14.1: The New World
The New World
Indigenous visual arts traditions in the Americas span thousands of years, representing cultures from Mesoamerica to the Arctic.
Gain an appreciation for the breadth and diversity of indigenous peoples and cultures of the Americas
Key Points
- The New World refers to the western hemisphere, especially the Americas, after the European “age of discovery” beginning in the early 16th century. The indigenous peoples of the Americas are the pre-Columbian (before European contact) inhabitants of North America, Mesoamerica , and South America as well as Greenland.
- Scientists believe that the most recent migration of humans from Eurasia to the Americas took place around 12,000 years ago via a land bridge that connected the two continents.
- Indigenous populations across the Americas created monumental architecture, large-scale cities, chiefdoms, states, and empires.
- Indigenous Americans also created pottery, paintings, jewelry, weaving and textiles, sculptures , basketry, carvings, beadwork, and other objects that comprise a major category in world art history.
Key Terms
- Mesoamerica : A region and cultural area in the Americas, extending approximately from central Mexico to Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica, where pre-Columbian societies flourished before the Spanish colonization of the Americas in the 15th and 16th centuries.
- pre-Columbian : The inhabitants, societies, and culture of the Americas prior to European contact, colonization, and influence; literally “pre-(Christopher)Columbus.”
- Paleo-Indians : The first people to inhabit the Americas from Eurasia more than 11,000 years ago.
- New World : The term used by Europeans to describe the western hemisphere, specifically the Americas, during the “age of discovery” beginning in the early 16th century.
- indigenous peoples : People defined in international or national legislation as having a set of specific rights based on their historical ties to a particular territory, and their cultural or historical distinctiveness from other populations that are often politically dominant.
The New World refers to the western hemisphere, especially the Americas, which was almost entirely unknown to Europeans before the “age of discovery” beginning in the early 16th century. The Italian explorer Christopher Columbus (1451-1506) was one of the earliest and most well-known of these European explorers; the first of his four famous voyages from Spain to the Americas began in 1492.
The indigenous peoples of the Americas are the pre-Columbian inhabitants of North America, Mesoamerica, and South America as well as Greenland. There are almost as many terms for indigenous people in the Americas as there are geographic regions. For example, “pueblos indígenas” is a common term in Spanish-speaking countries such as Mexico, El Salvador, and Nicaragua. “Aborigen” is used in Argentina; “Amerindian” is used in Guyana. Indigenous peoples are commonly known in Canada as Aboriginal peoples, which includes First Nations, Inuit , and Métis peoples. Indigenous peoples of the United States are commonly known as Native Americans or American Indians, and Alaskan Natives.
Scientists believe that migrations of humans from Eurasia (the combined continental landmass of Europe and Asia) to the Americas first took place via Beringia, a land bridge which formerly connected the two continents across what is now the Bering Strait. The most recent migration probably took place around 12,000 years ago, but the earliest period remains somewhat of a mystery. These early Paleo-Indians soon spread throughout the continent, diversifying into many hundreds of culturally distinct nations and tribes. According to the oral histories of many indigenous peoples of the Americas, they have been living there since their genesis, represented in a wide range of traditional creation stories.
Indigenous Cultures
While some indigenous peoples of the Americas were traditionally hunter-gatherers — and many, especially in Amazonia, still are — many groups practiced aquaculture and agriculture. The impact of their agricultural endowment to the world is a testament to their time and work in reshaping and cultivating the flora indigenous to the Americas. While some societies depended heavily on agriculture, others practiced a mix of farming, hunting, and gathering. In some regions, the indigenous peoples created monumental architecture, large-scale cities, chiefdoms (with hierarchies based on kinship), states, and empires. Many parts of the Americas are still populated by indigenous Americans, and some countries have sizable populations, including Belize, Bolivia, Colombia, Ecuador, Greenland, Guatemala, Mexico, and Peru.
At least a thousand different indigenous languages are spoken in the Americas. Some, such as Quechua languages, Aymara, Guaraní, Mayan languages, and Nahuatl, count their speakers in millions. Many indigenous people also maintain aspects of their traditional cultural practices, including religion, social organization, and subsistence economies. Some indigenous peoples still live in relative isolation from Western society.
Cultural practices in the Americas seem to have been shared mostly within geographical zones where unrelated peoples adopted similar technologies and social organizations. An example of such a cultural area is Mesoamerica, where millennia of coexistence and shared development among the peoples of the region produced a fairly homogeneous culture with complex agricultural and social patterns. Another well-known example is the North American plains where until the 19th century, Native American groups such as Blackfoot, Crow,and Sioux existed as nomadic hunter-gatherers (primarily buffalo hunting).
Indigenous visual arts traditions in the Americas span thousands of years, representing cultures from Mesoamerica to the Arctic. Visual arts by indigenous peoples of the Americas comprise a major category in world art history. Their contributions include pottery, paintings, jewelry, weavings and textiles, sculptures, basketry, carvings, and beadwork. Much of this artwork provides insight into the values , beliefs, and ceremonial rituals of early cultures of the Americas. In the following chapter, we will examine in detail the artworks of indigenous groups throughout North and South America prior to 1300.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Map_of_America_by_Sebastian_Munster.JPG. Provided by : Wikipedia. Located at : https://commons.wikimedia.org/wiki/File:Map_of_America_by_Sebastian_Munster.JPG . License : Public Domain: No Known Copyright
- Ceramics of indigenous peoples of the Americas. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Ceramics_of_indigenous_peoples_of_the_Americas. License : Public Domain: No Known Copyright
- Mesoamerica. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Mesoamerica . License : CC BY-SA: Attribution-ShareAlike
- History of Mesoamerica (Paleo-Indian). Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/History_of_Mesoamerica_(Paleo-Indian) . License : CC BY-SA: Attribution-ShareAlike
- Visual arts by indigenous peoples of the Americas. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Visual_arts_by_indigenous_peoples_of_the_Americas. License : CC BY-SA: Attribution-ShareAlike
- Ceramics of indigenous peoples of the Americas. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Ceramics_of_indigenous_peoples_of_the_Americas. License : CC BY-SA: Attribution-ShareAlike
- Indigenous peoples of the Americas. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Indigenous_peoples_of_the_Americas%23Culture. License : CC BY-SA: Attribution-ShareAlike
- New World. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/New_World. License : CC BY-SA: Attribution-ShareAlike


11.5: Counting Principles
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Learning Objectives
- Solve counting problems using the Addition Principle.
- Solve counting problems using the Multiplication Principle.
- Solve counting problems using permutations involving n distinct objects.
- Solve counting problems using combinations.
- Find the number of subsets of a given set.
- Solve counting problems using permutations involving n non-distinct objects.
A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.
Using the Addition Principle
The company that sells customizable cases offers cases for tablets and smartphones. There are \(3\) supported tablet models and \(5\) supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are \(8\) total options, as we can see in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\)
THE ADDITION PRINCIPLE
According to the Addition Principle , if one event can occur in \(m\) ways and a second event with no common outcomes can occur in \(n\) ways, then the first or second event can occur in \(m+n\) ways.
Example \(\PageIndex{1}\): Using the Addition Principle
There are \(2\) vegetarian entrée options and \(5\) meat entrée options on a dinner menu. What is the total number of entrée options?
Solution
We can add the number of vegetarian options to the number of meat options to find the total number of entrée options.
There are \(7\) total options.
Exercise \(\PageIndex{1}\)
A student is shopping for a new computer. He is deciding among \(3\) desktop computers and \(4\) laptop computers. What is the total number of computer options?
- Answer
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\(7\)
Using the Multiplication Principle
The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are \(2\) appetizer options, \(3\) entrée options, and \(2\) dessert options on a fixed-price dinner menu, there are a total of \(12\) possible choices of one each as shown in the tree diagram in Figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\)
The possible choices are:
- soup, chicken, cake
- soup, chicken, pudding
- soup, fish, cake
- soup, fish, pudding
- soup, steak, cake
- soup, steak, pudding
- salad, chicken, cake
- salad, chicken, pudding
- salad, fish, cake
- salad, fish, pudding
- salad, steak, cake
- salad, steak, pudding We can also find the total number of possible dinners by multiplying.
We could also conclude that there are \(12\) possible dinner choices simply by applying the Multiplication Principle.
\(\# \text{of appetizer options} \times \# \text {of entree options} \times \# \text {of dessert options}\)
\(2 × 3 × 2=12\)
THE MULTIPLICATION PRINCIPLE
According to the Multiplication Principle, if one event can occur in \(m\) ways and a second event can occur in \(n\) ways after the first event has occurred, then the two events can occur in \(m×n\) ways. This is also known as the Fundamental Counting Principle.
Example \(\PageIndex{2}\): Using the Multiplication Principle
Diane packed \(2\) skirts, \(4\) blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.
Solution
To find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.
There are \(16\) possible outfits.
Exercise \(\PageIndex{2}\)
A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are \(3\) types of breakfast sandwiches, \(4\) side dish options, and \(5\) beverage choices. Find the total number of possible breakfast specials.
- Answer
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There are \(60\) possible breakfast specials.
Finding the Number of Permutations of \(n\) Distinct Objects
The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.
Finding the Number of Permutations of \(n\) Distinct Objects Using the Multiplication Principle
To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.
There are four options for the first place, so we write a \(4\) on the first line.
After the first place has been filled, there are three options for the second place so we write a \(3\) on the second line.
After the second place has been filled, there are two options for the third place so we write a \(2\) on the third line. Finally, we find the product.
There are \(24\) possible permutations of the paintings.
How to: Given \(n\) distinct options, determine how many permutations there are.
- Determine how many options there are for the first situation.
- Determine how many options are left for the second situation.
- Continue until all of the spots are filled.
- Multiply the numbers together.
Example \(\PageIndex{3}\): Finding the Number of Permutations Using the Multiplication Principle
At a swimming competition, nine swimmers compete in a race.
- How many ways can they place first, second, and third?
- How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)
- How many ways can all nine swimmers line up for a photo?
Solution
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Draw lines for each place.
There are \(9\) options for first place. Once someone has won first place, there are \(8\) remaining options for second place. Once first and second place have been won, there are \(7\) remaining options for third place.
Multiply to find that there are \(504\) ways for the swimmers to place.
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Draw lines for describing each place.
We know Ariel must win first place, so there is only \(1\) option for first place. There are \(8\) remaining options for second place, and then \(7\) remaining options for third place.
