---|
| 2000 | 0.5 |
| 2002 | 20.0 |
| 2003 | 33.0 |
| 2004 | 47.0 |
To create a scatter plot
- Enter your \(X\) data into list L1 and your \(Y\) data into list L2.
- Press 2nd STATPLOT ENTER to use Plot 1. On the input screen for PLOT 1, highlight On and press ENTER. (Make sure the other plots are OFF.)
- For TYPE: highlight the very first icon, which is the scatter plot, and press ENTER.
- For Xlist:, enter L1 ENTER and for Ylist: L2 ENTER.
- For Mark: it does not matter which symbol you highlight, but the square is the easiest to see. Press ENTER.
- Make sure there are no other equations that could be plotted. Press Y = and clear any equations out.
- Press the ZOOM key and then the number 9 (for menu item "ZoomStat") ; the calculator will fit the window to the data. You can press WINDOW to see the scaling of the axes.
Exercise \(\PageIndex{1}\)
Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:
| \(X\) (hours practicing jump shot) | \(Y\) (points scored in a game) |
|---|---|
| 5 | 15 |
| 7 | 22 |
| 9 | 28 |
| 10 | 31 |
| 11 | 33 |
| 12 | 36 |
Construct a scatter plot and state if what Amelia thinks appears to be true.
- Answer
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Figure \(\PageIndex{2}\)
Yes, Amelia’s assumption appears to be correct. The number of points Amelia scores per game goes up when she practices her jump shot more.
A scatter plot shows the direction of a relationship between the variables. A clear direction happens when there is either:
- High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.
- High values of one variable occurring with low values of the other variable.
You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or to some other type of function. For a linear relationship there is an exception. Consider a scatter plot where all the points fall on a horizontal line providing a "perfect fit." The horizontal line would in fact show no relationship.
When you look at a scatter plot, you want to notice the overall pattern and any deviations from the pattern. The following scatterplot examples illustrate these concepts.
In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line, except in the case of a horizontal line where there is no relationship. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If \(x\) is the independent variable and \(y\) the dependent variable, then we can use a regression line to predict \(y\) for a given value of \(x\)
Summary
Scatter plots are particularly helpful graphs when we want to see if there is a linear relationship among data points. They indicate both the direction of the relationship between the \(x\) variables and the \(y\) variables, and the strength of the relationship. We calculate the strength of the relationship between an independent variable and a dependent variable using linear regression.


 Most prepositions such as above , below , and behind usually indicate a location in the physical world, but some prepositions such as during , after , and until show location in time. Some prepositions, such as about and of can show a person's relation to an idea. Some preposition use is idiomatic, meaning that the usage is not based in a rule; rather, the usage is based in tradition.
In, At, and On
The prepositions in , at , and on are used to indicate both location and time, but they are used in specific ways.
| Preposition | Time | Example | Place | Example |
|---|---|---|---|---|
| in | year | in 1942 | country | in Zimbabwe |
| month | in August | state | in California | |
| season | in the summer | city | in Chicago | |
| time of day (not with night ) | in the afternoon | |||
| Degree | Example | Other | Example | |
| currency | in Yen | |||
| language | in German |
| Preposition | Time | Example | Place | Example |
|---|---|---|---|---|
| on* | day | on Monday | surfaces | on the table |
| date | on May 23 | streets | on 124th Street | |
| specific days/dates | on Monday | modes of transportation | on the bus | |
| Degree | Example | Other | Example | |
| communication | on the radio | |||
| concerning | on magic |
| Preposition | Time | Example | Place | Example |
|---|---|---|---|---|
| at | time | at five o’clock | addresses | at 1600 Pennsylvania Avenue |
| with night | at night | location | at Rooney’s Grill | |
| Degree | Example | Other | Example | |
| temperature | at 0 degrees Celcius | idiomatic | She is good at dancing |
The table above also points out uses other than time and place.
Exercise 1
On a separate piece of paper, edit the following letter from a resident to her landlord by correcting errors with in, at , and on .
Dear Mrs. Salazar,
I am writing this letter to inform you that I will be vacating apartment 2A in 356 Maple Street at Wednesday, June 30, 2010. I will be cleaning the apartment at the Monday before I leave. I will return the keys to you on 5 p.m., sharp, at June 30. If you have any questions or specific instructions for me, please contact me in my office. I have enjoyed living at Austin, Texas, but I want to explore other parts of the country now.
Sincerely,
Milani Davis
The table below has examples of the usage of other common prepositions.
Table 12.15.2 -- Common Preposition Usage
| Preposition | Space | Time | Degree (measurement) | Other |
| about |
all around The dog ran about the yard. |
approximately It is about 1:00 p.m. |
approximately It is about 70 degrees Farenheit. |
concerning This is a book about mathematics. |
| against |
contact I plywood leaned against the house. |
conflict I was against what Karen wanted to do. |
conflict The fight was five against three. |
internal He was taken against his will. |
|
external The team won against all odds. |
||||
| above |
higher than The tree towers above the house. |
more than The temperature is above freezing. |
idiomatic She is above reproach. He is above suspicion. |
|
| around |
state The fence is around the dog run. |
approximately I scored around 1250 on the SAT. |
approximately I rang around four miles before pooping out. |
|
|
action The dogs ran around the dog park. |
||||
| before |
in front of In the painting, the buildings were before the mountains. |
earlier than Explorers reached the South Pole before astronauts reached the moon. |
||
| below |
lower than The submarine was below the surface of the water. |
less than His test scores were below average. |
||
| between |
at some point in relation to two entities The ball was stuck between the garbage can and the fence. |
at some point between two times They were going to meet up between 7 and 8 p.m. |
at some point between two measurements I weight between 150 and 160 pounds. |
idiomatic We an keep this secret between you and me, right? |
| by |
near The cafe near the corner is the best. |
no later than Be home for dinner by 5:30 p.m. |
duplication The inchworm moved inch by inch. |
without help I can do my work by myself. |
|
degree of failure The arrow missed the target by a mile. |
alone He went shopping by himself. |
|||
| for |
goal We set out for Australia in our yacht. |
duration We have been together for three years. |
exchange I bought the top for $12. |
reason People travel to France for the art. |
|
distance I drove for 500 miles. |
goal/purpose We hunted for boar. |
|||
| from |
a starting point We ran the race from the beginning to the end. |
duration I worked from 8 to 5:30. |
beginning of range The temperature tomorrow will range from 48 to 58 degrees. |
source The Egyptians made paper from papyrus. |
|
origin I moved here from Chicago. |
cause The ground was charred from fire. |
|||
| of |
names of geographic locations or institutions I plan to transfer to the University of California. |
before I woke up at a quarter of six. |
portion I would like a cup of tea. |
possessive A friend of mine recently went on the trip of a lifetime. |
|
source The table was made of metal. |
||||
| over |
above She wore a shawl over her shoulders. |
spanning time Over the break, I did a thorough house cleaning. |
more than I had to wait more than one-half hour for my doctor. |
communication I read my news over the Internet. |
|
action above The pole vaulter made it over the pole. |
||||
| through |
penetrate The mountain lion snuck through the bushes. |
duration Through the years, I've learned a few things. |
endurance Through good times and bad, we make survive. |
|
| to |
direction Kevin drove to the mountains. |
until I slept to 7 a.m. |
amount I will listen to the extent of my patience. |
accompany I listened to the radio. |
| toward |
in the direction of I walked toward the Northwest. |
near It was toward 11 a.m. when things were stolen from my car. |
measurement The snow began to melt as the temperature headed toward 40F. |
goal People need to work together toward civil discourse. |
| under |
below Moles burrow below the surface. |
less than I will be there in under fifteen minutes |
less than My dinner cost under $10. |
condition She admitted to the crime under duress. |
| with |
alongside, near The painting was hung parallel with window. |
together I went with Amy to the movies. |
equal standing or ability John kept the lead with Jordan. |
in regard to I am happy with my purchase. |
|
manner They danced with style. |
A beginner to prepositions may want to watch the following video, which shows the meaning of the words visually. This video mainly focuses on space and time meanings of prepositions.
