6.9: Summary
- Born globals
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Companies that operate internationally from the day that they are created.
- Christianity
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Faith based on the life, teachings, death, and resurrection of Jesus.
- Clusters
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Representing countries that share similar cultural characteristics.
- Cultural Intelligence
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Refers to the individuals’ capabilities to function and manage effectively in culturally diverse settings.
- Cultural paradox
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Insights from an understanding of culture may not necessarily coincide with reality in that culture.
- Cultural stereotyping
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Occurs when one assumes that all people within a culture act, think, and behave the same way.
- DHL Global Connected Index
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Index tracking the flow of capital, information, trade, and human resources and representing the degree of globalization.
- E-commerce
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Buying and selling of products using the Internet.
- Education
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Socializing experiences that prepare individuals to act in society.
- Emerging market multinationals
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Influential companies from emerging markets that are competing headon with established multinationals and rewriting the rules of competition by using new business models.
- Emerging markets
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Those markets in countries that present tremendous potential for multinational.
- Expatriate
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Foreign employee who moves and works in another country for an extended period of time.
- Exporting
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International entry mode where a company sends a product to an international market and fills the order like a domestic order.
- Foreign direct investment (FDI)
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Refers to deliberate efforts of a country or company to invest in another country through the form of ownership positions in companies in another country.
- Foreign direct investment (FDI)
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Involves a company investing in another country through the construction of facilities and buildings in another country.
- Global strategy
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Where all operations and activities are managed fairly similarly worldwide.
- Globalization
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Worldwide phenomenon whereby the countries of the world are becoming more interconnected and where trade barriers are disappearing.
- GLOBE project
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More recent cultural project involving 170 researchers who collected data on 17,000 managers from 62 countries around the world. .
- High-rigor cross-cultural training
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Methods of training where participants are much more actively engaged in the training process and can learn some tacit aspects of cross-cultural differences.
- Hinduism
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Represented by all those who honor the ancient scriptures called the Vedas.
- Hofstede model of national culture
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Project involving survey of over 88,000 employees in IBM subsidiaries from 72 countries.
- Immigration
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Movement of people from their home country to other countries; will continue to grow worldwide.
- Individualism
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Degree to which a society focuses on the relationship of the individual to the group.
- International franchising
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Where a company will license the complete business model.
- International strategic alliances
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Two or more companies from different countries enter into an agreement to conduct joint business activities.
- Islam
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Religion whose essence is described in the Qur’an as the submission to the will of Allah (God).
- Licensing
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Contractual agreement whereby a company is given the right to another company’s trademarks, know-how, and other intangible assets in return for a royalty or a fee.
- Local strategy
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Company’s operations are adapted to fit some specific countries.
- Low-rigor cross-cultural training
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Training where individuals are exposed to critical information to help them understand the realities of a different culture but are not actively engaged in their learning.
- Masuclinity
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Degree to which a society emphasizes traditional masculine qualities, such as advancement and earnings.
- Political risk
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Degree to which political decisions can impact a business’s ability to survive in a country.
- Postarrival cross-cultural training
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Training provided after the expatriate has arrived to the intended destination.
- Power distance
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Refers to the degree to which societies accept power differences and authority in society.
- Predeparture cross-cultural training
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Learning opportunities provided prior to departure.
- Regional strategy
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Where the multinational adapts activities and operations to regional requirementst.
- Religion
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Shared set of beliefs, activities, and institutions based on faith in supernatural forces.
- Social institution
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Complex of positions, roles, norms, and values lodged in particular types of social structures and organizing relatively stable patterns of human resources with respect to fundamental problems in . . . sustaining viable societal structures within a given environment.
- Social stratification
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Degree to which social benefits are unequally distributed; those patterns are perpetuated for life.
- Tariffs
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Extra charges that are added to the price of international products in the form of additional taxes or higher prices as a way to give domestic companies a price advantage while also protecting these companies from foreign competition.
- Trade agreements
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Popular policy instruments that countries agree on to eliminate cross-border barriers to trade and to promote global integration.
- Uncertainty avoidance
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Refers to the degree to which people in a society are comfortable with uncertainty and unpredictable situations.
- Uppsala model of internationalization
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Model that argues that as firms learn more about a specific market, they become more committed by investing more resources into that market.
Summary of Learning Outcomes
6.2 Importance of International Management
1. Why is it important to understand and appreciate the importance of international management in today’s world?
Any serious management student needs to understand and appreciate the importance of international management in today’s global business environment. Whether you work for a domestic company or a foreign company, you will likely need to interact with someone from another country or do business in another country. Understanding international management is therefore critical to address future challenges.
In this section, you learned about the many factors contributing to making globalization a reality: the growth of trade between countries, the growing importance of foreign direct investment, the growing competition from emerging market multinationals, and the globalization-fueling pervasiveness of the Internet. These factors all contribute to making the business world more global.
6.3 Hofstede's Cultural Framework
2. What is culture, and how can culture be understood through Hofsetde’s cultural framework?
Given the importance of globalization, any serious international management students will need to be able to understand the cultural aspects of a society in which they may find themselves and will need to learn how to adapt to various cultural conditions. The most popular cultural framework, the Hofstede scheme, was developed by Geert Hofstede, a Dutch social scientist who surveyed over 88,000 employees in 72 countries in which IBM had subsidiaries. He developed this cultural model primarily on the basis of differences in values and beliefs regarding work goals. This effort resulted in four main dimensions: power distance (the degree to which societies accept power differences and authority in society), individualism (the degree to which a society focuses on the relationship of the individual to the group), uncertainty avoidance (the degree to which people in a society are comfortable with uncertainty and unpredictable situations), and masculinity (degree to which a society emphasizes traditional masculine qualities such as advancement and earnings).