Multiply to find that there are \(56\) ways for the swimmers to place if Ariel wins first.
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Draw lines for describing each place in the photo.
There are \(9\) choices for the first spot, then \(8\) for the second, \(7\) for the third, \(6\) for the fourth, and so on until only \(1\) person remains for the last spot.
There are \(362,880\) possible permutations for the swimmers to line up.
Analysis
Note that in part c, we found there were \(9!\) ways for \(9\) people to line up. The number of permutations of \(n\) distinct objects can always be found by \(n!\).
Exercise \(\PageIndex{3A}\)
A family of five is having portraits taken. Use the Multiplication Principle to find the following:
- How many ways can the family line up for the portrait?
- How many ways can the photographer line up \(3\) family members?
- How many ways can the family line up for the portrait if the parents are required to stand on each end?
- Answer a
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\(120\)
- Answer b
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\(60\)
- Answer c
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\(12\)
Finding the Number of Permutations of n Distinct Objects Using a Formula
For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of \(n\) objects and we want to choose \(r\) objects from the set in order, we write \(P(n,r)\). Another way to write this is \(nP_r\), a notation commonly seen on computers and calculators. To calculate \(P(n,r)\), we begin by finding \(n!\), the number of ways to line up all nn objects. We then divide by \((n−r)!\) to cancel out the \((n−r)\) items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is \(6×5×4=120\). Using factorials, we get the same result.
\(\dfrac{6!}{3!}=\dfrac{6·5·4·3!}{3!}=6·5·4=120 \)
There are \(120\) ways to select \(3\) officers in order from a club with \(6\) members. We refer to this as a permutation of \(6\) taken \(3\) at a time. The general formula is as follows.
\[P(n,r)=\dfrac{n!}{(n−r)!}\]
Note that the formula stills works if we are choosing all \(n\) objects and placing them in order. In that case we would be dividing by \((n−n)!\) or \(0!\), which we said earlier is equal to \(1\). So the number of permutations of \(n\) objects taken \(n\) at a time is \(\dfrac{n!}{1}\) or just \(n!\).
Note: FORMULA FOR PERMUTATIONS OF N DISTINCT OBJECTS
Given \(n\) distinct objects, the number of ways to select \(r\) objects from the set in order is
\[P(n,r)=\dfrac{n!}{(n−r)!}\]
How to: Given a word problem, evaluate the possible permutations.
- Identify \(n\) from the given information.
- Identify \(r\) from the given information.
- Replace \(n\) and \(r\) in the formula with the given values.
- Evaluate.
Example \(\PageIndex{4}\): Finding the Number of Permutations Using the Formula
A professor is creating an exam of \(9\) questions from a test bank of \(12\) questions. How many ways can she select and arrange the questions?
Solution
Substitute \(n=12\) and \(r=9\) into the permutation formula and simplify.
\[\begin{align*} P(n,r)&=\dfrac{n!}{(n-r)!}\\ P(12,9)&=\dfrac{12!}{(12-9)!}\\ &=\dfrac{12!}{3!}\\ &=79,833,600 \end{align*}\]
There are \(79,833,600\) possible permutations of exam questions!
Analysis
We can also use a calculator to find permutations. For this problem, we would enter \(12\), press the \(nP_r\) function, enter \(9\), and then press the equal sign. The \(nP_r\) function may be located under the MATH menu with probability commands.
Q&A
Could we have solved the example above using the Multiplication Principle?
Yes. We could have multiplied \(15⋅14⋅13⋅12⋅11⋅10⋅9⋅8⋅7⋅6⋅5⋅4\) to find the same answer.
Exercise \(\PageIndex{4A}\)
A play has a cast of \(7\) actors preparing to make their curtain call. Use the permutation formula to find the following.
How many ways can the \(7\) actors line up?
- Answer
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\(P(7,7)=5,040\)
Exercise \(\PageIndex{4B}\)
How many ways can \(5\) of the \(7\) actors be chosen to line up?
- Answer
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\(P(7,5)=2,520\)
Find the Number of Combinations Using the Formula
So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations . A selection of \(r\) objects from a set of \(n\) objects where the order does not matter can be written as \(C(n,r)\). Just as with permutations, \(C(n,r)\) can also be written as \(nC_r\). In this case, the general formula is as follows.
\[C(n,r)=\dfrac{n!}{r!(n−r)!} \label{combo}\]
An earlier problem considered choosing \(3\) of \(4\) possible paintings to hang on a wall. We found that there were \(24\) ways to select \(3\) of the \(4\) paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings \(1, 2, 3\) would be the same as selecting paintings \(2, 3, 1\). To find the number of ways to select \(3\) of the \(4\) paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order \(3\) paintings. There are \(3!=3·2·1=6\) ways to order \(3\) paintings. There are \(\dfrac{24}{6}\), or \(4\) ways to select \(3\) of the \(4\) paintings. This number makes sense because every time we are selecting \(3\) paintings, we are not selecting \(1\) painting. There are \(4\) paintings we could choose not to select, so there are \(4\) ways to select \(3\) of the \(4\) paintings.
Note: FORMULA FOR COMBINATIONS OF N DISTINCT OBJECTS
Given \(n\) distinct objects, the number of ways to select \(r\) objects from the set is
\[C(n,r)=\dfrac{n!}{r!(n−r)!}\]
How to: Given a number of options, determine the possible number of combinations.
- Identify \(n\) from the given information.
- Identify \(r\) from the given information.
- Replace \(n\) and \(r\) in the formula with the given values.
- Evaluate.
Example \(\PageIndex{5}\): Finding the Number of Combinations Using the Formula
A fast food restaurant offers five side dish options. Your meal comes with two side dishes.
- How many ways can you select your side dishes?
- How many ways can you select \(3\) side dishes?
Solution
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We want to choose \(2\) side dishes from \(5\) options.
\(C(5,2)=\dfrac{5!}{2!(5−2)!}=10\)
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We want to choose \(3\) side dishes from \(5\) options.
\(C(5,3)=\dfrac{5!}{3!(5−3)!}=10\)
Analysis
We can also use a graphing calculator to find combinations. Enter 5, then press \(nC_r\), enter 3, and then press the equal sign. The \(nC_r\), function may be located under the MATH menu with probability commands.
Q&A
Is it a coincidence that parts (a) and (b) in the example above have the same answers?
No. When we choose r objects from n objects, we are not choosing \((n–r)\) objects. Therefore, \(C(n,r)=C(n,n–r)\).
Exercise \(\PageIndex{5}\)
An ice cream shop offers \(10\) flavors of ice cream. How many ways are there to choose \(3\) flavors for a banana split?
- Answer
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\(C(10,3)=120\)
Finding the Number of Subsets of a Set
We have looked only at combination problems in which we chose exactly rr objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers \(5\) toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is \(C(5,0)=1\) way to order a pizza with no toppings. There are \(C(5,1)=5\) ways to order a pizza with exactly one topping. If we continue this process, we get
\(C(5,0)+C(5,1)+C(5,2)+C(5,3)+C(5,4)+C(5,5)=32\)
There are \(32\) possible pizzas. This result is equal to \(2^5\).
We are presented with a sequence of choices. For each of the nn objects we have two choices: include it in the subset or not. So for the whole subset we have made nn choices, each with two options. So there are a total of \(2·2·2·…·2\) possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.
Note: FORMULA FOR THE NUMBER OF SUBSETS OF A SET
A set containing n distinct objects has \(2^n\) subsets.
Example \(\PageIndex{6}\): Finding the Number of Subsets of a Set
A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?
Solution
We are looking for the number of subsets of a set with \(4\) objects. Substitute \(n=4\) into the formula.
\[\begin{align*} 2^n&=2^4\\ &=16 \end{align*}\]
There are \(16\) possible ways to order a potato.
Exercise \(\PageIndex{6}\)
A sundae bar at a wedding has \(6\) toppings to choose from. Any number of toppings can be chosen. How many different sundaes are possible?
- Answer
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\(64\) sundaes
Finding the Number of Permutations of n Non-Distinct Objects
We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of \(12\) stickers. If all of the stickers were distinct, there would be \(12!\) ways to order the stickers. However, \(4\) of the stickers are identical stars, and \(3\) are identical moons. Because all of the objects are not distinct, many of the \(12!\) permutations we counted are duplicates. The general formula for this situation is as follows.
\[\dfrac{n!}{r_1!r_2!…r_k!}\]
In this example, we need to divide by the number of ways to order the \(4\) stars and the ways to order the \(3\) moons to find the number of unique permutations of the stickers.
Note: FORMULA FOR FINDING THE NUMBER OF PERMUTATIONS OF \(N\) NON-DISTINCT OBJECTS
If there are nn elements in a set and \(r_1\) are alike, \(r_2\) are alike, \(r_3\) are alike, and so on through \(r_k\), the number of permutations can be found by
\[\dfrac{n!}{r_1!r_2!…r_k!}\]
Example \(\PageIndex{7}\): Finding the Number of Permutations of \(n\) Non-Distinct Objects
Find the number of rearrangements of the letters in the word DISTINCT.
Solution
There are \(8\) letters. Both \(I\) and \(T\) are repeated \(2\) times. Substitute \(n=8\), \(r_1=2\), and \(r_2=2\) into the formula.
\(\dfrac{8!}{2!2!}=10,080\)
There are \(10,080\) arrangements.
Exercise \(\PageIndex{7}\)
Find the number of rearrangements of the letters in the word CARRIER.
- Answer
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\(840\)
Media
Access these online resources for additional instruction and practice with combinations and permutations.
Key Equations
| number of permutations of \(n\) distinct objects taken \(r\) at a time | \(P(n,r)=\dfrac{n!}{(n−r)!}\) |
| number of combinations of \(n\) distinct objects taken \(r\) at a time | \(C(n,r)=\dfrac{n!}{r!(n−r)!}\) |
| number of permutations of \(n\) non-distinct objects | \(\dfrac{n!}{r1!r2!…rk!}\) |
Key Concepts
- If one event can occur in \(m\) ways and a second event with no common outcomes can occur in \(n\) ways, then the first or second event can occur in \(m+n\) ways. See Example \(\PageIndex{1}\).
- If one event can occur in \(m\) ways and a second event can occur in \(n\) ways after the first event has occurred, then the two events can occur in \(m×n\) ways. See Example \(\PageIndex{2}\).
- A permutation is an ordering of \(n\) objects.
- If we have a set of \(n\) objects and we want to choose \(r\) objects from the set in order, we write \(P(n,r)\).