Prepositions after Verbs
Prepositions often follow verbs to create expressions with distinct meanings. In fact, some verbs require prepositions after them. These expressions are sometimes called prepositional verbs. It is important to remember that these prepositions cannot be separated from the verb.
| Verb + Preposition | Meaning | Example |
|---|---|---|
| agree with | to agree with something or someone | My husband always agrees with me. |
| apologize for | to express regret for something, to say sorry about something | I apologize for being late. |
| apply for | to ask for something formally | I will apply for that job. |
| believe in | to have a firm conviction in something; to believe in the existence of something | I believe in educating the world’s women. |
| care about | to think that someone or something is important | I care about the health of our oceans. |
| consist of | to be made up of certain things | My lunch consists of a turkey sandwich and an apple. |
| detract from | to make less appealing | The workload detracts from the salary that the job offers. |
| hear about | to be told about something or someone | I heard about the teachers’ strike. |
| look after | to watch or to protect someone or something | Will you look after my dog while I am on vacation? |
| part with | to give away or be separated from something | I parted with a bookshelf of books when I moved. |
| rely on | to depend on | I rely on my alarm clock going off in the morning. |
| substitute for | to replace something temporarily | The flour substitutes for cornstarch in the gravy recipe. |
| talk about | to discuss something | We will talk about the importance of recycling.sus |
| speak to, with | to talk to/with someone | I will speak to his teacher tomorrow. |
| wait for | to await the arrival of someone or something | I will wait for my package to arrive. |
Tip
It is a good idea to memorize these combinations of verbs plus prepositions. Write them down in a notebook along with the definition and practice using them when you speak.
Exercise 2
On a separate sheet of paper, complete the following sentences by choosing the correct preposition with the verb.
- Charlotte does not ________ (apologize for, believe in) aliens or ghosts.
- It is impolite to ________ (hear about, talk about) people when they are not here.
- Herman said he was going to ________ (believe in, apply for) the internship.
- Jonas would not ________ (talk about, apologize for) eating the last piece of cake.
- I ________ (care about, agree with) the environment very much
Prepositions after Adjectives
Similar to prepositions after verbs, prepositions after adjectives create expressions with distinct meanings unique to English. Remember, like prepositional verbs, these expressions also cannot be separated.
| Adjective + Preposition | Meaning | Example |
|---|---|---|
| afraid of | to fear something | I am afraid of big dogs that bare their teeth. |
| angry at, about | to feel or show anger toward (or about) someone or something | I am angry about the oil spill in the ocean. |
| confused about | to be unable to think with clarity about someone or something. | Shawn was confused about the concepts presented in class. |
| content with | to be satisfied with | I am content with the grade in my math class. |
| to be dependent on | to be reliant on something | The ex-smoker was dependent on her oxygen tank to breathe. |
| disappointed in, with | to feel dissatisfaction with someone or something | I was disappointed in my husband because he voted for that candidate. |
| dressed in | to clothe the body | He was dressed in a pin-striped suit. |
| to be free from/or | not shackled to something someone finds burdensome | I was free from my obligations for the summer. |
| happy for | to show happiness for someone or something | I was happy for my sister who graduated from college. |
| interested in | giving attention to something, expressing interest | I am interested in musical theater. |
| jealous of | to feel resentful or bitter toward someone or something (because of their status, possessions, or ability) | I was jealous of her because she always went on vacation. |
| to be sorry for | to have pity or regret | I am sorry for the things I said. |
| thankful for | to express thanks for something | I am thankful for my wonderful friends. |
| tired of | to be disgusted with, have a distaste for | I was tired of driving for hours without end. |
| worried about | to express anxiety or worry about something | I am worried about my father’s health. |
Note:
The following adjectives are always followed by the preposition at :
- Good
She is really good at chess.
- Excellent
Henry is excellent at drawing.
- Brilliant
Mary Anne is brilliant at playing the violiin.
Exercise 3
On a separate sheet of paper, complete the following sentences by writing the correct preposition with adjective.
- Meera was deeply ________ (interested in, thankful for) marine biology.
- I was ________ (jealous of, disappointed in) the season finale of my favorite show.
- Jordan won the race, and I am ________ (happy for, interested in) him.
- The lawyer was ________ (thankful for, confused about) the details of the case.
- Chloe was ________ (dressed in, tired of) a comfortable blue tunic.
Idiomatic Preposition Usage
The following phrases -- many of them noun phrases -- are expressed in a particular way and should be memorized. There isn't a rule or particular that determines the preposition to use; thus, the whole phrase is an idiom and should be memorized together.
Table 12.15.5 -- Idiomatic Prepositions
| Idiomatic Phrase | Example |
| in the end | In the end , the whole disagreement was a misunderstanding. |
| talked about ( not talked on ) | The president talked about current events. |
| in case of | In case of fire, break the glass. |
| in charge of | The principal was in charge of the high school. |
| in favor of | Most students were in favor of starting school at 9 a.m. |
| on account of | We were late on account of a broken down bus.* |
| on behalf of | We present this plaque on behalf of the city council. |
| in the course of | In the course of completing the project, we discovered some other problems. |
| in the habit of | I am in the habit of emailing people back within a day. |
| in the name of | Sue donated flowers in the name of her aunt, who had died. |
| on the advice of | I am quitting vaping on the advice of my doctor. |
| on the basis of | I am cancelling this subscription on the basis of cost. |
| on the strength of | The correlation was made on the strength of the data. |
*Note that a noun or noun phrase must be used after on account of as the object of the preposition. Sometimes student try to follow this phrase with an entire sentence, which is not grammatically correct.
Prepositional Phrases
Most of the time, prepositions begin prepositional phrases, which give more information about something in a sentence. As mentioned above, sometimes the verb you use requires a prepositional phrase after the verb. The subject of a sentence can never be the noun in a prepositional phrase. The subject is part of the core of the sentence, and the prepositional phrase is outside of that core. For more information , please see "Word Order and Sentence Structure" in section 12.2.
Example \(\PageIndex{1}\)
Subject verb preposition article noun(object of prep) infinitive gerund
Karen drove to the mountains to go skiing.
Sentence core prepositional phrase verbal phrase
Prepositions always must be followed by a noun in a prepositional phrase. Many times that noun will also be preceded by an article and/or an adjective.
Exercise 4
Underline the prepositional phrases in the following sentences. Do not underline any words that are not part of a prepositional phrase.
- Under the moonlight, the coyotes howled and yipped.
- I must finish my paper by Tuesday night.
- I felt on top of the world when I learned I had one a prize!
- Bill and Jordan thought they could play some basketball on Thursday after class.
- Chihuahuas and pit bulls are popular dogs, but unfortunately, too many of them fill up animal shelters throughout the country.
- Marta and Denise work toward understanding each others' points of view.
Exercise 5
Underline the prepositional phrases in the following sentences. Do not underline any words that are not part of a prepositional phrase.
1. Monica told us about her trip.
2. I hope we have sunshine throughout the summer.
3. The panther climbed up the tree.
4. The little boy was standing behind his mother's legs.
5. We stayed awake until dawn.
Contributors and Attributions
CC LICENSED CONTENT, SHARED PREVIOUSLY:
- Adapted from College ESL Writers: Applied Grammar and Composing Strategies for Success . A uthored by: Barbara Hall and Elizabeth Wallace. P rovided by: GALILEO Open Learning Materials. L icense: CC BY-NC-SA (3.0): Attribution
- Adapted from Writing for Success . Provided by: The Saylor Foundation. License: CC-NC-SA 3.0 .