6.4 The GLOBE Framework
3. How are regions of the world categorized using the GLOBE framework, and how does this categorization enhance understanding of cross-cultural leadership?
The GLOBE project cultural framework is a much more recent effort that involved 170 researchers who collected data on 17,000 managers from 62 countries around the world. The focus of the GLOBE project was to understand how national cultures have preferences for different leadership styles. One of the strengths of the GLOBE project is that it clusters societies that share similar characteristics. The seven important clusters of the GLOBE project are the Anglo cluster, the Confucian Asia cluster, the Germanic Europe cluster, the Latin America cluster, the Middle East cluster, the Nordic Europe cluster, and the Sub-Saharan cluster. Each cluster rates differently the styles of leadership that the GLOBE researchers considered. The six leadership profiles are charismatic types (degree to which the leader can inspire and motivate others), participative type (degree to which leaders involve others in decision making), humane-oriented type (degree to which the leader shows compassion and generosity), autonomous (degree to which the leader reflects independent and individualistic leadership), and self-protective (degree to which the leader is self-centered and uses a face-saving approach). The various clusters show preferences for specific leadership styles that are consistent with the cultural aspects emphasized in each cluster.
6.5 Cultural Stereotyping and Social Institutions
4. Why is an understanding of cultural stereotyping important, and what can students do to prepare for cultural stereotyping by looking at social institutions?
While the Hofstede and GLOBE culture frameworks are certainly useful and can provide a solid basis for understanding cultural differences, relying solely on cultural dimensions can lead to problems when managers are confronted with cultural paradoxes (when reality doesn’t coincide with expectations based on cultural dimensions) and cultural stereotyping (when it is assumed that everyone within the same culture acts and behaves similarly).
To broaden your understanding of cultural differences, you must also take into account a country’s social institutions.
While there are a large number of social institutions that can impact international business, we examined three main types of social institutions that affect how people act and behave: social stratification (degree to which social benefits are unequally distributed and those patterns are perpetuated for life), education (the socializing experiences which prepare individuals to act in society), and religion (the shared set of beliefs, activities, and institutions based on faith in supernatural forces).
6.6 Cross-Cultural Assignments
5. What steps can you undertake to be better prepared for cross-cultural assignments?
While the above sections provided you with many diagnostic tools to understand how to evaluate crosscultural differences, this section presented you with the ways to prepare for cross-cultural assignments. The goal of any training is to increase cultural intelligence, the ability to function and manage effectively in culturally diverse settings. To understand what companies can do to increase cultural intelligence, you learned about various types of training: low-rigor training (where individuals are exposed to critical information but are not necessarily actively engaged in their learning) and high-rigor training (methods of training where participants are much more actively engaged in the training process). You also learned that multinationals can also provide training before someone goes on an international assignment or while someone is already on the assignment.
6.7 Strategies for Expanding Globally
6. What are the main strategies that companies can use to go international?
As companies explore expanding into international markets, they adopt one of three main strategies, each of which has its advantages and disadvantages depending on the company’s and country’s characteristics. The three strategies are 1) the global strategy, in which all operations and activities are managed fairly similarly worldwide; 2) the regional strategy, in which the multinational adapts activities and operations to regional requirements; and 3) the local strategy, in which the company’s operations are adapted to fit some specific countries.
6.8 The Necessity of Global Markets
7. Why might it be necessary for a company to go international, and how might it accomplish this goal?
In the final section of the chapter, you first read about the need for companies to go international and learned that some markets present strong potential while others have floundered.
Companies can go international in many ways: exporting (an entry mode where a company sends a product to an international market and fills the order like a domestic order), licensing and franchising (a contractual agreement whereby a company is given the right to another company’s trademarks, know-how, and other intangible assets in return for a royalty or a fee), strategic alliances (where two or more companies from different countries enter into an agreement to conduct joint business activities), and foreign direct investment (which involves a company investing in another country through the construction of facilities and buildings in another country).
With each of these methods of entry, there is a trade-off between the cost of a means of entry and the amount of control a company has over its operations. For example, exporting is usually the cheapest way to go international but offers the company the least amount of control. Born globals do not have to think about how or when to go global because they are international from the day they are created.
- Why is international management a critical area that all management students should be aware of?
- Briefly describe the main cultural dimensions of Hofstede’s framework. Where does the U.S. stand on each of the dimensions?
- What is power distance? What are the implications of power distance for how management is conducted in different societies?
- How is the GLOBE project different from the Hofstede project of cultural dimensions? What are the main findings of the study?
- What are country clusters? Pick any three clusters and discuss some of the leadership preferences for each cluster.
- Compare and contrast low-rigor versus high-rigor cross-cultural training. Provide some examples of each type of training.
- What is predeparture cross-cultural training? What is postarrival cross-cultural training? Which method works best and why?
- What is a global strategy? When do companies prefer a global strategy?
- Compare and contrast a global, regional, and local strategy. Discuss some advantages and disadvantages of each method.
- What are the various means available to companies to go international? When is an exporting strategy most appropriate?
Management Skills Application Exercises
- Your manager is currently considering negotiations in China, and given your international management knowledge, you are asked for advice. How would you approach preparing to advise her? What type of information will you provide to her?
- You have a new product, and you are considering exporting the product. Visit the government’s export website at https://www.export.gov/article?id=Wh...-should-export . What are some of the key issues to take into consideration when exporting?
- You are in charge of HR in your company, and the company will soon need to send an expatriate to Japan. You will need to design a training program to better prepare your employee. What type of training program will you provide? What essential elements will the program include?
- You are considering investing in Saudi Arabia and know that the business culture there is influenced by religion. Using your knowledge of Islam, what should you expect? How can you better prepare for the investment?