- Permutation problems can be solved using the Multiplication Principle or the formula for \(P(n,r)\). See Example \(\PageIndex{3}\) and Example \(\PageIndex{4}\).
- A selection of objects where the order does not matter is a combination.
- Given \(n\) distinct objects, the number of ways to select \(r\) objects from the set is \(C(n,r)\) and can be found using a formula. See Example \(\PageIndex{5}\).
- A set containing \(n\) distinct objects has \(2n\) subsets. See Example \(\PageIndex{6}\).
- For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See Example \(\PageIndex{7}\).
Contributors and Attributions
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Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


7.4: Modeling with Exponential Functions
Students will be able to:
- Make predictions by using the exponential growth (or decay) model
- Use the doubling-time and half-life models to make predictions
- Estimate the doubling time and half-life by using the Rule of 70
in order to apply mathematical modeling to solve real-world applications.
The next growth we will examine is exponential growth. Linear growth occurs by adding the same amount in each unit of time. Exponential growth happens when an initial population increases by the same percentage or factor over equal time increments or generations. This is known as relative growth and is usually expressed as a percentage. For example, let’s say a population is growing by 5% each year. This means that 5% of the current population is added each year to the total. If the population was originally 10,000 in the first year, then 5% of 10,000 would be added which is 500, giving a new population of 10,500. In the second year, another 5% would be added, but this time you would take 5% of 10,500 which is 525, giving a new population of 11,025. While the percentage growth is the same each year, the actual number of people added changes, which makes this growth different from the linear model discussed earlier!
A quantity grows exponentially if it grows by a constant factor or rate for each unit of time.
In this graph, the blue straight line represents linear growth and the red curved line represents exponential growth.
A city is growing at a rate of 1.6% per year. The initial population in 2010 is \(P_{0} = 125,000\). Calculate the city’s population over the next few years.
Solution
The relative growth rate is 1.6%. This means an additional 1.6% is added on to 100% of the population that already exists each year. This is a growth factor of 101.6%.
- Population in 2011 \[125,000(1.016)^{1} = 127,000 \nonumber\]
- Population in 2012 \[127,000(1.016) = 125,000(1.016)^{2} = 129,032 \nonumber\]
- Population in 2013 \[129,032(1.016) = 125,000(1.016)^{3} = 131,097 \nonumber\]
We can create an equation for the city’s growth. Each year the population is 101.6% more than the previous year.
\[P(t) = 125,000(1+0.016)^{t} \nonumber\]
Note: The graph of the city growth follows an exponential growth model
St. Louis, Missouri has declined in population at a rate of 1.6 % per year over the last 60 years. The population in 1950 was 857,000. Find the population in 2014. (Wikipedia, n.d.)
Solution
\(P_{0} = 857,000\)
The relative rate of decrease (known as a decay rate) is 1.6%. This means 1.6% of the population is subtracted from 100% of the population that already exists each year. This is a factor of 98.4%.
- Population in 1951 \[857,000(0.984)^{1} = 843,288\nonumber\]
- Population in 1952 \[843,228(0.984) = 857,000(0.984)^{2} = 829,795\nonumber\]
- Population in 1953 \[828,795(0.984) = 857,000(0.984)^{3} = 816,519\nonumber\]
We can create an equation for the city’s growth. Each year the population is 1.6% less than the previous year.
\[P(t) = 857,000(1-0.016)^{t}\]
So the population of St. Louis Missouri in 2014, when \(t = 64\), is:
\[\begin{align*} P(64) &= 857,000(1-0.016)^{64} \\[4pt] &=857,000(0.984)^{64} \\[4pt] &= 305,258 \end{align*}\]
Note: The graph of the population of St. Louis, Missouri over time follows a declining exponential growth model.
\(\begin{align*}&P(t) = P_{0}(1+r)^{t} \\[4pt]&P_{0} \text{ is the initial population.} \\[4pt]&r \text{ is the relative growth or decay rate}. \\[4pt]&(1+r) \text{ is the growth multiplier}. \\[4pt]&t \text{ is the time unit.} \\[4pt]&r \text{ is positive if the population is increasing (in cases of growth) and negative if the population is decreasing (in cases of decay).} \end{align*}\)
The average inflation rate of the U.S. dollar over the last five years is 1.7% per year. If a new car cost $18,000 five years ago, how much would it cost today? (U.S. Inflation Calculator, n.d.)
Solution
To solve this problem, we use the exponential growth model with r = 1 .7%.
\(P_{0} = 18,000\) and \(t=5\)
\[\begin{align*}P(t) &= 18,000(1+0.017)^{t} \\[4pt] P(5) &= 18,000(1+0.017)^{5} = 19,582.91 \end{align*}\]
This car would cost \($19,582.91\) today.
In May of 2014 there were 15 cases of Ebola in Sierra Leone. By August, there were 850 cases. If the virus is spreading at the same rate (exponential growth), how many cases will there be in February of 2015? (McKenna, 2014)
Solution
To solve this problem, we have to find three things; the growth rate per month, the exponential growth model, and the number of cases of Ebola in February 2015. First calculate the growth rate per month. To do this, use the initial population\(P_{0} = 15\), in May 2014. Also, in August, three months later, the number of cases was 850 so,\(P(3) = 850\).
Use these values and the exponential growth model to solve for \(r\).
\[\begin{align*}P(t) &= P{0}(1+r)^{t} \\[4pt] 850 &= 15(1+r)^{3} \\[4pt] 56.67 &= (1+r)^{3} \\[4pt] \sqrt[3]{56.67} &= \sqrt[3]{(1+r)^{3}} \\[4pt] 3.84 &= 1+r \\[4pt] 2.84 &= r \end{align*}\]
The growth rate is 284% per month. Thus, the exponential growth model is:
\[P(t)=15(1+2.84)^{t} = 15(3.84)^{t} \nonumber\]
Now, we use this to calculate the number of cases of Ebola in Sierra Leone in February 2015, which is 9 months after the initial outbreak so, \(t=9\)
\[P(9)= 15(3.84)^{9} = 2,725,250\]
If this same exponential growth rate continues, the number of Ebola cases in Sierra Leone in February 2015 would be 2,725,250.
This is a bleak prediction for the community of Sierra Leone. Fortunately, the growth rate of this deadly virus should be reduced by the world community and World Health Organization by providing the needed means to fight the initial spread.
Note: The graph of the number of possible Ebola cases in Sierra Leone over time follows an increasing exponential growth model.
According to a new forecast, the population of Puerto Rico is in decline. If the population in 2010 is 3,978,000 and the prediction for the population in 2050 is 3,697,000, find the annual percent decrease rate. (Bloomberg Businessweek, n.d.)
Solution
To solve this problem we use the exponential growth model . We need to solve for r.
\[\begin{align*}P(t) &= P_{0}(1+r)^{t} \text{where } t = 40 \text{ years}\\[4pt] P(40) &= 3,697,000 \text{and} P_{0} = 3,978,000 \\[4pt] 3,697,000 &= 3,978,000(1+r)^{40} \\[4pt] 0.92936 &= (1+r)^{40} \\[4pt] \sqrt[40]{0.92936} &= \sqrt[40]{(1+r)^{40}} \\[4pt] 0.99817 &= 1+r \\[4pt] -0.0018 &= r \end{align*}\]
The annual percent decrease is 0.18%.
Let’s say that on April 1st I say I will give you a penny, on April 2nd two pennies, four pennies on April 3rd, and that I will double the amount each day until the end of the month. How much money would I have agreed to give you on April 30? With \(P_{0} = $0.01\), we get the following table:
|
Day |
Dollar Amount |
|---|---|
|
April 1 = \(P_{0}\) |
0.01 |
|
April 2 = \(P_{1}\) |
0.02 |
|
April 3 = \(P_{2}\) |
0.04 |
|
April 4 = \(P_{3}\) |
0.08 |
|
April 5 = \(P_{4}\) |
0.16 |
|
April 6 = \(P_{5}\) |
0.32 |
|
…. |
…. |
|
April 30 = \(P_{29}\) |
? |
In this example, the money received each day is 100% more than the previous day. If we use the exponential growth model \(P(t) = P_{0}(1+r)^{t}\) with r = 1, we get the doubling time model.
\[P(t) = P_{0}(1+1)^{t} = P_{0}(2)^{t} \nonumber\]
We use it to find the dollar amount when \(t = 29\) which represents April 30
\[P(29) = 0.01(2)^{29} = $5,368,709.12 \nonumber\]
Surprised? That is a lot of pennies.
Doubling Time Model
A water tank up on the San Francisco Peaks is contaminated with a colony of 80,000 E. coli bacteria. The population doubles every five days. We want to find a model for the population of bacteria present after \(t\) days. The amount of time it takes the population to double is five days, so this is our time unit. After \(t\) days have passed, then \(t/5\) is the number of time units that have passed. Starting with the initial amount of 80,000 bacteria, our doubling model becomes:
\[P(t) = 80,000(2)^{\frac{t}{5}} \nonumber\]
Using this model, how large is the colony in two weeks’ time? We have to be careful that the units on the times are the same; 2 weeks = 14 days.
Solution
\[P(14) = 80,000(2)^{\frac{14}{5}} = 557,152 \nonumber\]
The colony is now 557,152 bacteria.
If \(D\) is the doubling time of a quantity (the amount of time it takes the quantity to double) and\(P_{0}\) is the initial amount of the quantity then the amount of the quantity present after \(t\) units of time is \(P(t) = P_{0}(2)^{\frac{t}{D}}\)
The doubling time of a population of flies is eight days. If there are initially 100 flies, how many flies will there be in 17 days?
Solution
To solve this problem, use the doubling time model with \(D=8\) and \(P_{0} = 100\) so the doubling time model for this problem is:
\[P(t) = 100(2)^{t/8} \nonumber\]
When \(t = 17\, days\),
\[P(17) = 100(2)^{\frac{17}{8}} = 436 \nonumber\]
There are 436 flies after 17 days.
Note: The population of flies follows an exponential growth model.
Suppose that a bacteria population doubles every six hours. If the initial population is 4000 individuals, how many hours would it take the population to increase to 25,000?