ALL RIGHTS RESERVED CONTENT:
- Learn Prepositions Easily . Authored by: Jennifer Banks. License: All Rights Reserved. License Terms: Standard YouTube License.
This page last updated on June 8, 2020.


5.8: Safety Considerations Across the Life Span
To promote safety for patients of all ages, nurses should be knowledgeable about safety risks according to age and developmental stages because the types and frequencies of accidents vary among age groups. Information from the Centers for Disease Control (CDC) regarding safety tips for each age group is summarized in the following subsections. [1]
Infants and Preschoolers
Drowning is the leading cause of death in children aged 1-3 years. Motor vehicle accidents, falls, choking, and accidental poisoning are also safety concerns for this age group. Infants and toddlers are curious, but they lack the judgement to recognize the dangers of their actions, so childproofing the home and providing adult supervision are essential for this developmental age group. [2] See Figure 5.7 [3] for an image of an infant car seat used to protect infants in the event of a motor vehicle accident. Nurses help educate parents about the proper use, positioning, and installation of car seats.
School-Aged Children
In children aged 4-11, motor vehicle injuries are a major cause of unintentional injury, along with drowning and poisoning. This age group is more aware of dangers and limitations, but adult supervision is still important. The nurse should educate parents of school-aged children about safety seats, booster seats, or shoulder seat belts while riding in the car. [4]
Bicycle accidents are also a common concern in this age group. Many bike accidents involve the head or face because of the lack of helmet use. Nurses provide health teaching to school-aged children regarding bicycle safety and helmet use. See Figure 5.8 [5] for an image of a girl wearing a bike helmet.
Because this age group is beginning to enjoy more independence, basic instructions and education on how to recognize and respond to potentially dangerous situations with strangers should also be provided. Parents should also be educated about the AMBER alert system that can be activated if a child is missing and believed to be kidnapped or in danger. This AMBER alert system uses the resources of law enforcement and the media to notify the public about a possible abduction or a missing child in danger. [6]
Nurses must also be aware of signs of maltreatment and child abuse because millions of children are affected each year. Child abuse includes physical, sexual, emotional abuse, and neglect. According to the American Academy of Child and Adolescent Psychiatry (AACAP), after abuse or violence, many children develop mental health problems, including depression and posttraumatic stress disorder. These children may also have serious medical problems, learning problems, and problems getting along with friends and family members. Every state has laws that require health care professionals to report suspected child abuse no matter what form this abuse takes. [7]
Adolescents
Motor vehicle accidents are the number one cause of death for adolescents. Teens aged 16-19 are three times more likely to be in a fatal crash than drivers older than age 20. Adolescent males are twice as likely to die in a motor vehicle accident than females of the same age. Texting while driving is a common cause of distracted driving and accidents in adolescents. Because much of an adolescent’s time is spent away from the home, it is difficult for parents to control many of the decisions that adolescents make. Nurses educate teenagers to use seat belts, obey speed limits, and never use a cell phone or text while driving. [8] See Figure 5.9 [9] for an image reminding teenage drivers to not text and drive.
Traumatic brain injuries (TBI) may occur in this age group due to participation in sports and recreation-related activities. TBI results from a blow, jolt, or hit to the head that causes a disruption in blood function or flow to the brain. Nurses should always be alert for indications of a concussion when a sports injury has occurred. Signs of a concussion requiring immediate medical attention include the following:
- Headache, vomiting, balance problems, fatigue, or drowsiness
- A dazed and confused appearance or difficulty concentrating or remembering; confusion
- Emotional irritability, nervousness, or a change in personality
The CDC has comprehensive information and education materials for parents, coaches, players, and health care providers as part of their “Heads Up” program. [10]
Substance abuse is another significant concern in the adolescent population and includes substances such as tobacco, alcohol, illicit drugs, prescription medication, over-the-counter medications, and bath salts. The National Institutes of Health provides many resources for educating teens and their parents about substance abuse. [11]
Adults
Intimate partner violence and substance abuse are common safety issues in the adult population.
Intimate Partner Violence
Intimate partner violence (IPV) is widespread in the United States and is the most prevalent adult safety issue. Intimate partner violence includes physical or sexual violence, stalking, and psychological or coercive aggression by current or former intimate partners. Victims can be female or male, and sexual orientation can be heterosexual or LGBTQ+. The nurse is often the initial health care professional in contact with a victim of IPV. Prompt recognition of a potential or actual threat to patient and staff safety is crucial. It is often the nurse’s assessment that plays an important role in identifying a patient experiencing IPV. Compassion and understanding are important to show to this vulnerable population. Effective communication is necessary to help victims come forward and share their experiences of abuse. IPV is a complex issue, and the patient may not initially consider leaving the abuser as an option. See Figure 5.10 [12] for an image of a sign in a community demonstrating support against domestic violence.
See the following hyperlinks for tools and resources to share with patients experiencing IPV. For example, the Danger Assessment Tool is a self-administered survey that is free to use and is available in several languages. [13] Nurses can refer patients experiencing IPV to the National Center on Domestic Violence, the Trauma and Mental Health database for resources, [14] and the National Domestic Violence Hotline for free, confidential support. [15] Most importantly, nurses should assist patients experiencing IPV to create a safety plan.
View the tools and resources available at these hyperlinks to share with individuals experiencing intimate partner violence:
Substance Abuse
Substance abuse is defined by the World Health Organization (WHO) as a maladaptive pattern of using alcohol and/or drugs despite it causing persistent social, occupational, psychological, or physical problems that can be physically hazardous. Substance abuse continues to be a safety issue that affects adults across all socioeconomic levels. In America over 450,000 people died between 1999 and 2018 as a result of an opioid overdose. [16] The abuse of prescription pain medication (such as oxycontin and fentanyl) and heroin is a national crisis that plagues social and economic welfare. Substance abuse not only affects an individual, but also causes harm to their family members. Early identification of substance abuse, rehabilitation interventions, and continued support are key for helping the individual, as well as their family members, in the recovery process. See Figure 5.11 [17] for an image of a heroin needle found in a community setting.
Older Adults
According to the Centers for Disease Control and Prevention, falls and motor vehicle accidents are leading causes of injury in older adults. However, several other issues pose significant hazards for this population, such as fires, accidental overdosing on medications (due to poor eyesight and confusion), elder abuse, and financial exploitation. [18] In most reported cases of elder abuse, a caregiver or a person in trusted relationship is the perpetrator. For various reasons such as fear and disappointment, most of these cases go unreported. Abuse, including neglect and exploitation, is experienced by about 1 in 10 people aged 60 and older who live at home. From 2002 to 2016, more than 643,000 older adults were treated in the emergency department for nonfatal assaults and over 19,000 homicides occurred. [19] Read an example of an older adult experiencing financial exploitation in the following box.
Consider the story of John, a 92-year-old male who lost his wife over a year ago and has been lonely ever since. He lives alone in a large home in the country. John hired a repairman to fix his roof. The repairman befriended John, bringing him homemade cookies and pies and even running errands for him. The repairman often stayed for coffee, and the two of them spent time talking about fishing and gardening. The repairman convinced John to take out a reverse mortgage to pay for additional improvements on his home. Then, knowing John’s bank account numbers and login information, the repairman stole $250,000 that John received for his reverse mortgage.
Most victims of elder abuse are frequently seen in the emergency department several times before they are admitted to the hospital. Nurses must be alert to any indications of elder abuse, such as suspicious injuries or behaviors, and report suspected incidents to local adult protective services agencies. Commons signs of elder abuse or maltreatment include the following: [20]
- Bruises, cuts, burns, or broken bones that are unexplainable or suspiciously explained
- Malnourishment or weight loss
- Poor hygiene, an unkempt appearance, unclean clothing, or dirty, matted hair
- Foul odor from clothing or body
- Anxiety, depression, or confusion
- Unexplained transactions or loss of money
- Withdrawal from family members or friends
Download the Elder Mistreatment Assessment tool from The Hartford Institute for Geriatric Nursing. [21]
View additional resources related to elder abuse using the hyperlinks in the following box.