- You have designed a new surfboard that presents significant advantages over current models. How can you determine whether you are ready to launch your company as a born global
Managerial Decision Exercises
- You are the CEO of a company that produces high-end laptops for gaming. You have heard that there is interest in your product from international customers. You need to decide how to enter the international market. What issues do you take into consideration when deciding among the different means of entry? Which approach do you think would work best?
- Visit Hofstede’s cultural dimensions website at www.hofstede-insights.com/mo...tionalculture/. Pick two countries and compare them with the United States on the cultural dimensions. How would you manage cultural differences in these countries?
- Your company is interesting in exploring international expansion into Africa. Visit the African Union’s website at https://au.int/en/ . What are some of the countries included in the African Union? How easy will it be to approach entering these markets?
- Your company is interested in exploring investments in several sectors in Zimbabwe. Because of political instability, you are obviously very reluctant given the risks of doing business in that country. Using data from www.marsh.com/us/campaigns/p...-map-2017.html, discuss the concept of political risk. When can you decide that investing in Zimbabwe is a good idea?
- You will be sending one of your employees to several new countries for short-term assignments. You need to decide between low-rigor cross-cultural training and high-rigor cross-cultural training. Which method would work best? How would you decide between the two, and what elements would your training involve?
Critical Thinking Case
SAP and the American CEO of a German Multinational
SAP is a German multinational specializing in enterprise application software. The company was founded by five engineers, and the company is now the world’s leading business software maker. Through its software, SAP helps its customers streamline production processes. SAP also provides forecasting services to its customers to help them predict customer trends. It currently has over 87,000 employees in 130 countries assisting over 335,000 customers worldwide.
For the first time in its history, the company is currently headed by an American, Bill McDermott. McDermott’s training was in sales, and that provided him with significant expertise to become SAP’s current CEO. In 2010, SAP was facing declining revenue worldwide and needed a turnaround. Initially, McDermott was co-CEO with Jim Hagemann Snabe, a Danish executive who was one of the company’s cofounders. The arrangement worked well, and when Snabe retired in 2014, McDermott became CEO.
McDermott’s success came from the many changes he instituted to better adapt to cultural differences. For instance, he quickly discovered that sales were not very effective in the United States because the salespeople were more interested in focusing on the engineering aspects of SAP’s products at the expense of listening to American customers. Such experiences led to the development of more customer-focused innovation and a more empathetic approach to customer needs, things McDermott strongly believes in.
In visiting his German counterparts, McDermott also saw other potential sources of cross-cultural conflict. For instance, he saw that presentations in the United States were much more effective if the presentation quickly engaged the audience and got them excited. In contrast, a German audience preferred a more disciplined, fact-based presentation. McDermott also discovered key differences between the way U.S. companies are managed in comparison to German companies. For example, he found that while U.S. public companies are pressured by quarterly results, SAP was much more interested in 30-year cycles as opposed to 90-day stock price movements.
McDermott’s success at managing cross-cultural differences is no surprise. When he was a teenager, he purchased a distressed deli shop in Long Island. Long Island was already a melting pot of immigrants, and he learned how to deal with a diverse group of customers. When he was first hired at 27 to sell Xerox copy machines, he found that American customers do not have a long time for a sales pitch. He learned to be quick and to the point. In contrast, in Asia, he found that you had to focus on developing relationships rather than focus on the product. At the age of 29, he was asked to turn around business in Puerto Rico. There he found employee morale to be very low because of cost-cutting measures. Rather than blindly implementing American management, he listened to the local employees and implemented many measures to improve operations. For instance, he worked to improve customer service. Most importantly, he reinstated a Christmas party that had been canceled as a cost-cutting measure. This lifted morale and led to the turnaround.
McDermott has many important lessons for aspiring cross-cultural leaders. He advises that leaders be respectful of cross-cultural differences. Additionally, because SAP has one global vision, he can have all employees focus on that vision. He therefore also suggests that leaders and managers adopt a compelling vision that can be readily shared with all employees. He also believes that the customer experience is what is critical. Finally, he recommends that the savvy manager be human and empathetic and show humility.
Critical Thinking Questions
- What are some of the sources of McDermott’s excellence at managing cross-cultural differences? How did his experience managing a deli store at a young age help him develop cross-cultural management skills?
- What are some of the cross-cultural differences he discovered? Using your knowledge of culture, explain some of these differences.
- What is your assessment of his lessons for cross-cultural managers? Relate these lessons to the GLOBE findings of the effective global leader.
Sources: Geoff Colvin, “ A CEO’s plan to defy disruption,” Fortune , November 2014, pp. 36; Michal Lev-Ram, “Inside SAP’s radical make-over,” Fortune, April 9th, 2012, Issue 5, pp. 35-38; Bill McDermott, “SAP’s CEO on being the American head of a German multinational,” Harvard Business Review , 2016, November, https://hbr.org/2016/11/saps-ceo-on-...-multinational ; SAP Corporate Website https://www.sap.com/index.html .


21.4.2: The Complement of a Set
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Learning Outcomes
- Determine the complement of a set.
- Write the complement of a set using set notation.