Solution
\(P_{0} = 4000\) and \(D = 6\), so the doubling time model for this problem is:
\[P(t) = 4000(2)^{\frac{t}{6}}\nonumber\]
Now, find t when \(P(t) = 25,000\)
\[25,000 = 4000(2)^{\frac{t}{6}}\nonumber\]
\[\dfrac{25,000}{4000} = \dfrac{\cancel{4000}(2)^{\frac{t}{6}}}{\cancel{4000}}\nonumber\]
\[6.25 = (2)^{\frac{t}{6}}\nonumber\]
Now, we know that the exponent on the base of 2 has to be between 2 and 3 since
\[(2)^{2} = 4\nonumber\]
\[(2)^{3} = 8\nonumber\]
Since the exponent on the base of 2 is \[\frac{t}{6}\] this means t must be between 12 and 18 to ensure the exponent is between 2 and 3. By guessing different values of t, we can estimate the answer until we find a value of t that gives the proper results when substituted into the exponent. The best estimate for t is 16 since:
\[(2)^{\frac{t}{6}} = (2)^{\frac{16}{6}} = 6.3496 \nonumber\]
Which is very close to 6.25. Using technology would allow us to get a more accurate answer. Our conclusion is it would take approximately 16 hours for the population of the bacteria to increase to 25,000.
A certain cancerous tumor doubles in size every six months. If the initial size of the tumor is four cells, how many cells will there in three years? In seven years?
Solution
To calculate the number of cells in the tumor, we use the doubling time model. Change the time units to be the same. The doubling time is six months = 0.5 years.
\[P(t) = 4(2)^{\frac{t}{0.5}} \nonumber\]
When \(t = 3\) years
\[P(3) = 4(2)^{\frac{3}{0.5}} = 256 \text{cells} \nonumber\]
When \(t = 7\) years
\[P(7) = 4(2)^{\frac{7}{0.5}} = 65,536 \text{cells} \nonumber\]
Rule of 70
There is a simple formula for approximating the doubling time of a population. It is called the rule of 70 and it is an approximation for growth rates less than 15%. Do not use this formula if the growth rate is 15% or greater.
For a quantity growing at a constant percentage rate (not written as a decimal), \(R\), per time period, the doubling time is approximately given by
\[D \approx \dfrac{70}{R}\]
A bird population on a certain island has an annual growth rate of 2.5% per year. Approximate the number of years it will take the population to double. If the initial population is 20 birds, use it to find the bird population of the island in 17 years.
Solution
To solve this problem, first approximate the population doubling time.
Doubling time \(D \approx \dfrac{70}{2.5} = 28\) years.
With the bird population doubling in 28 years, we use the doubling time model to find the population is 17 years.
\[P(t) = 20(2)^{\frac{t}{28}} \nonumber\]
When \(t = 17\) years
\[P(16) = 20(2)^{\frac{17}{28}} = 30.46 \nonumber\]
There will be 30 birds on the island in 17 years.
Suppose that a certain city’s population doubles every 12 years. What is the approximate annual growth rate of the city?
Solution
By solving the doubling time model for the growth rate, we can solve this problem.
\[\begin{align*} D &\approx \dfrac{70}{R} \\ R \cdot D &\approx \dfrac{70}{\cancel{R}} \cdot \cancel{R} \\ RD &\approx 70 \\ \dfrac{R\cancel{D}}{D} &\approx \dfrac{70}{D} \\ \text{Annual growth rate R} &\approx \dfrac{70}{D} \\ R &= \dfrac{70}{12} = 5.83 \% \end{align*}\]
The annual growth rate of the city is approximately 5.83%
Exponential Decay and Half-Life Model
The half-life of a material is the time it takes for a quantity of material to be cut in half. This term is commonly used when describing radioactive metals like uranium or plutonium. For example, the half-life of carbon-14 is 5730 years, which means that 500 mg of carbon-14 will decay to 250 mg in 5370 years.
If a substance has a half-life, this means that half of the substance will be gone in a unit of time. In other words, the amount decreases by 50% per unit of time. Using the exponential growth model with a decrease of 50%, we have
\[P(t) = P_{0}(1-0.5)^{t} = P_{0}(\dfrac{1}{2})^{t}\]
If \(H\) is the half-life of a quantity (the amount of time it takes the quantity to be cut in half) and \(P_{0}\) is the initial amount of the quantity then the amount of the quantity present after t units of time is
\[P(t) = P_{0}(\dfrac{1}{2})^{\frac{t}{H}}\]
Let’s say a substance has a half-life of eight days. If there are 40 grams present now, how much is left after three days?
Solution
We want to find a model for the quantity of the substance that remains after t days. The amount of time it takes the quantity to be reduced by half is eight days, so this is our time unit. After t days have passed, then t8 is the number of time units that have passed. Starting with the initial amount of 40, our half-life model becomes:
\[P(t) = 40(\dfrac{1}{2})^{\frac{t}{8}} \nonumber\]
With \(t=3\)
\[P(3) = 40(\dfrac{1}{2})^{\frac{3}{8}} = 30.8 \nonumber\]
There are 30.8 grams of the substance remaining after three days.
Lead-209 is a radioactive isotope. It has a half-life of 3.3 hours. Suppose that 40 milligrams of this isotope is created in an experiment, how much is left after 14 hours?
Solution
Use the half-life model to solve this problem.
\(P_{0} = 40\) and \(H = 3.3\), so the half-life model for this problem is:
\[P(t) = 40(\dfrac{1}{2})^{\frac{t}{3.3}} \nonumber\]
With \(t=14\) hours,
\[P(14) = 40(\dfrac{1}{2})^{\frac{14}{3.3}} = 2.1 \nonumber\]
There are 2.1 milligrams of Lead-209 remaining after 14 hours.
Note: The milligrams of Lead-209 remaining follows a decreasing exponential growth model.
Nobelium-259 has a half-life of 58 minutes. If you have 1000 grams, how much will be left in two hours?
Solution
We solve this problem using the half-life model. Before we begin, it is important to note the time units. The half-life is given in minutes and we want to know how much is left in two hours. Convert hours to minutes when using the model: two hours = 120 minutes.
\(P_{0} = 1000\) and \(H = 58\) minutes, so the half-life model for this problem is:
\[P(t) = 1000(\dfrac{1}{2})^{\frac{t}{58}} \nonumber\]
With \(t=120\) minutes,
\[P(120) = 1000(\dfrac{1}{2})^{\frac{120}{58}} = 238.33 \nonumber\]
There are 238 grams of Nobelium-259 is remaining after two hours.
Radioactive carbon-14 is used to determine the age of artifacts because it concentrates in organisms only when they are alive. It has a half-life of 5730 years. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.
Solution
In this problem, we want to estimate the age of the scrolls. In 1947, 76% of the carbon-14 remained. This means that the amount remaining at time t divided by the original amount of carbon-14, \(P_{0}\), is equal to 76%. So, \(\dfrac{P(t)}{P_{0}} = 0.76\) we use this fact to solve for t .
\[\begin{align*} P(t) &= P_{0}(\dfrac{1}{2})^{\frac{t}{5730}} \\ \dfrac{P(t)}{P_{0}} &= (\dfrac{1}{2})^{\frac{t}{5730}} \\ 0.76 &= (\dfrac{1}{2})^{\frac{t}{5730}} \end{align*}\]
Now, we know that the exponent on the base of 1/2 has to be between 0 and 1 since
\[(\dfrac{1}{2})^{0} = 1\nonumber\]
\[(\dfrac{1}{2})^{1} = 0.5\nonumber\]
Since the exponent on the base of 1/2 is \[\frac{t}{5730}\] this means t must be between 0 and 5730 to ensure the exponent is between 0 and 1. By guessing different values of t, we can estimate the answer until we find a value of t that gives the proper results when substituted into the exponent. The best estimate for t is 2270 since:
\[(\dfrac{1}{2})^{\frac{t}{5730}} = (\dfrac{1}{2})^{\frac{2270}{5730}} = 0.76 \nonumber\]
Which is almost exactly what we needed. Using technology would allow us to get a more accurate answer and more importantly, it would make it much easier to find the correct value since there are many choices for t between 0 and 5730.
Our conclusion is the Dead Sea Scrolls are approximately 2270 years old.
There is simple formula for approximating the half-life of a population. It is called the rule of 70 and is an approximation for decay rates less than 15%. Do not use this formula if the decay rate is 15% or greater.
For a quantity decreasing at a constant percentage (not written as a decimal), R , per time period, the half-life is approximately given by:
\[\text{Half-life } H \approx \dfrac{70}{R}\]
The population of wild elephants is decreasing by 7% per year. Approximate the half -life for this population. If there are currently 8000 elephants left in the wild, how many will remain in 25 years?
Solution
To solve this problem, use the half-life approximation formula.
\[\text{Half-Life } H \approx \dfrac{70}{7} = 10 \text{ years} \nonumber\]
\(P_{0} = 7000\) and \(H = 10\) years, so the half-life model for this problem is:
\[P(t) = 7000(\dfrac{1}{2})^{\frac{t}{10}} \nonumber\]
When \(t=25\),
\[P(25) = 7000(\dfrac{1}{2})^{\frac{25}{10}} = 1237.4 \nonumber\]
There will be approximately 1237 wild elephants left in 25 years.
Note: The population of elephants follows a decreasing exponential growth model.


16.1: Ethical Principles in Scientific Research
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Some of the expected tenets of ethical behaviour that are widely accepted within the scientific community are as follows:
Voluntary participation and harmlessness. Subjects in a research project must be aware that their participation in the study is voluntary, that they have the freedom to withdraw from the study at any time without any unfavourable consequences, and they will not be harmed as a result of their participation or non-participation in the project. One of the most flagrant violations of the voluntary participation principle is the forced medical experiments conducted by Nazi researchers on prisoners of war during World War II, as documented in the post-War Nuremberg Trials—these experiments also originated the term ‘crimes against humanity’. Lesser known violations include the Tuskegee syphilis experiments conducted by the U.S. Public Health Service from 1932–1972, in which nearly 400 impoverished African‑American men suffering from syphilis were denied penicillin even after it was accepted as an effective treatment for syphilis. Instead, subjects were presented with false treatments such as spinal taps. Even if subjects face no mortal threat, they should not be subjected to personal agony as a result of their participation. In 1971, psychologist Philip Zimbardo created the Stanford Prison Experiment, where Stanford students recruited as subjects were randomly assigned roles such as prisoners or guards. When it became evident that student prisoners were suffering psychological damage as a result of their mock incarceration, and student guards were exhibiting sadism that would later challenge their own self-image, the experiment was terminated.