Additional resources for older adults suspected as being victims of elder abuse:
- Centers for Disease Control and Prevention. (2020, October 8). Parent information . https://www.cdc.gov/parents/index.html ↵
- Centers for Disease Control and Prevention. (2020, October 8). Parent information . https://www.cdc.gov/parents/index.html ↵
- "ARISE Newborn in Car Seat 144049.jpg" by ARISE project is licensed under CC BY 4.0 ↵
- Centers for Disease Control and Prevention. (2020, October 8). Parent information . https://www.cdc.gov/parents/index.html ↵
- “ 539478754.jpg ” by thechatat is used under license from Shutterstock.com ↵
- Amber Alert. https://amberalert.ojp.gov / ↵
- American Academy of Child & Adolescent Psychiatry. (2020, April). Trauma and child abuse resource center . https://www.aacap.org/AACAP/Families_and_Youth/Resource_Centers/Child_Abuse_Resource_Center/Home.aspx ↵
- Centers for Disease Control and Prevention. (2020, October 8). Parent information . https://www.cdc.gov/parents/index.html ↵
- “ texting-while-driving/man-texting-while-driving-md.jpg ” by unknown at QuoteInspector.com is licensed under CC BY-ND 4.0 . ↵
- Centers for Disease Control and Prevention. (2020, November 20). Helmet safety . https://www.cdc.gov/headsup/helmets/index.html ↵
- National Institute on Drug Abuse. (n.d.). NIDA for teens . https://teens.drugabuse.gov/ ↵
- “ Domestic_violence_free-zone.jpg ” by Ben Pollard is licensed under CC BY-SA 2.0 ↵
- Danger Assessment. https://www.dangerassessment.org/DATools.aspx ↵
- National Center on Domestic Violence, Trauma & Mental Health. (n.d.). National domestic violence organizations . http://www.nationalcenterdvtraumamh.org/resources/national-domestic-violence-organizations/ ↵
- National Domestic Violence Hotline. https://www.thehotline.org/ ↵
- Centers for Disease Control and Prevention. (2020, December 17). Opioid overdose . https://www.cdc.gov/drugoverdose/ ↵
- “ Heroin syringe ” by Thomas Marthinsen is licensed under CC BY-NC-SA 2.0 ↵
- Centers for Disease Control and Pevention (n.d.) Injury Prevention and Control . https://www.cdc.gov/injury/index.html ↵
- Centers for Disease Control and Prevention (n.d.) Elder Abuse . https://www.cdc.gov/violenceprevention/elderabuse/ ↵
- Nursing Home Abuse Center. (2020, January 8). Signs of elder abuse . https://www.nursinghomeabusecenter.com/elder-abuse/signs / ↵
- The Hartford Institute for Geriatric Nursing, New York University, Rory Meyers School of Nursing. (n.d.). Assessment tools for best practices of care for older adults. https://hign.org/consultgeri-resources/try-this-series ↵


11.10: Exercise- Loops
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In this exercise, we’ll introduce the concept of loops , which will make your life much easier by allowing you to automate the processing of multiple participants. There are several kinds of loops, but the most common is called a for loop . It allows you to repeat the same sequence of steps multiple times, but with one variable changing (e.g., a variable that indicates which participant should be processed). We’re going to go through several examples in this exercise, starting simple and working our way up to a script that processes the data from multiple participants.
As usual, let’s start by quitting EEGLAB, typing clear all , and restarting EEGLAB. Then open Script3.m by double-clicking on it in the Current Folder pane. Run the script (e.g., by clicking the Run button) and see what it does. You should see that it prints a list of numbers between 1 and 10. In a later script, these will be the ID numbers of our participants. Note that the number 5 is missing. This is meant to indicate that Subject 5 is being excluded (because that subject had too many artifacts and was excluded from the final analyses).
Now take a look at the script. The main body of the script is this:
for subject = [ 1 2 3 4 6 7 8 9 10 ]
display(subject);
end
In Matlab, a for loop begins with for and ends with end . The lines of code between the for and end lines are the body of the loop. This body will be executed multiple times, once for each element specified in the array on the for line. The for line defines a variable (which we’ve named subject in this example) and specifies an array of values that will be stored in this variable as we go through the loop ( [ 1 2 3 4 6 7 8 9 10 ] in this example).
In Script3.m , the body of the loop is a single line consisting of the display(subject) command, which just prints the value of the variable named subject that we specified on the for line. Note that it’s conventional to indent the body of a loop using tabs. The tabs are ignored by Matlab, but they make it easier to see the structure of a script. Not required, but highly recommended!
Each time we go through the loop, the variable named subject is set to a new value in the array of values following the equals sign. When the loop starts, subject will be set to 1 (because 1 is the first value in the array). The display(subject) line will then execute, and it will display a value of 1 because that’s the value of the subject variable. Then the end line occurs, telling Matlab to go back to the start of the loop and set subject to the next value in the array (2). The display(subject) line will then execute, but this time it will display a value of 2 because that’s now the value of the subject variable.
Matlab will keep repeating the loop, setting subject to 3, then to 4, then to 6, etc. There is no 5 in our array of values, so subject will never take on that value. The array is just a list of values, and any sequence of values will work. For example, we could use [ 3 1 5 9 ] and then subject would take on that set of values in that order (3, then 1, then 5, then 9). We could even use non-integer numbers (e.g., [ 5.23 6.1 -5.442 10021.2 ] ) or character strings (e.g., [ ‘S1’ ‘S2’ ‘S3’ ‘S5’ ] ). Matlab is much more flexible than most programming languages in this regard. Spend some time playing with the array in the script so that you get a good sense of how it works.
Now close Script3.m and open Script3b.m , which is a slightly more sophisticated version of the same set of ideas. Run the script to see what it does. You should see a set of lines starting with this:
Processing Subject 1
Processing Subject 2
Processing Subject 3
Now look at the script. You’ll notice two main changes from the first script. First, instead of providing an explicit list of subject IDs in the for line, we’ve defined a variable named SUB that stores this array:
SUB = [ 1 2 3 4 6 7 8 9 10 ]; %Array of subject IDs
We then specify this variable as our array in the for statement:
for subject = SUB
This approach embodies my #1 principle of writing good code: All values used by a script should be defined as variables at the top of the script . I’ll say more about this principle later in the chapter.
The second change to the script is that it uses the fprintf command instead of the display command to print the value of the subject variable. The fprintf command is much more powerful and flexible. It takes a little time to learn to use it, but it’s well worth the time. Here’s the specific version used in our script:
fprintf('Processing Subject %d\n', subject);
The first parameter in the fprintf command is a formatting string . It contains plain text that is printed by the routine, and it also includes formatting statements for variables that appear as subsequent parameters. The %d tells the command to print a whole number (the value of the subject variable), and the \n tells it to print a newline (a return). You can do much more than this with fprintf , but we’re keeping it simple for now (see the fprintf documentation for details).
Once you understand how this script works, close it and open Script3c.m . If you run it, you’ll see that the output has more information than provided by the previous script. For each iteration of the loop, it indicates how many times we’ve gone through the loop, plus the subject’s ID. This allows us to see that the fifth subject has an ID of 6, not 5.
Look at the script to see how it works. Near the top, you’ll see that we use a Matlab function called length to define a variable named num_subjects that stores the number of subjects in the SUB array:
num_subjects = length(SUB);
We’ve then used this new variable to define the array of values for the loop, which we’ve defined as 1:num_subjects . Type 1:num_subjects on the command line. You’ll see that it is equivalent to a list of integers between 1 and num_subjects (rather than 1 through 10, skipping 5). As a result, we’re no longer looping through the subject IDs. As a result, I’ve changed the name of the variable in the for statement to subject_index .