We saw in the section "Represent an Inequality as an Interval on a Number Line" how to graph the complement for a set defined by an inequality. Complements come up very often in statistics, so it is worth revisiting this, but instead of graphically we will focus on set notation. Recall that the complement of a set is everything that is not in that set. Sometimes it is much easier to find the probability of a complement than of the original set, and there is an easy relationship between the probability of an event happening and the probability of the complement of that event happening.
\[P\left(A\right)=1-P\left(not\:A\right) \nonumber \]
Example \(\PageIndex{1}\)
Find the complement of the set:
\[A=\left\{x\mid x<4\right\} \nonumber \]
Solution
The complement of the set of all numbers that are less than 4 is the set of all numbers that are at least as big as 4. Notice that the number 4 is not in the set A, since the inequality is strict (does not have an "="). Therefore the number 4 is in the complement of the set A. In set notation:
\[A^c=\left\{x\mid x\ge4\right\} \nonumber \]
Example \(\PageIndex{2}\)
When computing probabilities the complement is sometimes much easier than the original set. For example suppose you roll a die 6 times and want to find the probability that the number 3 comes up at least once. Find the complement of this event.
Solution
First note that the event of at least once means that there could be one 3, two 3's, three 3's, four 3's, five 3's, or six 3's. It turns out that this would be a burden to deal with each of these possibilities. However the complement is quite easy. The complement of getting at least one 3 is that you go no 3's.
Example \(\PageIndex{3}\)
Suppose that we want to find the probability that at least 20 people in the class have done their homework. Find the complement of this event.
Solution
Sometimes it is easiest to list nearby outcomes and then determine the outcomes that satisfy the event. Finally, to find the complement, you select the rest. First list numbers near 20:
\[...,\:17,\:18,\:19,\:20,\:21,\:22,\:... \nonumber \]
Now, the ones that are at least 20 are all the ones including 20 and to the right of 20:
\[20,\:21,\:22,\:... \nonumber \]
These are the large numbers. The complement includes all the small numbers.
\[...,\:17,\:18,\:19 \nonumber \]
We can write this in set notation as:
\[\left\{x\mid x\le19\right\} \nonumber \] or equivalently \[\left\{x\mid x < 20\right\} \nonumber \]
Example \(\PageIndex{4}\)
Suppose a number is picked at random from the whole numbers from 1 to 10. Let A be the event that a number is both even and less than 8. Find the complement of A.
Solution
First, the set of numbers that are both even and less than 8 is:
\[A\:=\:\left\{2,\:4,\:6\right\} \nonumber \]
The complement of this set is all the numbers from 1 to 10 that are not in A:
\[A^c=\left\{1,\:3,\:5,\:7,\:8,\:9,\:10\right\} \nonumber \]
Exercise
Suppose that two six sided dice are rolled. Let the A be the event that either the first die is even or the sum of the dice is greater than 5 or both have occurred. Find the complement of A.


1.5: Images and Inverses
Functions can be used to define subsets of given sets.
Let \(f: X \rightarrow Y\) and \(W \subseteq X\) . The image of \(W\) under \(f\) , written \(f[W]\) , is the set \[\{f(x) \mid x \in W\} .\] So if \(f: X \rightarrow Y\) , then \[\operatorname{Ran}(f)=f[X] .\] EXAMPLE 1.30. Suppose \(f\) is the real function \(f(x)=x^{2}+3\) . Let \(W=\{-2,2,3\}\) , and \(Z=(-1,2)\) . Then \(f[W]=\{7,12\}\) , and \(f[Z]=[3,7)\) .
In applications of mathematics, functions often describe numerical relationships between measurable observations. So if \(f: X \rightarrow Y\) and \(a \in X\) , then \(f(a)\) is the predicted or actual measurement associated with \(a\) . In this context, one is often interested in determining which elements of \(X\) are associated with a value, \(b\) , in the codomain of \(f\) .
Let \(f: X \rightarrow Y\) and \(b \in Y\) . Then the inverse image of \(b\) under \(f, f^{-1}(b)\) , is the set \[\{x \in X \mid f(x)=b\} .\] This set is also called the pre-image of \(b\) under \(f\) .
Note that if \(b \notin \operatorname{Ran}(f)\) , then \(f^{-1}(b)=\emptyset\) . If \(f\) is an injection, then for any \(b \in \operatorname{Ran}(f), f^{-1}(b)\) has a single element.
Let \(f: X \rightarrow Y\) and \(Z \subseteq Y\) . The inverse image of \(Z\) under \(f\) , or the pre-image of \(Z\) under \(f\) , is the set \[f^{-1}[Z]=\{x \in X \quad \mid f(x) \in Z\}\] We use \(f^{-1}[]\) to mean the inverse image of a subset of the codomain, and \(f^{-1}\) ( ) for the inverse image of an element of the codomain - both are subsets of the domain of \(f\) . If \(Z \cap \operatorname{Ran}(f)=\emptyset\) , then \[f^{-1}[Z]=\emptyset .\] ExAMPLE 1.32. Let \(f\) be as in Figure \(1.33\) Then \(f[\{b, c\}]=\{1,3\}\) , and \(f^{-1}[\{1,3\}]=\{a, b, c, d\}\) .
Let \(g\) be the real function \(g(x)=x^{2}+3\) . If \(b \in \mathbb{R}\) and \(b>3\) , then \[g^{-1}(b)=\{\sqrt{b-3},-\sqrt{b-3}\} .\] If \(b=3\) , then \(g^{-1}(3)=\{0\}\) . If \(b<3\) , then \(g^{-1}(b)\) is empty.
Let \(h\) be the real function \(h(x)=e^{x}\) . If \(b \in \mathbb{R}\) and \(b>0\) , then \[h^{-1}(b)=\left\{\log _{e}(b)\right\} .\] For instance, \[h^{-1}(1)=\{0\} .\] Because \(h\) is strictly increasing, the inverse image of any element of the codomain \((\mathbb{R})\) is either a set with a single element or the empty set.
Let \(I=(a, b)\) , where \(a, b \in \mathbb{R}\) and \(0<a<b\) (that is \(I\) is the open interval with end points \(a\) and \(b\) ). Then \[h^{-1}[I]=\left(\log _{e}(a), \log _{e}(b)\right) .\] We have discussed the construction of new functions from existing functions using algebraic operations and composition of functions. Another tool for building new functions from known functions is the inverse function.