If an instructor asks their students to fill out a questionnaire and informs them that their participation is voluntary, students should not fear that their non-participation may hurt their grade in class in any way. For instance, it in unethical to provide bonus points for participation and no bonus points for non-participation, because it places non-participants at a distinct disadvantage. To avoid such circumstances, the instructor could provide an alternate task for non-participants so that they can recoup the bonus points without participating in the research study, or by providing bonus points to everyone irrespective of their participation or non-participation. Furthermore, all participants must receive and sign an Informed Consent form that clearly describes their right to refuse participation, as well as their right to withdraw, before their responses in the study can be recorded. In a medical study, this form must also specify any possible risks to subjects from their participation. For subjects under the age of 18, this form must be signed by their parent or legal guardian. Researchers must retain these informed consent forms for a period of time (often three years) after the completion of the data collection process in order comply with the norms of scientific conduct in their discipline or workplace.
Anonymity and confidentiality. To protect subjects’ interests and future wellbeing, their identity must be protected in a scientific study. This is done using the dual principles of anonymity and confidentiality. Anonymity implies that the researcher or readers of the final research report or paper cannot identify a respondent by their response. An example of anonymity in scientific research is a postal survey in which no identification numbers are used to track who is responding to the survey and who is not. In studies of deviant or undesirable behaviours, such as drug use or illegal music downloading by students, truthful responses may not be obtained if subjects are not assured of anonymity. Further, anonymity assures that subjects are insulated from law enforcement or other authorities who may have an interest in identifying and tracking such subjects in the future.
In some research designs such as face-to-face interviews, anonymity is not possible. In other designs, such as a longitudinal field survey, anonymity is not desirable because it prevents the researcher from matching responses from the same subject at different points in time for longitudinal analysis. Under such circumstances, subjects should be guaranteed confidentiality , in which the researcher can identify a person’s responses, but promises not to divulge that person’s identify in any report, paper, or public forum. Confidentiality is a weaker form of protection than anonymity, because in most countries, researchers and their subjects do not enjoy the same professional confidential relationship privilege as is granted to lawyers and their clients. For instance, two years after the Exxon Valdez supertanker spilled ten million barrels of crude oil near the port of Valdez in Alaska, communities suffering economic and environmental damage commissioned a San Diego research firm to survey the affected households about increased psychological problems in their family. Because the cultural norms of many Native Americans made such public revelations particularly painful and difficult, respondents were assured their responses would be treated with confidentiality. However, when this evidence was presented in court, Exxon petitioned the court to subpoena the original survey questionnaires—with identifying information—in order to cross-examine respondents regarding answers they had given to interviewers under the protection of confidentiality, and their request was granted. Fortunately, the Exxon Valdez case was settled before the victims were forced to testify in open court, but the potential for similar violations of confidentiality still remains.
In another extreme case, Rik Scarce—a graduate student at Washington State University—was called before a grand jury to identify the animal rights activists he observed for his 1990 book, Eco ‑w arriors: Understanding the radical environmental movement . In keeping with his ethical obligations as a member of the American Sociological Association, Scarce refused to answer grand jury questions, and was forced to spend 159 days at Spokane County Jail. To protect themselves from similar travails, researchers should remove any identifying information from documents and data files as soon as they are no longer necessary.
Disclosure. Usually, researchers are obliged to provide information about their study to potential subjects before data collection to help them decide whether or not they wish to participate. For instance, who is conducting the study, for what purpose, what outcomes are expected, and who will benefit from the results. However, in some cases, disclosing such information may potentially bias subjects’ responses. For instance, if the purpose of a study is to examine the extent to which subjects will abandon their own views to conform with ‘groupthink’, and they participate in an experiment where they listen to others’ opinions on a topic before voicing their own, then disclosing the study’s purpose before the experiment will likely sensitise subjects to the treatment. Under such circumstances, even if the study’s purpose cannot be revealed before the study, it should be revealed in a debriefing session immediately following the data collection process, with a list of potential risks or harm to participants during the experiment.
Analysis and reporting. Researchers also have ethical obligations to the scientific community on how data is analysed and reported in their study. Unexpected or negative findings should be disclosed in full, even if they cast some doubt on the research design or the findings. Similarly, many interesting relationships are discovered after a study is completed, by chance or data mining. It is unethical to present such findings as the product of deliberate design. In other words, hypotheses should not be designed in positivist research after the fact based on the results of data analysis, because the role of data in such research is to test hypotheses, and not build them. It is also unethical to ‘carve’ their data into different segments to prove or disprove their hypotheses of interest, or to generate multiple papers claiming different datasets. Misrepresenting questionable claims as valid based on partial, incomplete, or improper data analysis is also dishonest. Science progresses through openness and honesty, and researchers can best serve science and the scientific community by fully disclosing the problems with their research, so that they can save other researchers from similar problems.


11.4: Engaging in Reader-Centered Writing
Reader-Centered Writing
Writing can be conceptualized as writer-centered or reader-centered . Things like diaries and journals are primarily writer-centered, in that they are written for the benefit of the writer. Your schoolwork may also have been somewhat writer-centered, in that often your goal was to “show what you know” and thereby “get a good grade.” Many communications, especially technical ones, require that you shift this mindset and write for the benefit of your reader—or design the content and structure of your communication for your “reader.” This mindset should be informed by an understanding of your audience. Use these guidelines and ask yourself the following questions:
- Who is my target audience? Are they internal or external readers? Upstream, downstream or lateral from you? Do I have multiple readers?
- What is their perspectives on the topic, on me, and on the document I will write? What are they expecting to do with the document? What is the document meant to accomplish? Why has it been requested? What is my role and relationship to my readers? What does the reader need to know? Already know? What does my reader NOT need to have explained?
- What is my goal or purpose in writing to these readers? What am I trying to communicate? What do I want them to do as a result of reading this document? How can I plan the content to meet my readers’ needs?
- What is my reader’s goal? Why does this audience want or need to read this document?
Getting a clear understanding of your audience is important in communicating effectively. It also enables you to imagine your audience as you write and revise. Keep asking yourself whether what you have said would be clear to your audience. How could you say it better?
Choose one of the topics below. Then perform an audience analysis, using the questions above to gain an understanding of the needs of different audiences. Write a profile of your intended reader(s) and consider what sort of information they will need and why?
-
You have been asked to write a report on
Maintaining Internet Privacy
for
a) A new internet user who just signed up for internet service
b) A start up e-commerce website developer -
Prepare a document on
Food-born Diseases
for
a) Restaurant workers (servers and kitchen staff)
b) For a health inspector training course -
Provide information on a proposed
New Bus Shelter Design
to
a) Mayor’s office
b) Contractor
c) Newspaper reporter writing an article on the issue
Professional Tone
“Tone” refers to the attitude that a document conveys towards the topic and/or the reader. You have likely read something that sounded angry, or optimistic, or humorous, or cynical, or enthusiastic. These words characterize the tone. Technical communication tends to avoid displaying an obvious emotion, and instead strives for a neutral tone.
Tone is created through word choice (diction), word order (syntax), sentence construction, and viewpoint. Consider a piece of academic writing that you may have read. It creates a formal tone through its use of specialized terminology, sophisticated vocabulary, complex sentence structures, and third person voice. This style suits the genre because it is directed at experts and scholars in the field, and seeks to convey complex information densely and objectively, with an emphasis on reason, logic, and evidence.
Now consider a piece of business writing that you may have read. The tone may be sightly less formal but not colloquial. The language is direct and plain, and the sentences are shorter and more straightforward. It may make use of the second person (“you”). This style suits business writing because it is directed at colleagues, management, or clients who are seeking information clearly and quickly and who may need to take action on it.
Writing Constructively
Striking the appropriate tone involves understanding your purpose, context, and audience. It also involves an understanding that workplaces are often hierarchical, and that cooperation and teamwork are required. Therefore, it is important to consider how you want your reader to feel, and what may make your reader feel that way. Your goal is to write constructively, which means to use positive phrasing to convey your message to your reader. Table \(\PageIndex{1}\) illustrates the differences between destructive/negative and constructive/positive feelings the reader may experience as a result of the tone used in a document.
| Negative | Constructive |
|---|---|
| misunderstood | understood |
| outraged | conciliatory |
| disgusted | pleased |
| guilty | capable |
| belittled | empowered |
| patronized | respected |
| defensive | proud |
| chastised | valued |
| humiliated | honoured |
| excluded | a sense of belonging |
| resentment | contentment |
Considering how your reader may feel after reading your document is an important part of revision. Did your tone come across like you hoped it would? Could it be misconstrued? Often this is where peer reviewing can be helpful. Asking a colleague to review your document before sending it off to its intended audience is a common professional practice.
Sometimes, you will need to communicate information that is unpleasant, such as delivering bad news or rejecting a request. Communicating constructively is possible—and arguably even more important—in these situations. Regardless of message, how can you ensure you are communicating constructively?
- Adopt an adult-to-adult approach: that is to say, avoid talking down to your reader in a patronizing tone, and likewise avoid sounding petulant or unwilling to take responsibility. Aim to communicate respectfully, responsibly, confidently, and cooperatively — as one responsible adult to another.
- Be courteous: focus on the reader as much as possible. Use “you” unless it results in blaming (one effective use of passive verbs is to avoid assigning blame: “mistakes were made”). Use traditionally accepted forms of courtesy and politeness. Use gender-neutral phrasing and plural forms, unless you are referring to a specific person and you know their gender.
- Focus on the positive: emphasize what you can do rather than what you can’t. Try to avoid negative wording and phrasing (no, not, never, none, isn’t, can’t, don’t, etc .). Focus on what can be improved.
- Be genuine: apologize if you have made a mistake. Take responsibility and promise to do better. Be authentic in your expression. Avoid sounding like marketing material (ad-speak). Make reasonable claims that can be backed with evidence.
Consider the following perspectives:
TABLE \(\PageIndex{2}\): Perspectives
| Writer-Centered (I, we) | Reader-Centered (you) |
|---|---|
| If I can answer any questions, I’ll be happy to do so. | If you have any questions, please ask. |
| We shipped the order this morning. | Your order was shipped this morning. |
| I’m happy to report that … | You’ll be glad to know that … |
| Negative Phrasing | Constructive Phrasing |
|---|---|
| We cannot process your claim because the necessary forms have not been completed | Your claim can be processed as soon as we receive the necessary forms |
| We do not take phone calls after 3:00 pm on Fridays | You try … |
| We closed your case because we never received the information requested in our letter of April … |
A colleague has asked you to review his email before sending. What revisions to content, tone, and style would you suggest?