Each time through the loop, we get the subject ID by finding the element in SUB that corresponds to subject_index and store it as a text string in a variable named ID . For example, when subject_index is 5, we get the 5 th element of SUB , which is 6 (because SUB skips subject 5). SUB is an array of numbers, but as you’ll see in the next script, it’s useful to store the ID as a text string. We therefore use a Matlab function called num2str to convert the number to a string before storing it in ID . Note that the format string for the fprintf command uses %s to indicate that this command should print a string variable for ID .
Why it Pays to Include Good Comments and Meaningful Variable Names in Your Scripts
When you’re in the middle of writing a script to process the data for an experiment, you will get very focused on getting the job done . That is, you just want to script to work so that you can get to the next step of the project (and ultimately to the point of submitting a paper for publication). However, the fastest route to a goal is not always the straightest: If you focus too much on the immediate goal of getting the script to work, you may actually slow your progress toward the final goal of getting the paper submitted. It really pays to take your time when writing a script and write the code in a way that will be optimal in the long run.
In practice, this means following good coding practices that reduce the likelihood of errors, like defining all important values as variables at the top of the script. Errors can really slow you down if you don’t realize the error until you’re near the point of submitting the paper and now need to repeat all the analyses, change all the figures, and update the text of your paper. It’s also important to realize that you will probably need to come back to your script many months after you’ve written it (e.g., when you’re writing your Method section or you realize you need to reanalyze your data), and you will save yourself a lot of time if you write your code in a way that’s easy to read later.
There are two straightforward ways of making your code more readable. The first is to use variable names that have an obvious meaning. For example, I could have used something like ns as the name of the variable that holds the number of subjects, but instead I used num_subjects . The second is to add lots of comments. For examples, take a look at the example scripts I created for this chapter. Of course, these scripts were designed to be read by other people, so I put more work into that comments than I might have for a regular script that I wasn’t planning to share. But I still include tons of comments in scripts that I don’t plan to share. It’s a gift to my future self, because I know I will probably need to come back to those scripts after months or even years, and the comments will make my life much easier then. And when I come back to a script with good comments, I always try to thank my past self for the gift.
You should also keep in mind that it’s becoming more and more common to make your data and scripts available online when you publish a paper. This means that you’re never really writing scripts just for yourself. Other researchers are going to be looking at your scripts, so you don’t want the scripts to be embarrassing, and you want the other researchers to be able to understand your code. If you make the code easy to understand, this increases the likelihood that the other researchers will follow up on your research, which means that your research will have a larger impact. And aren’t you doing research so that it has an impact?
I’ve found that people often spend a huge amount of time polishing their scripts (making them more logical and adding lots of comments) right before they’re going to submit the paper for publication. They often find mistakes, and then they end up having to change their figures and the statistics in their Results sections. It’s really inefficient. It makes much more sense to write your scripts with the intent of sharing them—including clear logic and lots of comments—right from the beginning. This takes a lot of discipline, because when you’re writing the script you just want to get the job done. But this approach will save you a lot of time and agony in the long run.


4.4: Multiplication and Division in Base Systems
- Multiply and divide in bases other than 10.
- Identify errors in multiplying and dividing in bases other than 10.
Just as in Addition and Subtraction in Base Systems, once we decide on a system for counting, we need to establish rules for combining the numbers we’re using. This includes the rules for multiplication and division. We are familiar with those operations in base 10. How do they change if we instead use a different base? A larger base? A smaller one?
In this section, we use multiplication and division in bases other than 10 by referencing the processes of base 10, but applied to a new base system.
Multiplication in Bases Other Than 10
Multiplication is a way of representing repeated additions, regardless of what base is being used. However, different bases have different addition rules. In order to create the multiplication tables for a base other than 10, we need to rely on addition and the addition table for the base. So let’s look at multiplication in base 6.
Multiplication still has the same meaning as it does in base 10, in that is 4 added to itself six times, .
So, let’s apply that to base 6. It should be clear that 0 multiplied by anything, regardless of base, will give 0, and that 1 multiplied by anything, regardless of base, will be the value of “anything.”
Step 1: So, we start with the table below:
| * | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 0 | 2 | 4 | |||
| 3 | 0 | 3 | ||||
| 4 | 0 | 4 | ||||
| 5 | 0 | 5 |
Step 2: Notice
| * | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 0 | 2 | 4 | 10 | ||
| 3 | 0 | 3 | 10 | |||
| 4 | 0 | 4 | ||||
| 5 | 0 | 5 |
Step 3:
Next, we need to fill in
| * | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 0 | 2 | 4 | 10 | 12 | |
| 3 | 0 | 3 | 10 | |||
| 4 | 0 | 4 | 12 | |||
| 5 | 0 | 5 |
Step 4:
Finally,
| * | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 0 | 2 | 4 | 10 | 12 | 14 |
| 3 | 0 | 3 | 10 | |||
| 4 | 0 | 4 | 12 | |||
| 5 | 0 | 5 | 14 |
Step 5:
A similar analysis will give us the remainder of the entries. Here is
This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that
| * | 0 | 1 | 2 | 3 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 2 | 0 | 2 | 4 | 10 | 12 | 14 |
| 3 | 0 | 3 | 10 | 13 | 20 | 23 |
| 4 | 0 | 4 | 12 | 20 | 24 | 32 |
| 5 | 0 | 5 | 14 | 23 | 32 | 41 |
Notice anything about that bottom line? Is that similar to what happens in base 10?
To summarize the creation of a multiplication in a base other than base 10, you need the addition table of the base with which you are working. Create the table, and calculate the entries of the multiplication table by performing repeated addition in that base. The table needs to be drawn only the one time.
Create the multiplication table for base 7.
- Answer
-
Step 1: Let’s apply the process demonstrated and outlined above to find the base 7 multiplication table. It should be clear that 0 multiplied by anything, regardless of base, will give 0, and that 1 multiplied by anything, regardless of base, will be the value of “anything.” So, we start with the table below:
* 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 3 0 3 6 4 0 4 5 0 5 6 0 6 Step 2: Notice
2 × 2 = Example 4.29,4 . So, we add that to our table:6 + 2 = 11 6 + 2 = 11 * 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 11 3 0 3 6 4 0 4 11 5 0 5 6 0 6 Step 3: Next, we need to fill in
. Using repeated addition,2 × 5 2 × 5 if we use our base 7 addition rules. So, we add that to our table:2 × 5 = 2 + 2 + 2 + 2 + 2 = 11 + 2 = 13 2 × 5 = 2 + 2 + 2 + 2 + 2 = 11 + 2 = 13 * 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 11 13 3 0 3 6 4 0 4 11 5 0 5 13 6 0 6 Step 4: Finally,
. And so we add that to our table:2 × 6 = 2 + 2 + 2 + 2 + 2 + 2 + 2 = 13 + 2 = 15 2 × 6 = 2 + 2 + 2 + 2 + 2 + 2 + 2 = 13 + 2 = 15 * 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 11 13 15 3 0 3 6 4 0 4 11 5 0 5 13 6 0 6 15 Step 5: A similar analysis will give us the remainder of the entries. Here is
demonstrated:4 7 × 5 7 4 7 × 5 7 4 7 × 5 7 = 4 7 + 4 7 + 4 7 + 4 7 + 4 7 = 11 7 + 11 7 + 4 7 = 22 7 + 4 7 = 26 7 4 7 × 5 7 = 4 7 + 4 7 + 4 7 + 4 7 + 4 7 = 11 7 + 11 7 + 4 7 = 22 7 + 4 7 = 26 7 This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that
4 7 + 4 7 = Example 4.29. Using those addition rules, the rest of the table is given below:11 7 * 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 11 13 15 3 0 3 6 12 15 21 24 4 0 4 11 15 22 26 33 5 0 5 13 21 26 34 42 6 0 6 15 24 33 42 51
Create the multiplication table for base 4.