Let \(f: X \mapsto Y\) be a bijection. Then the inverse function of \(f, f^{-1}: Y \rightarrow X\) , is the function with graph \[\{(b, a) \in Y \times X \mid(a, b) \in \operatorname{graph}(f)\} .\] The function \(f^{-1}\) is defined by "reversing the arrows". For this to make sense, \(f: X \rightarrow Y\) must be bijective. Indeed, if \(f\) were not surjective, then there would be an element \(y\) of \(Y\) that is not in the range of \(f\) , so cannot be mapped back to anything in \(X\) . If \(f\) were not injective, there would be elements \(z\) of \(Y\) that were the image of distinct elements \(x_{1}\) and \(x_{2}\) in \(X\) . One could not define \(f^{-1}(z)\) without specifying how to choose a particular pre-image. Both these problems can be fixed. If \(f\) is injective but not surjective, one can define \(g: X \mapsto \operatorname{Ran}(f)\) by \[g(x)=f(x)\] for all \(x \in X\) . Then \(g^{-1}: \operatorname{Ran}(f) \mapsto X\) . If \(f\) is not injective, the problem is trickier; but if we can find some subset of \(X\) on which \(f\) is injective, we could restrict our attention to that set.
Let \(f\) be the real function \(f(x)=x^{2}\) . The function \(f\) is not an bijection, so it does not have an inverse function. However the function \[\begin{aligned} g:[0, \infty) & \rightarrow[0, \infty) \\ x & \mapsto x^{2} \end{aligned}\] is a bijection. In this case, \[g^{-1}(y)=\sqrt{y} .\]
Let \(f\) be the real function, \(f(x)=e^{x}\) . You know from calculus that \(f\) is an injection, and that \(\operatorname{Ran}(f)=\mathbb{R}^{+}\) . Hence \(f\) is not a surjection, since the implicit codomain of a real function is \(\mathbb{R}\) . The function \[\begin{aligned} g: \mathbb{R} & \rightarrow \mathbb{R}^{+} \\ x & \mapsto e^{x} \end{aligned}\] is a bijection and \[g^{-1}(x)=\log _{e}(x)\] Warning: For \(f: X \mapsto Y\) a bijection we have assigned two different meanings to \(f^{-1}(b)\) . In Definition 1.31, it means the set of points in \(X\) that get mapped to \(b\) . In Definition 1.36, it means the inverse function, \(f^{-1}\) , of the bijection \(f\) applied to the point \(b \in Y\) . However, if \(f\) is a bijection, so that the second definition makes sense, then these definitions are closely related. Suppose \(a \in \operatorname{Dom}(f)\) and \(f(a)=b\) . According to Definition 1.31, \(f^{-1}(b)=\{a\}\) and by Definition \(1.36\) \(f^{-1}(b)=a\) . In practice the context will make clear which definition is intended. DEFINITION. Identity function, id \(\left.\right|_{X}\) Let \(X\) be a set. The identity function on \(X\) , id \(\left.\right|_{X}: X \mapsto X\) , is the function defined by \[\left.\operatorname{id}\right|_{X}(x)=x .\] If \(f: X \rightarrow Y\) is a bijection, then \(f^{-1}\) is the unique function such that \[f^{-1} \circ f=\left.\operatorname{id}\right|_{X}\] and \[f \circ f^{-1}=\left.\mathrm{id}\right|_{Y} .\] Because \(f(x)=x^{2}\) is not an injection, it has no inverse, even after restricting the codomain to be the range. Therefore in order to "invert" \(f\) , we considered a different function \(g(x)\) , which was equal to \(f\) on a subset of the domain of \(f\) , and was an injection. In Example 1.37, we accomplished this by defining the function \(g(x)=x^{2}\) with domain \(\{x \in \mathbb{R} \mid x \geq 0\}\) . Many of the functions that we need to invert for practical and theoretical reasons happen not to be injections, and hence do not have inverse functions. One way to address this obstacle is to consider the function on a smaller domain.
Given a function, \(f: X \rightarrow Y\) we may wish to define an "inverse" of \(f\) on some subset of \(W \subseteq X\) for which the restriction of \(f\) to \(W\) is an injection.
Let \(f: X \rightarrow Y\) and \(W \subseteq X\) . The restriction of \(f\) to \(W\) , written \(\left.f\right|_{W}\) , is the function \[\begin{aligned} \left.f\right|_{W}: W & \rightarrow Y \\ x & \mapsto f(x) . \end{aligned}\] So if \(f: X \rightarrow Y\) and \(W \subseteq X\) , then \[\operatorname{graph}\left(\left.f\right|_{W}\right)=[W \times Y] \cap[\operatorname{graph}(f)] .\] EXAMPLE 1.40. Let \(f(x)=(x-2)^{4}\) . Let \(W=[2, \infty)\) . Then \[\left.f\right|_{W}: W \rightarrow[0, \infty)\] is a bijection. ExAmPLE 1.41. Let \(f\) be the real function, \(f(x)=\tan (x)\) . Then \[\operatorname{Dom}(f)=\{x \in \mathbb{R} \mid x \neq \pi / 2+k \pi, k \in \mathbb{Z}\},\] and \[\operatorname{Ran}(f)=\mathbb{R} .\] The function \(f\) is periodic with period \(\pi\) , and is therefore not an injection. Nonetheless, it is important to answer the question, "At what angle(s), \(x\) , does \(\tan (x)\) equal a particular value, \(a \in \mathbb{R}\) ".