From: Jake Burns
To: J. Parsons, Project Coordinator
Date: 12 December 2015
Subject: Two Problems
Hi Ms. P
Say, we may need to increase the budget on this project by $12,000. Sam screwed up when he calculated material costs. Now we don’t have enough budgeted to add the additional G3 servers with the 36GB 15k hot pluggable hard drives. I know you don’t know what all that means, but trust me. WE NEED THOSE SERVER UPGRADES!!!
Also, I would like to talk about getting my office moved closer to the rest of the IT department. All the running back and forth is disturbing other employees. I am so far away from everyone that I figure I must need to change deodorant or something. ; )
JB
How do you think the following memo will make the recipients feel? How would you revise the following memo to more constructively address the problem?
From: Ann Onymous
To: All Employees
Date: Feb. 3, 2011
Subject: Littering
For some time now, smoking has been strictly prohibited within five metres of the Main Building entrance. Do NOT smoke anywhere near the doors!
Some of you still insist on smoking and have been doing so inside this area. As a result, the areas near the rear exit and around the picnic tables are constantly littered with smoking-related debris (filter tips, half-smoked cigarettes, empty lighters, etc.), creating an eyesore and making more work for my staff, who have to keep cleaning up this mess.
Starting Monday, sand buckets will be provided outside the read doors and in the picnic area. Use them!
For further reading, see “ Communication in the Workplace: What Can NC State Students Expect? ” a study based on the responses of over 1000 professionals from various fields, including engineering, on how important business, technical and scientific communication is to their work.


19.18: Job Descriptions in Small Business
- Discuss the importance of job descriptions in small business
In a small business, the employees often wear a lot of hats.  Angela in marketing might hire a new part-time employee, and she might put on her HR hat and collect and file all the paperwork herself. Zane in operations might go out to install a product, then see an opportunity to temporarily be a salesperson and convince an onlooker that he, too, needs that product.
Employees often find it enjoyable to step out of their regular roles and try something new, but when small businesses don’t draft job descriptions, when they don’t think through the position and what will constitute their employee’s eight-hour day, it can sometimes spell disaster. Let’s talk about the ways that job descriptions provide a safety net for small business employers.
- Hiring the right employee. First and foremost, a small business wants to hire a new employee with skills that fit business needs. Job descriptions should include a thoughtful list of tasks that the small business owner expects to be completed. The list can be researched and cross-checked against similar job descriptions to ensure that the HR professional isn’t out there looking for a “unicorn,” and that employees fitting the description actually exist.
- Set clear expectations for the new employee. This may seem obvious, as it’s the major purpose of a job description. Put an employee wearing many hats with a newbie who’s eager to please, and pretty soon the new person is working 80 hours a week and isn’t sure what they’re really supposed to be doing. A job description that sets clear expectations for an employee can help them make choices about the many tasks that come their way.
- Defend the business against discrimination charges. The small business owner needs to keep the ADA and discrimination laws in mind when drafting a job description for new employees. The job description can be used to outline “essential functions” for the role, like how much time the employee will spend on her feet or how much weight she needs to be able to lift. Thinking about these qualifications up front can aid a small business employer in making sure a rejection letter isn’t misinterpreted as discrimination, and the job description can provide reference in a court of law to prove that an employee didn’t have skills to perform the job should a discrimination charge be leveled.
- Classifying employees as exempt or non-exempt. Job duties should be aligned with FLSA exemptions of the overtime rule. The job description and job title should match the reality of the job. If a high school degree is all that’s necessary for a position, a master’s shouldn’t be required in the description. Putting the job description in writing and offering it up to public consumption allows for an additional level of honesty. This is another way that the HR professional can protect the small business from discrimination suits and overtime claims from employees that shouldn’t be classified as exempt.
- Protect the small business from overpayment. Job descriptions allow an HR professional to do salary analysis, matching skills and job duties to their worth in the local job market. Compensation analysis not only saves the small business from overpaying to bring on an employee with the necessary skills, it ensures they don’t undervalue those skills, either. A quality employee comes at market price.
- Evaluate employees and help them improve. A job description that sets clear expectations about the tasks to be performed is the basis for employee performance evaluation. When sitting down to talk with an employee about his or her performance, the job description is a reminder of the initial expectations and can be the foundation of that discussion.
Job descriptions are important for legal reasons as much as they are important for the happiness and overall success of the employee brought into the small business. They can be easily overlooked or dismissed as unnecessary work, but documenting the needs and requirements of a position can help save a small business owner time and money as much as it helps her find a great employee.
Contributors and Attributions
- Job Descriptions in Small Business. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Authored by : Clkr Free Vectors. Provided by : Pixabay. Located at : pixabay.com/vectors/superhero-super-hero-hero-figure-304713/%20%20%20. License : CC0: No Rights Reserved . License Terms : Pixabay License


Section 3.1: Prelude to Evaluation Research
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?QuestionPro refers to evaluation research, also known as program evaluation, as research purpose instead of a specific method. Evaluation research is the systematic assessment of the worth or merit of time, money, effort and resources spent in order to achieve a goal. Additionally, evaluation research is a type of applied research, and so it is intended to have some real-world effect. Many methods like surveys and experiments can be used to do evaluation research. The process of evaluation research consisting of data analysis and reporting is a rigorous, systematic process that involves collecting data about organizations, processes, projects, services, and/or resources. Evaluation research enhances knowledge and decision-making, and leads to practical applications.
The three case studies found in this section provide real-world evaluation research for you to see how community psychology practitioners conduct this work in community settings. In Lessons from Conducting an Equity-Focused, Participatory Needs Assessment , Brown et al. describe their process of engaging in community-based collaborative work with the LGBTQIA community in North Texas with a partnership that consisted of a full-time community psychology practitioner, an academic partner, and other stakeholders.
Program Evaluation: A Fundamental Component in Effective Community Practice contributed by Dr. Patricia O’Connor, expands the traditional single-case study format to include multiple mini-case studies from which “lessons learned” are highlighted through evaluation-based practice of community psychology (CP). In this study, CP practitioners and relevant stakeholders who work together to design and implement needed community-based programs.
Our third case story, Showing up and Standing with: An Intersectional Approach to a Participatory Evaluation of a Housing First Program on O’ahu , contributed by Dr. Anna Pruitt takes us to the lovely O’ahu and captures the work conducted in an ongoing five-year participatory evaluation partnership between Housing First program participants, staff, and community psychologist evaluators in the multicultural context of the Island of O‘ahu in Hawai‘i. Using an intersectional lens (Crenshaw, 1989; Weber, 2009), this case study explores the challenges and successes of building this partnership among individuals from diverse racial and ethnic backgrounds with varying degrees of power, housing experiences, and mental and physical health issues.


<IP_ADDRESS>: WorldCat@OSU
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WorldCat@OSU searches the holdings of libraries from all over the world (including Ohio State University Libraries and OhioLINK libraries), as well as content from thousands of journals and millions of electronic books and web-accessible documents.
WorldCat@OSU is good for quick searches on a topic, as a starting point, and for interdisciplinary topics. However, despite its size, it is not all-inclusive. It does not search all journals and databases, and full-text searching is limited.
To access WorldCat@OSU, choose the Search All tab on the OSU Libraries’ main page at http://library.osu.edu . (Off-campus users will be asked to sign in with an OSU username and password or proceed as a guest for fewer options.)


13.2: Newton's Law of Universal Gravitation
- List the significant milestones in the history of gravitation
- Calculate the gravitational force between two point masses
- Estimate the gravitational force between collections of mass
We first review the history of the study of gravitation, with emphasis on those phenomena that for thousands of years have inspired philosophers and scientists to search for an explanation. Then we examine the simplest form of Newton’s law of universal gravitation and how to apply it.
The History of Gravitation
The earliest philosophers wondered why objects naturally tend to fall toward the ground. Aristotle (384–322 BCE) believed that it was the nature of rocks to seek Earth and the nature of fire to seek the Heavens. Brahmagupta (598~665 CE) postulated that Earth was a sphere and that objects possessed a natural affinity for it, falling toward the center from wherever they were located.
The motions of the Sun, our Moon, and the planets have been studied for thousands of years as well. These motions were described with amazing accuracy by Ptolemy (90–168 CE), whose method of epicycles described the paths of the planets as circles within circles. However, there is little evidence that anyone connected the motion of astronomical bodies with the motion of objects falling to Earth—until the seventeenth century.
Nicolaus Copernicus (1473–1543) is generally credited as being the first to challenge Ptolemy’s geocentric (Earth-centered) system and suggest a heliocentric system, in which the Sun is at the center of the solar system. This idea was supported by the incredibly precise naked-eye measurements of planetary motions by Tycho Brahe and their analysis by Johannes Kepler and Galileo Galilei. Kepler showed that the motion of each planet is an ellipse (the first of his three laws, discussed in Kepler’s Laws of Planetary Motion ), and Robert Hooke (the same Hooke who formulated Hooke’s law for springs) intuitively suggested that these motions are due to the planets being attracted to the Sun. However, it was Isaac Newton who connected the acceleration of objects near Earth’s surface with the centripetal acceleration of the Moon in its orbit about Earth.
Finally, in Einstein’s Theory of Gravity , we look at the theory of general relativity proposed by Albert Einstein in 1916. His theory comes from a vastly different perspective, in which gravity is a manifestation of mass warping space and time. The consequences of his theory gave rise to many remarkable predictions, essentially all of which have been confirmed over the many decades following the publication of the theory (including the 2015 measurement of gravitational waves from the merger of two black holes).
Newton’s Law of Universal Gravitation
Newton noted that objects at Earth’s surface (hence at a distance of R E from the center of Earth) have an acceleration of g, but the Moon, at a distance of about 60 R E , has a centripetal acceleration about (60) 2 times smaller than g. He could explain this by postulating that a force exists between any two objects, whose magnitude is given by the product of the two masses divided by the square of the distance between them. We now know that this inverse square law is ubiquitous in nature, a function of geometry for point sources. The strength of any source at a distance r is spread over the surface of a sphere centered about the mass. The surface area of that sphere is proportional to r 2 . In later chapters, we see this same form in the electromagnetic force.
Newton’s law of gravitation can be expressed as
\[\vec{F}_{12} = G \frac{m_{1} m_{2}}{r^{2}} \hat{r}_{12} \label{13.1}\]
where \(\vec{F}_{12}\) is the force on object 1 exerted by object 2 and \(\hat{r}_{12}\) is a unit vector that points from object 1 toward object 2.