Create the multiplication table for base 12.
- Answer
-
Let’s apply the repeated addition to base 12. Here is
demonstrated:7 12 × 9 12 7 12 × 9 12 7 12 × 9 12 = 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 = 12 12 + 12 12 + 12 12 + 12 12 + 7 12 = 48 12 + 7 12 = 53 12 7 12 × 9 12 = 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 + 7 12 = 12 12 + 12 12 + 12 12 + 12 12 + 7 12 = 48 12 + 7 12 = 53 12 This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that
7 12 + 7 12 = Example 4.31. Using those addition rules, the rest of the table is given below:12 12 * 0 1 2 3 4 5 6 7 8 9 A B 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 A B 2 0 2 4 6 8 A 10 12 14 16 18 1A 3 0 3 6 9 10 13 16 19 20 23 26 29 4 0 4 8 10 14 18 20 24 28 30 34 38 5 0 5 A 13 18 21 26 2B 34 39 42 47 6 0 6 10 16 20 26 30 36 40 46 50 56 7 0 7 12 19 24 2B 36 41 48 53 5A 65 8 0 8 14 20 28 34 40 48 54 60 68 74 9 0 9 16 23 30 39 46 53 60 69 76 83 A 0 A 18 26 34 42 50 5A 68 76 84 92 B 0 B 1A 29 38 47 56 65 74 83 92 A1 Table 4.9
Create the multiplication table for base 14.
The multiplication table in base 2 below is as minimal as the addition table in the solution for Table 4.6. Since the product of 1 with anything is itself, the following multiplication table is found.
| * | 0 | 1 |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
As with the addition table, we can use the multiplication tables and the addition tables to perform multiplication of two numbers in bases other than base 10. The process is the same, with the same carry rules and placeholder rules.
-
Calculate
.45 6 × 24 6 45 6 × 24 6 -
Calculate
.101 2 × 110 2 101 2 × 110 2
- Answer
-
-
Step 1:
Use the base 6 multiplication table (Table 4.8) and, when necessary, the base 6 addition table (Table 4.4).
Set up this calculation using columns:
4 5 x 2 4 Step 2: Multiply the 1s digits, 5 and 4, using the base 6 multiplication table (Table 4.8). There we see the result is 32 6 . So, we enter the 2 and carry the 3.
3 4 5 x 2 4 2 Table 4.8]), then . So, we enter the 31.24 6 + 3 6 = 31 6 24 6 + 3 6 = 31 6 3 4 5 x 2 4 3 1 2 Step 4: Now we move on to the 2 in the “tens” place in the bottom value. We multiply the 2 6 and the 5 6 , and we get 14 6 . So, we enter the 4 and carry the 1.
Step 5: Next up, we multiply the 2 and the 4, and then add 1. This gives us
. We enter those on that second line.12 6 + 1 6 = 13 6 12 6 + 1 6 = 13 6 1 4 5 x 2 4 3 1 2 1 3 4 0 Step 6: Now we add down the columns.
1 4 5 x 2 4 1 3 1 2 1 3 4 0 2 0 5 2 Step 7: The 3 and the 3 add to 10 in base 6, so we enter the 0 and carry the 1. We now have the result:
45 6 × 24 6 = 2052 6 45 6 × 24 6 = 2052 62.
Step 1: Use the base 2 multiplication table (Table 4.10) and, when necessary, the base 2 addition table in the solution for Example 4.29. Set up this calculation using columns:
1 0 1 x 1 1 0 Step 2: Using the pattern established above, and the processes from multiplication from base 10, we find the following:
1 0 1 x 1 1 0 0 0 0 1 0 1 1 0 1 Step 3: Adding down the columns results in the following:
1 0 1 x 1 1 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 So,
.101 2 × 110 2 = 11110 2 101 2 × 110 2 = 11110 2
-
Step 1:
Use the base 6 multiplication table (Table 4.8) and, when necessary, the base 6 addition table (Table 4.4).
1. Calculate \(43_6 \times 52_6\).
2. Calculate \(11101_2 \times 11_2\).
Summarizing the process of multiplying two numbers in different bases, the multiplication table is referenced. Using that table, the multiplication is carried out in the same manner as it is in base 10. The addition rules for the base will also be referenced when carrying a 1 or when adding the results for each digit’s multiplication line.
Calculate
- Answer
-
Step 1: Use the base 12 multiplication table in the solution for Example 4.39 and, when necessary, the base 12 addition table in the solution for Table 4.7. Set up this calculation using columns:
3 A × 7 4 Step 2: First, the 4 is multiplied by 3A, resulting in the first line.
3 A × 7 4 1 3 4 Step 3: Now we move on to the 7 in the “tens” place in the bottom value.
5 3 A x 7 4 1 3 4 2 2 A 0 Step 4: Now we add down the columns.
3 A x 7 4 1 3 4 2 2 A 0 2 4 1 4 Step 5: The 3 and the A add to 11 in base 12, so we enter the 1 and carry the 1.
We now have the result:
3 A 12 × 74 12 = 2414 12 3 A 12 × 74 12 \(3 \mathrm{~A}_{12} \times 74_{12}=2414_{12}\)= 2414 12
Calculate \(\mathrm{B} 3_{12} \times 47_{12}\).
Division in Bases Other Than 10
Just as with the other operations,
division in a base
other than 10, the process of division in a base other than 10 is the same as the process when working in base 10. For instance,
-
Calculate
.14 6 ÷ 5 6 14 6 ÷ 5 6 -
Calculate
5 A 12 ÷ 7 12 5 A 12 ÷ 7 12
- Answer
-
-
Looking at the multiplication table for base 6 (Table 4.8), we see that
. Using that, we know that5 6 × 2 6 = 12 6 5 6 × 2 6 = 12 6 .14 6 ÷ 5 6 = 2 6 14 6 ÷ 5 6 = 2 6 -
Looking at the multiplication table for base 12 in the solution for Example 4.39, we see that
. Using that, we know that7 12 × A 12 = 5 A 12 7 12 × A 12 = 5 A 12 .5 A 12 ÷ 7 12 = A 12 5 A 12 ÷ 7 12 = A 12
-
Looking at the multiplication table for base 6 (Table 4.8), we see that
Calculate \(10_6 \div 3_6\)
Calculate \(50_{12} \div \mathrm{A}_{12}\).
Errors in Multiplying and Dividing in Bases Other Than Base 10
The types of errors encountered when multiplying and dividing in bases other than base 10 are the same as when adding and subtracting. They often involve applying base 10 rules or symbols to an arithmetic problem in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base.
Explain the error in the following calculation, and determine the correct answer:
- Answer
-
Since the problem is in base 6, the symbol set available is 0, 1, 2, 3, 4, and 5. The 8 in the answer is clearly not a legal symbol for base 6. Looking back to the base 6 multiplication table (Table 4.8), we see that
.4 6 × 2 6 = 12 6 4 6 × 2 6 = 12 6
Explain the error in the following calculation and determine the correct answer: \(13_4 \times 21_4=54_4\)
The second type of error is using a base 10 rule when the numbers are not in base 10. For instance, in base 17,
Explain the error in the following calculation. Determine the correct answer:
- Answer
-
If this problem was a base 10 problem, this would be the correct answer. However, in base 12, \(8_{12} \times 7_{12}\) is not 56 , but is instead 48 . To correct this error, carefully use the multiplication table for base 12. If properly used, the correct answer would be \(18_{12} \times 7_{12}=\mathrm{B} 8_{12}\).