This is mathematically equivalent to asking, \[\text { "What is } \arctan (a) \text { ?". }\] In calculus this need was met by restricting the domain to a largest interval, \(I\) such that \[\left.f\right|_{I}: I \mapsto \mathbb{R}\] For any \(k \in \mathbb{Z}\) , \[\left(\frac{(2 k+1) \pi}{2}, \frac{(2 k+3) \pi}{2}\right)\] is such an interval. In order to define a specific function, the simplest of these intervals is selected, and we define \[\operatorname{Tan}:=\left.\tan \right|_{(-\pi / 2, \pi / 2)} .\] Observe that \[\text { Tan : }(-\pi / 2, \pi / 2) \mapsto \mathbb{R} .\] So the function is invertible, that is, Tan has an inverse function, \[\operatorname{Arctan}=\operatorname{Tan}^{-1} \text {. }\]


8.6.1: Review Questions
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Review Questions
1 .
What assessment finding suggests that a patient may have fibrocystic breast changes?
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green-tinged nipple discharge
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ongoing breast pain
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firm, ropy feel of the breast tissue under the skin
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peau d’orange appearance of the skin
2 .
What nursing intervention would the nurse include in the preoperative care plan for a patient scheduled for an outpatient lumpectomy of a fibroadenoma?
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Administer pain and antiemetic medications.
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Help the patient arrange for a cab home after the procedure.
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Advise the patient to avoid food or drink for 2 hours before surgery.
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Review discharge instructions.
3 .
What is a patient statement that indicates effective teaching after a new diagnosis of nonlactational mastitis?
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“I will avoid wearing a bra until I feel better.”
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“I should take steps to stop smoking cigarettes.”
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“Applying an ice pack to my breasts can help reduce pain.”
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“I should take the prescribed antibiotics until I feel better.”
4 .
What is a risk factor for breast cancer?
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menarche at age 15
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family history of diabetes
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alcohol use of at least 2 drinks per day
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history of three vaginal births by age 25
5 .
What information would the nurse include when teaching a patient about core needle biopsy?
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It can be performed with or without ultrasound guidance.
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Local anesthetic is not needed before this procedure.
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It is usually performed in the outpatient surgery center.
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A special dye is injected into the area around the tumor.
6 .
The nurse is caring for a patient receiving chemotherapy. What nursing intervention would they provide?
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Administer antibiotics.
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Administer antiemetics.
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Encourage physical activity after infusion.
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Encourage eating a large meal.
7 .
What type of mastectomy removes the entire breast but leaves the chest wall intact?
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total simple mastectomy
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modified radical mastectomy
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skin-sparing mastectomy
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nipple-sparing mastectomy
8 .
The nurse would advise the patient to report what symptom immediately during a PD-1 (checkpoint inhibitor) infusion?
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nausea
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fatigue
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dizziness
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diarrhea
9 .
What medication would the nurse include when teaching a patient about aromatase inhibitors?
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anastrozole (Arimidex)
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fulvestrant (Faslodex)
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tamoxifen (Novaldex)
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pembrolizumab (Keytruda)
10 .
What hormonal medication can be used in a premenopausal patient without the concurrent use of ovarian suppression?
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letrozole (Femara)
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exemestane (Aromasin)
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elacestrant (Orserdu)
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tamoxifen (Soltamox)


9.2: Auto Lab Equipment Safety
Reflection: Auto Shop Equipment
Vehicles are used in everyday life as people drive to work, to see family, or just for the enjoyment of travel. In order for vehicles to be road-worthy, they must be in top operating condition. Failure of any vehicle’s component or system may result in injury or even death. Auto technicians receive vehicles brought in for routine maintenance and/or service. Technicians in automotive shops when servicing vehicles utilize numerous equipment that must be routinely inspected, operated correctly and safely.
Review the Automotive Shop Equipment discussed below and identify the standard that would compliment a safety protocol for the safe use of that equipment. Considering the hierarchy of controls what type of control is it? Reference workbook section 7: Tools and Equipment-Industrial Safety and Manufacturing and textbook chapter 12: Crane and Hoists for insight.
Automotive Shop Equipment-Hazards
There are several different areas in an auto shop. You must describe the areas by name and understand the basic rules that apply to each. It is important that you learn your shop layout and organization to improve work efficiency and safety. The auto shop includes the following work areas:
- Repair area
- Tool room
- Classroom
Repair Area
The repair area includes any location in the shop where repair operations are performed, such as in the shop stall, lift, alignment rack, and outside work area. It normally includes every area except the classroom, locker room, and tool room.
Shop Stall
A shop stall is a small work area where a vehicle can be parked for repairs. Sometimes, each stall is numbered and marked off with lines painted on the floor. Some auto shop facilities have an outside work area adjacent to the garage overhead doors. In good weather, this area can be used for auto repairs. Always raise the shop doors all the way and pull cars through the doors very slowly.
Lift
The lift is used to raise a vehicle into the air. It is necessary for working under the car (draining oil, greasing front end ball sockets, or repairing exhaust systems).
Remember these lift safety steps:
- Ask your instructor for a demonstration and get permission before using the lift. You must be trained and authorized to use the lift.
- Always read manufacturer's operating instructions closely before using a lift.  A-Most lifts have a mechanical safety catch. A large metal lug must be engaged fully into the metal post before working under the vehicle. Without the mechanical catch engaged, a hydraulic failure could make the vehicle drop instantly to the ground with deadly crushing force.
- Inspect the lift, the floor area around the lift.
- Position the vehicle's center of gravity (point of perfect balance) on the lift as described in the vehicle service manual. If a front-engine vehicle or a pickup truck with no weight in the bed, position the lift arms (if applicable) more to the front of the chassis. In a mid- or rear-engine car, place the lift pads more to the rear so the car will not slide or tilt off the lift.