As shown in Figure \(\PageIndex{1}\), the \(\vec{F}_{12}\) vector points from object 1 toward object 2, and hence represents an attractive force between the objects. The equal but opposite force \(\vec{F}_{21}\) is the force on object 2 exerted by object 1.
These equal but opposite forces reflect Newton’s third law, which we discussed earlier. Note that strictly speaking, Equation \ref{13.1} applies to point masses—all the mass is located at one point. But it applies equally to any spherically symmetric objects, where r is the distance between the centers of mass of those objects. In many cases, it works reasonably well for nonsymmetrical objects, if their separation is large compared to their size, and we take r to be the distance between the center of mass of each body.
The Cavendish Experiment
A century after Newton published his law of universal gravitation, Henry Cavendish determined the proportionality constant G by performing a painstaking experiment. He constructed a device similar to that shown in Figure \(\PageIndex{2}\), in which small masses are suspended from a wire. Once in equilibrium, two fixed, larger masses are placed symmetrically near the smaller ones. The gravitational attraction creates a torsion (twisting) in the supporting wire that can be measured.
The constant G is called the universal gravitational constant and Cavendish determined it to be G = 6.67 x 10 −11 N • m 2 /kg 2 . The word ‘universal’ indicates that scientists think that this constant applies to masses of any composition and that it is the same throughout the Universe. The value of G is an incredibly small number, showing that the force of gravity is very weak. The attraction between masses as small as our bodies, or even objects the size of skyscrapers, is incredibly small. For example, two 1.0-kg masses located 1.0 meter apart exert a force of 6.7 x 10 −11 N on each other. This is the weight of a typical grain of pollen.
Although gravity is the weakest of the four fundamental forces of nature, its attractive nature is what holds us to Earth, causes the planets to orbit the Sun and the Sun to orbit our galaxy, and binds galaxies into clusters, ranging from a few to millions. Gravity is the force that forms the Universe.
To determine the motion caused by the gravitational force, follow these steps:
- Identify the two masses, one or both, for which you wish to find the gravitational force.
- Draw a free-body diagram, sketching the force acting on each mass and indicating the distance between their centers of mass.
- Apply Newton’s second law of motion to each mass to determine how it will move.
Consider two nearly spherical Soyuz payload vehicles, in orbit about Earth, each with mass 9000 kg and diameter 4.0 m. They are initially at rest relative to each other, 10.0 m from center to center. (As we will see in Kepler’s Laws of Planetary Motion , both orbit Earth at the same speed and interact nearly the same as if they were isolated in deep space.) Determine the gravitational force between them and their initial acceleration. Estimate how long it takes for them to drift together, and how fast they are moving upon impact.
Strategy
We use Newton’s law of gravitation to determine the force between them and then use Newton’s second law to find the acceleration of each. For the estimate, we assume this acceleration is constant, and we use the constant-acceleration equations from Motion along a Straight Line to find the time and speed of the collision.
Solution
The magnitude of the force is
\[|\vec{F}_{12}| = F_{12} = G \frac{m_{1} m_{2}}{r^{2}} = (6.67 \times 10^{-11}\; N\; \cdotp m^{2}/kg^{2}) \frac{(9000\; kg)(9000\; kg)}{(10\; m)^{2}} = 5.4 \times 10^{-5}\; N \ldotp\]
The initial acceleration of each payload is
\[a = \frac{F}{m} = \frac{5.4 \times 10^{-5}\; N}{9000\; kg} = 6.0 \times 10^{-9}\; m/s^{2} \ldotp\]
The vehicles are 4.0 m in diameter, so the vehicles move from 10.0 m to 4.0 m apart, or a distance of 3.0 m each. A similar calculation to that above, for when the vehicles are 4.0 m apart, yields an acceleration of 3.8 x 10 −8 m/s 2 , and the average of these two values is 2.2 x 10 −8 m/s 2 . If we assume a constant acceleration of this value and they start from rest, then the vehicles collide with speed given by
\[v^{2} = v_{0}^{2} + 2a (x - x_{0}),\; where\; v_{0} = 0,\]
so
\[v = \sqrt{2(2.2 \times 10^{-9}\; N)(3.0\; m)} = 3.6 \times 10^{-4}\; m/s \ldotp\]
We use v = v 0 + at to find t = v/a = 1.7 x 10 4 s or about 4.6 hours.
Significance
These calculations—including the initial force—are only estimates, as the vehicles are probably not spherically symmetrical. But you can see that the force is incredibly small. Astronauts must tether themselves when doing work outside even the massive International Space Station (ISS), as in Figure \(\PageIndex{3}\), because the gravitational attraction cannot save them from even the smallest push away from the station.
What happens to force and acceleration as the vehicles fall together? What will our estimate of the velocity at a collision higher or lower than the speed actually be? And finally, what would happen if the masses were not identical? Would the force on each be the same or different? How about their accelerations?
- Answer
-
Add texts here. Do not delete this text first.
The effect of gravity between two objects with masses on the order of these space vehicles is indeed small. Yet, the effect of gravity on you from Earth is significant enough that a fall into Earth of only a few feet can be dangerous. We examine the force of gravity near Earth’s surface in the next section.
Find the acceleration of our galaxy, the Milky Way, due to the nearest comparably sized galaxy, the Andromeda galaxy (Figure \(\PageIndex{4}\)). The approximate mass of each galaxy is 800 billion solar masses (a solar mass is the mass of our Sun), and they are separated by 2.5 million light-years. (Note that the mass of Andromeda is not so well known but is believed to be slightly larger than our galaxy.) Each galaxy has a diameter of roughly 100,000 light-years (1 light-year = 9.5 x 10 15 m) .
Strategy
As in the preceding example, we use Newton’s law of gravitation to determine the force between them and then use Newton’s second law to find the acceleration of the Milky Way. We can consider the galaxies to be point masses, since their sizes are about 25 times smaller than their separation. The mass of the Sun (see Appendix D ) is 2.0 x 10 30 kg and a light-year is the distance light travels in one year, 9.5 x 10 15 m.
Solution
The magnitude of the force is
\[F_{12} = G \frac{m_{1} m_{2}}{r^{2}} = (6.67 \times 10^{-11}\; N\; \cdotp m^{2}/kg^{2}) \frac{[(800 \times 10^{9})(2.0 \times 10^{30}\; kg)]^{2}}{[(2.5 \times 10^{6})(9.5 \times 10^{15}\; m)]^{2}} = 3.0 \times 10^{29}\; N \ldotp\]
The acceleration of the Milky Way is
\[a = \frac{F}{m} = \frac{3.0 \times 10^{29}\; N}{(800 \times 10^{9})(2.0 \times 10^{30}\; kg)} = 1.9 \times 10^{-13}\; m/s^{2} \ldotp\]
Significance
Does this value of acceleration seem astoundingly small? If they start from rest, then they would accelerate directly toward each other, “colliding” at their center of mass. Let’s estimate the time for this to happen. The initial acceleration is ~10 −13 m/s 2 , so using v = at , we see that it would take ~10 13 s for each galaxy to reach a speed of 1.0 m/s, and they would be only ~0.5 x 10 13 m closer. That is nine orders of magnitude smaller than the initial distance between them. In reality, such motions are rarely simple. These two galaxies, along with about 50 other smaller galaxies, are all gravitationally bound into our local cluster. Our local cluster is gravitationally bound to other clusters in what is called a supercluster. All of this is part of the great cosmic dance that results from gravitation, as shown in Figure \(\PageIndex{5}\).


16.4: Integral Domains and Fields
Let us briefly recall some definitions. If \(R\) is a commutative ring and \(r\) is a nonzero element in \(R\text{,}\) then \(r\) is said to be a zero divisor if there is some nonzero element \(s \in R\) such that \(rs = 0\text{.}\) A commutative ring with identity is said to be an integral domain if it has no zero divisors. If an element \(a\) in a ring \(R\) with identity has a multiplicative inverse, we say that \(a\) is a unit . If every nonzero element in a ring \(R\) is a unit, then \(R\) is called a division ring . A commutative division ring is called a field .
Example \(16.12\)
If \(i^2 = -1\text{,}\) then the set \({\mathbb Z}[ i ] = \{ m + ni : m, n \in {\mathbb Z} \}\) forms a ring known as the Gaussian integers . It is easily seen that the Gaussian integers are a subring of the complex numbers since they are closed under addition and multiplication. Let \(\alpha = a + bi\) be a unit in \({\mathbb Z}[ i ]\text{.}\) Then \(\overline{\alpha} = a - bi\) is also a unit since if \(\alpha \beta = 1\text{,}\) then \(\overline{\alpha} \overline{\beta} = 1\text{.}\) If \(\beta = c + di\text{,}\)
Solution
then
\[ 1 = \alpha \beta \overline{\alpha} \overline{\beta} = (a^2 + b^2 )(c^2 + d^2)\text{.} \nonumber \]
Therefore, \(a^2 + b^2\) must either be \(1\) or \(-1\text{;}\) or, equivalently, \(a + bi = \pm 1\) or \(a + bi = \pm i\text{.}\) Therefore, units of this ring are \(\pm 1\) and \(\pm i\text{;}\) hence, the Gaussian integers are not a field. We will leave it as an exercise to prove that the Gaussian integers are an integral domain.
Example \(16.13\)
The set of matrices
\[ F = \left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \right\} \nonumber \]
Solution
with entries in \({\mathbb Z}_2\) forms a field.
Example \(16.14\)
The set \({\mathbb Q}( \sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}\) is a field. The inverse of an element \(a + b \sqrt{2}\) in \({\mathbb Q}( \sqrt{2}\, )\) is
Solution
\[ \frac{a}{a^2 - 2 b^2} +\frac{- b}{ a^2 - 2 b^2} \sqrt{2}\text{.} \nonumber \]
We have the following alternative characterization of integral domains.
Proposition \(16.15\). Cancellation Law
Let \(D\) be a commutative ring with identity. Then \(D\) is an integral domain if and only if for all nonzero elements \(a \in D\) with \(ab = ac\text{,}\) we have \(b=c\text{.}\)
- Proof
-
Let \(D\) be an integral domain. Then \(D\) has no zero divisors. Let \(ab = ac\) with \(a \neq 0\text{.}\) Then \(a(b - c) =0\text{.}\) Hence, \(b - c = 0\) and \(b = c\text{.}\)
Conversely, let us suppose that cancellation is possible in \(D\text{.}\) That is, suppose that \(ab = ac\) implies \(b=c\text{.}\) Let \(ab = 0\text{.}\) If \(a \neq 0\text{,}\) then \(ab = a 0\) or \(b=0\text{.}\) Therefore, \(a\) cannot be a zero divisor.