Explain the error in the following calculation. Determine the correct answer: \(49_{14} \times 9_{14}=441_{14}\)
Check Your Understanding
- To create the multiplication table for a given base, what should be used?
- What are the differences between multiplying in base 10 and multiplying in a different base?
- When dividing in a base other than base 10 , what table is referenced?
- Compute \(24_6 \times 53_6\).
- Compute \(32_{14} \div 4_{14}\).
- How do you know an error has occurred in a base 5 multiplication question if the answer obtained was 285 ?
- What are two common ways to determine an error is committed when computing in s base other than base \(10 ?\)


4.11: IV Glossary
Advocacy: The act or process of pleading for, supporting, or recommending a cause or course of action. [1]
Art of nursing : Unconditionally acceptance of the humanity of others, respecting their need for dignity and worth, while providing compassionate, comforting care. [2]
At-risk populations: Groups of people who share a characteristic that causes each member to be susceptible to a particular human response, such as demographics, health/family history, stages of growth/development, or exposure to certain events/experiences. [3]
Associated conditions: Medical diagnoses, injuries, procedures, medical devices, or pharmacological agents. These conditions are not independently modifiable by the nurse, but support accuracy in nursing diagnosis. [4]
Basic nursing care: Care that can be performed following a defined nursing procedure with minimal modification in which the responses of the patient to the nursing care are predictable. [5]
Caring relationship: A relationship described as one in which the whole person is assessed while balancing the vulnerability and dignity of the patient and family. [6]
Client: Individual, family, or group, which includes significant others and populations. [7]
Clinical judgment: The observed outcome of critical thinking and decision-making. It is an iterative process that uses nursing knowledge to observe and access presenting situations, identify a prioritized client concern, and generate the best possible evidence-based solutions in order to deliver safe client care. [8]
Clinical reasoning: A complex cognitive process that uses formal and informal thinking strategies to gather and analyze patient information, evaluate the significance of this information, and weigh alternative actions. [9]
Clustering data: Grouping data into similar domains or patterns.
Collaborative nursing interventions : Nursing interventions that require cooperation among health care professionals and unlicensed assistive personnel (UAP).
Coordination of care: While implementing interventions during the nursing process, includes components such as organizing the components of the plan with input from the health care consumer, engaging the patient in self-care to achieve goals, and advocating for the delivery of dignified and person-centered care by the interprofessional team. [10]
Critical thinking: Reasoning about clinical issues such as teamwork, collaboration, and streamlining workflow. [11]
Cue: Subjective or objective data that gives the nurse a hint or indication of a potential problem, process, or disorder.
Deductive reasoning: “Top-down thinking” or moving from the general to the specific. Deductive reasoning relies on a general statement or hypothesis—sometimes called a premise or standard—that is held to be true. The premise is used to reach a specific, logical conclusion.
Defining characteristics: Observable cues/inferences that cluster as manifestations of a problem-focused, health-promotion diagnosis, or syndrome. This does not only imply those things that the nurse can see, but also things that are seen, heard (e.g., the patient/family tells us), touched, or smelled. [12]
Delegation: The assignment of the performance of activities or tasks related to patient care to unlicensed assistive personnel while retaining accountability for the outcome. [13]
Dependent nursing interventions : Interventions that require a prescription from a physician, advanced practice nurse, or physician’s assistant.
Direct care: Interventions that are carried out by having personal contact with a patient.
Electronic Medical Record (EMR): An electronic version of the patient’s medical record.
Evidence-Based Practice (EBP): A lifelong problem-solving approach that integrates the best evidence from well-designed research studies and evidence-based theories; clinical expertise and evidence from assessment of the health care consumer’s history and condition, as well as health care resources; and patient, family, group, community, and population preferences and values. [14]
Expected outcomes: Statements of measurable action for the patient within a specific time frame and in response to nursing interventions. “SMART” outcome statements are specific, measurable, action-oriented, realistic, and include a time frame.
Functional health patterns: An evidence-based assessment framework for identifying patient problems and risks during the assessment phase of the nursing process.
Generalization: A judgment formed from a set of facts, cues, and observations.
Goals: Broad statements of purpose that describe the aim of nursing care.
Health teaching and health promotion: Employing strategies to teach and promote health and wellness. [15]
Independent nursing interventions: Any intervention that the nurse can provide without obtaining a prescription or consulting anyone else.
Indirect care: Interventions performed by the nurse in a setting other than directly with the patient. An example of indirect care is creating a nursing care plan.
Inductive reasoning: A type of reasoning that involves forming generalizations based on specific incidents.
Inference: Interpretations or conclusions based on cues, personal experiences, preferences, or generalizations.
Licensed Practical Nurses or Licensed Vocational Nurses (LPNs/LVNs): Nurses who have had specific training and passed a licensing exam. The training is generally less than that of a Registered Nurse. The scope of practice of an LPN/LVN is determined by the facility and the state’s Nurse Practice Act.
Medical diagnosis: A disease or illness diagnosed by a physician or advanced health care provider such as a nurse practitioner or physician’s assistant. Medical diagnoses are a result of clustering signs and symptoms to determine what is medically affecting an individual.
Nursing: Nursing integrates the art and science of caring and focuses on the protection, promotion, and optimization of health and human functioning; prevention of illness and injury; facilitation of healing; and alleviation of suffering through compassionate presence. Nursing is the diagnosis and treatment of human responses and advocacy in the care of individuals, families, groups, communities, and populations in the recognition of the connection of all humanity. [16]
Nursing care plan: Specific documentation of the planning and delivery of nursing care that is required by The Joint Commission.
Nursing process: A systematic approach to patient-centered care with steps including assessment, diagnosis, outcome identification, planning, implementation, and evaluation; otherwise known by the mnemonic “ADOPIE.”
Objective data: Data that the nurse can see, touch, smell, or hear or is reproducible such as vital signs. Laboratory and diagnostic results are also considered objective data.
Outcome : A measurable behavior demonstrated by the patient that is responsive to nursing interventions. [17]
PES Statement: The format of a nursing diagnosis statement that includes:
- Problem (P) – statement of the patient problem (i.e., the nursing diagnosis)
- Etiology (E) – related factors (etiology) contributing to the cause of the nursing diagnosis
- Signs and Symptoms (S) – defining characteristics manifested by the patient of that nursing diagnosis
Prescription: Orders, interventions, remedies, or treatments ordered or directed by an authorized primary health care provider. [18]
Primary data: Information collected from the patient.
Primary health care provider: Member of the health care team (usually a medical physician, nurse practitioner, etc.) licensed and authorized to formulate prescriptions on behalf of the client. [19]
Prioritization: The skillful process of deciding which actions to complete first, second, or third for optimal patient outcomes and to improve patient safety.
Quality improvement: The “combined and unceasing efforts of everyone — health care professionals, patients and their families, researchers, payers, planners, and educators — to make the changes that will lead to better patient outcomes (health), better system performance (care), and better professional development (learning).” [20]
Rapport: Developing a relationship of mutual trust and understanding.
Registered Nurse (RN): A nurse who has had a designated amount of education and training in nursing and is licensed by a state Board of Nursing.
Related factors: The underlying cause (etiology) of a nursing diagnosis when creating a PES statement.
Right to self-determination : Patients have the right to determine what will be done with and to their own person.
Scientific method: Principles and procedures in the discovery of knowledge involving the recognition and formulation of a problem, the collection of data, and the formulation and testing of a hypothesis.
Secondary data : Information collected from sources other than the patient.
Subjective data: Data that the patient or family reports or data that the nurse makes as an inference, conclusion, or assumption, such as “The patient appears anxious.”