- Raise vehicle slowly while keeping your fingers away from any moving parts on the lift. Raise the car about one inch off the ground. Double check your lift points and make sure the vehicle is perfectly level. Always place your head next to the shop floor so you can see the alignment of the liftpad and vehicle pinch welds or frame rails.
- When the vehicle is high enough to be worked on, make sure the lift's mechanical safety catch is engaged. Do not walk under the lift without the catch in the fully locked position
Figure \(\PageIndex{1}\): Diagram of Lift. (Source; Greg Ling-Long Beach City College)
Alignment Rack
The alignment rack is used when working on a car's steering and suspension systems. It may contain a special tool board and equipment used when replacing worn suspension and steering parts, or adjusting wheel alignment. When using an alignment rack, the car should be pulled onto the rack slowly and carefully. Someone should guide the driver and help keep the tires centered on the rack. As with other complicated and potentially dangerous equipment, obtain a full demonstration before using the alignment rack.
Tool room
The tool room is a shop area where larger pieces of repair equipment are stored. The tool room is used to store shop tools, small equipment, and supplies, such as nuts, bolts, and oil. It is normally located adjacent to (next to) the repair area or classroom. When working in the tool room, you will be responsible for keeping track of shop tools. Every tool checked out of the tool room must be recorded and called in or retrieved before the end of the class period.
Normally, the tools hang on the walls of the tool room for easy access. Each tool may have a painted silhouette, which indicates where it should be kept. Your instructor will detail specific tool room policies and procedures.
Classroom / Training Room
The classroom is usually an office-like area in a school or large repair facility, where students or mechanics can meet to increase their knowledge of auto service and repair. Either a certified instructor or the service manager presents the information to the students or experienced technicians. The classroom is used for seminars, demonstrations, and other technician training activities. It is often located adjacent to the repair area. Follow all posted safety rules for the classroom.
Flash Card-Key terms and Definitions
This interactive feature not available in print version of this workbook
Standard Mapping - Identify Overlapping Standards for Automotive Safety
Working in the automotive shop is a hazardous environment which requires safety to avoid injury or possible death. As a technician, you will be working with tools and equipment to service and repair vehicles. In this activity you will map the requirements for tool and equipment safety and specific safety warnings and procedures to safety standards to OSHA safety standards applicable to shop safety.
Standard Mapping Table
| General Industry Standard Subpart | Section | Shop Equipment or tool Safety Warning, Safety Procedure |
|---|---|---|
| 1910 Subpart F – Powered Platforms, Manlifts, and Vehicle-Mounted Work Platforms | 1910.67 – Vehicle-Mounted elevating and rotating work platforms | |
| 1910 Subpart H - Hazardous Materials | 1910.106 - Flammable liquids | |
| 1910 Subpart I - Personal Protective Equipment | 1910.132 - General requirements | |
| 1910 Subpart I - Personal Protective Equipment | 1910.133 - Eye and face protection | |
| 1910 Subpart I - Personal Protective Equipment | 1910.134 - Respiratory Protection | |
| 1910 Subpart I - Personal Protective Equipment | 1910.138 - Hand Protection | |
| 1910 Subpart K - Medical and First Aid | 1910.151 - Medical services and first aid | |
| 1910 Subpart L - Fire Protection | 1910.157 - Portable fire extinguishers | |
| 1910 Subpart L - Fire Protection | 1910.165 - Employee alarm systems | |
| 1910 Subpart N - Materials Handling and Storage | 1910.179 - Overhead and gantry cranes | |
| 1910 Subpart O - Machinery and Machine Guarding | 1910.212 - General requirements for all machines | |
| 1910 Subpart O - Machinery and Machine Guarding | 1910.217 - Mechanical power presses | |
| 1910 Subpart P - Hand and Portable Powered Tools and Other Hand-Held Equipment | 1910.242 - Hand and portable powered tools and equipment, general |
Video-Equipment Safety
How to use 2-post Lift
How it works?!? The CAR WIZARD shows how to operate a two-post car lift.
Vehicle Lift Failures
Case Study – Fatality
Discuss in your group the specifics of the case below. The details are summarized. View the figures in the report showing the shop equipment and usage of the vehicle lift. Discuss in your groups how you would have prevented this incident. Consider creating a JHA to support your case.
Are there any standards that mitigate or are a work around for servicing a vehicle while on a lift? What are they and how would you implement?
Subject: Accident: 14543334 - Vehicle Fell From Auto Lift, Mechanic Crushed Under Auto
Accident detail
Introduction
Employee #1, A mechanic, was repairing a tie-rod on a vehicle which was suspended over his head on an auto lift. He inadvertently struck the lift securing pad with his hammer, causing the telescoping movable securing lift arm to move. The vehicle fell from the auto lift to the garage floor. The mechanic was caught under the falling car and crushed to death. The auto lift was manufactured by The Dover Corp., Rotary Lift Division. The auto lift's ID number was E770 3022M; serial number 0500, part number P451. The lift’s model number was AB00137-SP80 MARK II. It had a 7,000 pound lift capacity. The manufacturer’s lift instructions were not followed during installation of the auto lift.
Employer: Gronski Enterprises 3905 Birney Avenue Scranton, PA 18505


5.2: Song Cycles
We will begin by looking at collections of songs that work together to tell a story that is emotionally complex, if not heavy in plot detail. Such a collection can be called a song cycle . A song cycle usually consists of about eight to twenty songs that use carefully crafted texts and music to present a cohesive narrative. Each song is distinct from the others and the order cannot be changed. While the term song cycle is most often applied to works from the art music world, it is valid across many genres. When a popular artist releases an album of songs that accomplish the purpose of a song cycle, however, the product is referred to as a concept album . The most important difference between a song cycle and concept album is that the former is most commonly conceived of with live performance in mind, whereas the latter is often developed in the studio and consumed as a recording. For this reason, a song cycle is more likely to have limited instrumentation, while producers of concept albums often have a wider variety of sound tools at their disposal. We will begin by considering a concept album that includes not only sounds but images and spoken poetry.