The following surprising theorem is due to Wedderburn.
Theorem \(16.16\)
Every finite integral domain is a field.
- Proof
-
Let \(D\) be a finite integral domain and \(D^\ast\) be the set of nonzero elements of \(D\text{.}\) We must show that every element in \(D^*\) has an inverse. For each \(a \in D^\ast\) we can define a map \(\lambda_a : D^\ast \rightarrow D^\ast\) by \(\lambda_a(d) = ad\text{.}\) This map makes sense, because if \(a \neq 0\) and \(d \neq 0\text{,}\) then \(ad \neq 0\text{.}\) The map \(\lambda_a\) is one-to-one, since for \(d_1, d_2 \in D^*\text{,}\)
\[ ad_1 = \lambda_a(d_1) = \lambda_a(d_2) = ad_2 \nonumber \]
implies \(d_1 = d_2\) by left cancellation. Since \(D^\ast\) is a finite set, the map \(\lambda_a\) must also be onto; hence, for some \(d \in D^\ast\text{,}\) \(\lambda_a(d) = ad = 1\text{.}\) Therefore, \(a\) has a left inverse. Since \(D\) is commutative, \(d\) must also be a right inverse for \(a\text{.}\) Consequently, \(D\) is a field.
For any nonnegative integer \(n\) and any element \(r\) in a ring \(R\) we write \(r + \cdots + r\) (\(n\) times) as \(nr\text{.}\) We define the characteristic of a ring \(R\) to be the least positive integer \(n\) such that \(nr = 0\) for all \(r \in R\text{.}\) If no such integer exists, then the characteristic of \(R\) is defined to be \(0\text{.}\) We will denote the characteristic of \(R\) by \(\chr R\text{.}\)
Example \(16.17\)
For every prime \(p\text{,}\) \({\mathbb Z}_p\) is a field of characteristic \(p\text{.}\)
Solution
By Proposition \(3.4\), every nonzero element in \({\mathbb Z}_p\) has an inverse; hence, \({\mathbb Z}_p\) is a field. If \(a\) is any nonzero element in the field, then \(pa =0\text{,}\) since the order of any nonzero element in the abelian group \({\mathbb Z}_p\) is \(p\text{.}\)
Lemma \(16.18\)
Let \(R\) be a ring with identity. If \(1\) has order \(n\text{,}\) then the characteristic of \(R\) is \(n\text{.}\)
- Proof
-
If \(1\) has order \(n\text{,}\) then \(n\) is the least positive integer such that \(n 1 = 0\text{.}\) Thus, for all \(r \in R\text{,}\)
\[ nr = n(1r) = (n 1) r = 0r = 0\text{.} \nonumber \]
On the other hand, if no positive \(n\) exists such that \(n1 = 0\text{,}\) then the characteristic of \(R\) is zero.
Theorem \(16.19\)
The characteristic of an integral domain is either prime or zero.
- Proof
-
Let \(D\) be an integral domain and suppose that the characteristic of \(D\) is \(n\) with \(n \neq 0\text{.}\) If \(n\) is not prime, then \(n = ab\text{,}\) where \(1 \lt a \lt n\) and \(1 \lt b \lt n\text{.}\) By Lemma \(16.18\), we need only consider the case \(n 1 = 0\text{.}\) Since \(0 = n 1 = (ab)1 = (a1)(b1)\) and there are no zero divisors in \(D\text{,}\) either \(a1 =0\) or \(b1=0\text{.}\) Hence, the characteristic of \(D\) must be less than \(n\text{,}\) which is a contradiction. Therefore, \(n\) must be prime.


36.1: Appendix B- Community/Public Health Nursing [C/PHN] Competencies
The core competencies of public health nursing are defined as the fundamental knowledge, skills, and attitudes that a person holds that allow for growth and success and guide all public health professionals. The definition and practice of public health nursing were updated in 2013 to address the evolving trends in the U.S. economy as well as the changing landscape of health practices, politics, and society.
According to public health professionals, the key elements of public health nursing practice include a focus on population-based health care needs with attention to many of the social determinants of health (SDOH), a comprehensive and systematic approach to assess populations with an emphasis on preventative strategies, and key interventions at the level of individuals, families, communities, countries, and across the globe.
American Association of Colleges of Nursing
The American Association of Colleges of Nursing (AACN) defines a population as a “discrete group that the nurse and others care for across settings at local, regional, national, and global levels.”
The AACN defines nursing competencies for managing the health of populations at the level of the entry-level nurse as well as for advanced practice nurses, called The Essentials.
American Nurses Association
The American Nurses Association (ANA) has defined the scope and standards for public health nursing. The ANA has published the third edition of its scope and standards for public health nursing, available for purchase at this link.
American Public Health Association
The American Public Health Association defines public health practice and public health nursing in its official statement.
Quad Council Coalition of Public Health Organizations
The Quad Council Coalition of Public Health Organizations was founded in 1988 to provide a united voice for the specialty of public health nursing practice. The coalition represents the joint efforts of the Alliance of Nurses for Healthy Environments (AHNE), the Association of Community Health Nursing Educators (ACHNE), the Association of Public Health Nurses (APHN), and the American Public Health Association – Public Health Nursing Section (APHA-PHN).
A thorough description of the Quad Council Coalition competencies is available at Quad Council Coalition Competency Review Task Force (2018) Community/Public Health Nursing Competencies .
These competencies are defined as the combination of observable and measurable knowledge, skills, abilities, and personal attributes that contribute to enhanced employee performance and ultimately result in organizational success. Core competencies are detailed in three tiers.
- Tier 1 competencies are those performed by the generalist public health nurse.
- Tier 2 competencies apply to public health nurses involved in management activities.
- Tier 3 competencies apply to public health nurses in executive or senior management roles.
The core competencies are described by the Quad Council within the following eight domains:
- Assessment and analytic skills
- Policy development/program planning skills
- Communication skills
- Cultural competency skills
- Community dimensions of practice skills
- Public health sciences skills
- Financial planning
- Evaluation and management skills
- Leadership and systems thinking skills
For example, assessment and analytic skills are included as competencies in domain 1 for the entry-level public nurse and include the following:
1A1: Assessing the health status and health literacy of individuals and families, including determinants of health, using multiple sources of data.
1A2a: Use of an ecological perspective and epidemiological data to identify health risks for a population.
1A2b. Identifying individual and family assets, needs, values, beliefs, resources and relevant environmental factors.
1A3. Selecting variables that measure health and public health conditions.
1A4. Using a data collection plan that incorporates valid and reliable methods and instruments for collection of qualitative and quantitative data to inform the service for individuals, families, and a community.
1A5. Interpreting valid and reliable data that impacts the health of individuals, families, and communities to make comparisons that are understandable to all who were involved in the assessment process.
The Council on Linkages between Academia and Public Health Practice
The Council on Linkages between Academia and Public Health Practice is a collaborative formed by 24 national organizations, including the American Nursing Association (ANA). This council developed core competencies for public health professionals in 2021. The core competencies were developed in eight domains that include specific responsibilities at three levels: front line, program management, and senior management. Complete mapping of these competencies is available at this link . The eight domains are as follows:
- Data Analytics and Assessment Skills
- Policy Development and Program Planning Skills
- Communication Skills
- Health Equity Skills
- Community Partnership Skills
- Public Health Sciences Skills
- Management and Finance Skills
- Leadership and Systems Thinking Skills
For public health nurses seeking to build their competencies through graduate education with a master’s degree, this link provides important considerations regarding the roles and competencies of the public health practitioner, a public health physician, and a public health nurse.
Community-Based Participatory Research: In order to engage and collaborate with communities, specific competencies include the following (adapted from www.ncbi.nlm.nih.gov/pmc/articles/PMC9219558/):
-
Involving community partners as members of the research team when identifying the research problem
- For example, public health nurses may request discussion of a health concern at a city chamber of commerce meeting.
-
Collaborating with the community during the creation of a study design as well as implementation to ensure accordance with local values and challenges
- When planning to assess a community, respected local leaders in the community can provide clarity in how to best access individuals, families, and sub-communities.
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Building upon strengths and challenges in the community to help build capacity within the community as well the opportunities for the researcher
- This competency may involve identifying and accessing current community organizations in order to build a coalition.
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The researchers and community partners convening to plan how a community-based health initiative will be disseminated and communicated to relevant audiences within the community served
- While flyers in local restaurants may be helpful, some communities may be better accessed through local television or a particular form of social media.


19: Walking and Working Surfaces
“An incident is just the tip of the iceberg, a sign of a much larger problem below the surface.” – Don Brown
Overview
Walking and working surfaces are many and vary widely. Yet the walking-working surfaces standard is sparse on specific safety requirements for every type of surface and existing direction is rather generic. This does not mean however that the safety standards are not comprehensive. The standard simply states upfront that: All places of employment, passageways, storerooms, service rooms, and walking-working surfaces are kept in a clean, orderly, and sanitary condition. This simple premise speaks volumes.
Slips, trips, and falls are one of the leading causes of workplace incidents, accidents, and fatalities. Working surfaces should not be the cause of these and every employer must understand what clean, orderly, and sanitary looks like for their specific environments.
The working-walking surfaces standard has within its scope safety requirements on ladders, stairways, scaffolds, and dock boards specifically, but must apply to every surface a worker is expected to perform work.
Chapter Objective:
- Understand the scope of the walking-working surfaces standard.
- Discuss fall prevention methods.
- Discuss fall protection methods.
Learning Outcome:
- Apply housekeeping best practices for maintaining walking-working surfaces.
- Understand and apply best fall prevention and protection methods for the job.
Standards: 1910 Subpart D Walking-Working Surfaces, 1910 Subpart F Powered Platforms, Man lifts, Vehicle Mounted Work Platforms
Key Terms:
Arrest, deceleration, guardrails, mid-rail, step bolt, travel restraint
Mini-Lecture: Walking-Working Surfaces
Required Time: 1 hour; Independent Study and reflection 1 ½ hour.
Thumbnail: Scissorlift, attribution Jahudleston, Pixabay