Unlicensed Assistive Personnel (UAP): Any unlicensed personnel trained to function in a supportive role, regardless of title, to whom a nursing responsibility may be delegated. [21]
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- Herdman, T. H., & Kamitsuru, S. (Eds.). (2018). Nursing diagnoses: Definitions and classification, 2018-2020 . Thieme Publishers New York. ↵
- Herdman, T. H., & Kamitsuru, S. (Eds.). (2018). Nursing diagnoses: Definitions and classification, 2018-2020. Thieme Publishers New York. ↵
- Wisconsin Administrative Code. (2018). Chapter N 6 standards of practice for registered nurses and licensed practical nurses. https://docs.legis.wisconsin.gov/code/admin_code/n/6.pdf ↵
- Walivaara, B., Savenstedt, S., & Axelsson, K. (2013). Caring relationships in home-based nursing care - registered nurses’ experiences. The Open Journal of Nursing, 7, 89-95. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3722540/pdf/TONURSJ-7-89.pdf ↵
- NCSBN. (n.d.). 2019 NCLEX-RN test plan . https://www.ncsbn.org/2019_RN_TestPlan-English.htm ↵
- NCSBN. (n.d.). NCSBN clinical judgment model . https://www.ncsbn.org/14798.htm ↵
- Klenke-Borgmann, L., Cantrell, M. A., & Mariani, B. (2020). Nurse educator’s guide to clinical judgment: A review of conceptualization, measurement, and development. Nursing Education Perspectives, 41 (4), 215-221. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (3rd ed.). American Nurses Association. ↵
- Klenke-Borgmann, L., Cantrell, M. A. , & Mariani, B. (2020). Nurse educator’s guide to clinical judgment: A review of conceptualization, measurement, and development. Nursing Education Perspectives, 41 (4), 215-221. ↵
- NANDA International. (n.d.). Glossary of terms. https://nanda.org/nanda-i-resources/glossary-of-terms / ↵
- American Nurses Association. (2013). ANA’s principles for delegation by registered nurses to unlicensed assistive personnel (UAP) . American Nurses Association. https://www.nursingworld.org/~4af4f2/globalassets/docs/ana/ethics/principlesofdelegation.pdf ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- Herdman, T. H., & Kamitsuru, S. (Eds.). (2018). Nursing diagnoses: Definitions and classification, 2018-2020. Thieme Publishers New York. ↵
- NCSBN. (n.d.). 2019 NCLEX-RN test plan . https://www.ncsbn.org/2019_RN_TestPlan-English.htm ↵
- NCSBN. (n.d.). 2019 NCLEX-RN test plan . https://www.ncsbn.org/2019_RN_TestPlan-English.htm ↵
- Batalden, P. B., & Davidoff, F. (2007). What is "quality improvement" and how can it transform healthcare?. BMJ Quality & Safety, 16 (1), 2–3. https://doi.org/10.1136/qshc.2006.022046 ↵
- NCSBN. (n.d.). 2019 NCLEX-RN test plan . https://www.ncsbn.org/2019_RN_TestPlan-English.htm ↵


4.S: Capacitance (Summary)
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Key Terms
| capacitance | amount of charge stored per unit volt |
| capacitor | device that stores electrical charge and electrical energy |
| dielectric | insulating material used to fill the space between two plates |
| dielectric breakdown | phenomenon that occurs when an insulator becomes a conductor in a strong electrical field |
| dielectric constant | factor by which capacitance increases when a dielectric is inserted between the plates of a capacitor |
| dielectric strength | critical electrical field strength above which molecules in insulator begin to break down and the insulator starts to conduct |
| energy density | energy stored in a capacitor divided by the volume between the plates |
| induced electric-dipole moment | dipole moment that a nonpolar molecule may acquire when it is placed in an electrical field |
| induced electrical field | electrical field in the dielectric due to the presence of induced charges |
| induced surface charges | charges that occur on a dielectric surface due to its polarization |
| parallel combination | components in a circuit arranged with one side of each component connected to one side of the circuit and the other sides of the components connected to the other side of the circuit |
| parallel-plate capacitor | system of two identical parallel conducting plates separated by a distance |
| series combination | components in a circuit arranged in a row one after the other in a circuit |
Key Equations
| Capacitance | \(\displaystyle C=\frac{Q}{V}\) |
| Capacitance of a parallel-plate capacitor | \(\displaystyle C=ε_0\frac{A}{d}\) |
| Capacitance of a vacuum spherical capacitor | \(\displaystyle C=4πε_0\frac{R_1R_2}{R_2−R_1}\) |
| Capacitance of a vacuum cylindrical capacitor | \(\displaystyle C=\frac{2πε_0l}{ln(R_2/R_1)}\) |
| Capacitance of a series combination | \(\displaystyle \frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+⋯\) |
| Capacitance of a parallel combination | \(\displaystyle C_P=C_1+C_2+C_3+⋯\) |
| Energy density | \(\displaystyle u_E=\frac{1}{2}ε_0E^2\) |
| Energy stored in a capacitor | \(\displaystyle U_C=\frac{1}{2}V^2C=\frac{1}{2}\frac{Q^2}{C}=\frac{1}{2}QV\) |
| Capacitance of a capacitor with dielectric | \(\displaystyle C=κC_0\) |
|
Energy stored in an isolated capacitor with dielectric |
\(\displaystyle U=\frac{1}{κ}U_0\) |
| Dielectric constant | \(\displaystyle κ=\frac{E_0}{E}\) |
| Induced electrical field in a dielectric | \(\displaystyle \vec{E_i}=(\frac{1}{κ}−1)\vec{E_0}\) |
Summary
8.2 Capacitors and Capacitance
- A capacitor is a device that stores an electrical charge and electrical energy. The amount of charge a vacuum capacitor can store depends on two major factors: the voltage applied and the capacitor’s physical characteristics, such as its size and geometry.
- The capacitance of a capacitor is a parameter that tells us how much charge can be stored in the capacitor per unit potential difference between its plates. Capacitance of a system of conductors depends only on the geometry of their arrangement and physical properties of the insulating material that fills the space between the conductors. The unit of capacitance is the farad, where \(\displaystyle 1F=1C/1V\).
8.3 Capacitors in Series and in Parallel
- When several capacitors are connected in a series combination, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances.
- When several capacitors are connected in a parallel combination, the equivalent capacitance is the sum of the individual capacitances.
- When a network of capacitors contains a combination of series and parallel connections, we identify the series and parallel networks, and compute their equivalent capacitances step by step until the entire network becomes reduced to one equivalent capacitance.
8.4 Energy Stored in a Capacitor
- Capacitors are used to supply energy to a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps.
- The energy stored in a capacitor is the work required to charge the capacitor, beginning with no charge on its plates. The energy is stored in the electrical field in the space between the capacitor plates. It depends on the amount of electrical charge on the plates and on the potential difference between the plates.
- The energy stored in a capacitor network is the sum of the energies stored on individual capacitors in the network. It can be computed as the energy stored in the equivalent capacitor of the network.
8.5 Capacitor with a Dielectric
- The capacitance of an empty capacitor is increased by a factor of \(\displaystyle κ\) when the space between its plates is completely filled by a dielectric with dielectric constant \(\displaystyle κ\).
- Each dielectric material has its specific dielectric constant.
- The energy stored in an empty isolated capacitor is decreased by a factor of κκ when the space between its plates is completely filled with a dielectric with dielectric constant \(\displaystyle κ\).
8.6 Molecular Model of a Dielectric
- When a dielectric is inserted between the plates of a capacitor, equal and opposite surface charge is induced on the two faces of the dielectric. The induced surface charge produces an induced electrical field that opposes the field of the free charge on the capacitor plates.
- The dielectric constant of a material is the ratio of the electrical field in vacuum to the net electrical field in the material. A capacitor filled with dielectric has a larger capacitance than an empty capacitor.
- The dielectric strength of an insulator represents a critical value of electrical field at which the molecules in an insulating material start to become ionized. When this happens, the material can conduct and dielectric breakdown is observed.