13.4: Three Strategies of Navigating Today's Hollywood
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- Kristen J. Warner
- University of California Press
Strategy 1: Blind Casting
The physical embodiment of visual difference rather than a qualitatively meaningful representation of difference, blind casting is an illusion of equality and parity in casting. In other words, it functions as a form of diversity you can count rather than a notion of diversity that accounts for the nature of the roles or content. Blind casting thus operates as a way to increase diversity in physical difference without investing in any associated cultural differences. Colorblind casting logic is useful to guilds like the Screen Actors Guild (SAG-AFTRA). Because the guild has no authority over or investment in the form or quality of the employment (that is, the guild does not regulate content), it cares less about the nature of the role than about counting that role as an employment gain for their members. Colorblind casting logic thus offers an easy method to assuage dual concerns: it makes available more job opportunities for the least employed sectors of the guild’s membership and imbues those job opportunities with an air of “respectability” because of the way colorblind logic evacuates any cultural specificity in its operating logic in favor of normative whiteness.
Yet, underpinned by neoliberal race logic, colorblind casting deploys a universal, “we are all the same” rhetoric that only superficially addresses the issues of diversity, employment, and racial representation in television and film. Its resolution relies solely on visible difference. Blind casting thus forces minority actors who desire employment to input cultural differences and output a standardized form of whiteness. Moreover, that this input/output practice has become so commonsensical makes acknowledging its existence a bit of a conundrum.  When I make visible the dissonance that occurs when blind casting places an actor of color in a narrative context that, say, isolates him or her from a larger community of color, I am the racist for making a fuss over what Stuart Hall calls “matter out of place.” 17 Rather than identifying how this process erases a community’s socio-historical specificity, the logic requires that we celebrate its evacuation of race as an issue at all. Diversity matters now only inasmuch as the networks (and guilds) can count the representation of visible difference.
Yet identifying “matter out of place” is one way to observe the failure of the blind casting strategy. If blind-casting roles for actors of color ultimately normalizes them to the degree that they become culturally illegible, the same effects can reveal the dangers of such an enterprise. When writers do not consider racial difference and history as part of their character’s backstory, they too often succumb to a set of unintended racially troped pitfalls. Consider the blind-cast role of the Black character Bonnie Bennett—a witch—in CW’s The Vampire Diaries ( TVD ). Bennett, whose original surname in the book series, McCullough, was changed to Bennett as a consequence of the casting decision, is a central character in the televised series. According to the original material, Bonnie McCullough is a fair-skinned redhead. Yet Bonnie Bennett signifies as an African American teenager. While it is a laudable effort on the part of the network and its executives to diversify TVD ’s ensemble cast, the failure to adjust the character’s backstory to account for the long history of racialized imagery of Black women and witchcraft opens the series to a number of accidental pitfalls. In the series, Bonnie belongs to a family of witches who historically served as slaves to the lead actors. Continuing in the tradition, Bonnie’s servitude to her white friends results in her sacrificing her life for theirs. Finally, unlike Bonnie’s female counterparts, who are immersed in teen sexuality and coupling—a vital convention of the teen drama—Bonnie is rarely paired with a love interest. Instead, her sole devotion is to those she serves. Collectively, these tropes raise troubling historical associations with Black representation and further perpetuate the sort of symbolic violence against Black female bodies that blind casting’s postracial ethos is intended to counter.
Despite the pitfalls, blind casting remains a viable option for employment when few other promising opportunities exist for actors of color. Moreover, because the parts are written normatively, many actors themselves celebrate the opportunities as “respectable” alternatives to race-specific casting calls, which often perpetuate troubling stereotypes.
Strategy 2: Racially Ambiguous Performance
A second strategy deployed by underrepresented groups to circumvent the industrial barriers to employment is self-fashioning as a racially ambiguous actor. Recalling the earlier statistic from SAG-AFTRA, racially unknown/other actors accounted for 4.1 percent of all roles in 2008. The category “racially unknown/ other” designates actors who did not select a racial or ethnic identity on the surveys SAG-AFTRA sends to identify the racial and ethnic makeup of its membership. According to interviews with guild representatives, members opt not to self-identify because they fear it will relegate them to the limited roles intended for a particular racialized group. Implicit in this trend is the practice of “passing” among those actors who believe they can be cast in roles with a racial identity other than their own.
For instance, the biracial identities of Jessica Szohr from CW’s Gossip Girl and Rashida Jones from NBC’s Parks and Recreation remain “unmarked” in these texts and others in which they appear. Similarly, Troian Bellisario’s racial ethnicity is ambiguous enough to allow her to pass as just one of the (white) girls in ABC Family’s Pretty Little Liars —even though reading her body suggests there is something “not quite white” about her character. Beyond indeterminate racial identities, racially or ethnically ambiguous performers find themselves cast as multiple races and ethnicities. Blair Redford’s ambiguous look allowed him to be Latino for ABC Family’s Switched at Birth and American Indian for ABC Family’s The Lying Game . While the logic behind this strategy might increase an actor’s employment opportunities by expanding the number of types through which his or her look is interpreted, it also privileges (oftentimes racist) assumptions about the look of a given racial group.
Racially ambiguous performers also amplify colorblindness’s insidious power. As a strategy, it not only makes race something that is “unseen” but detaches racialized bodies from their socio-historical contexts. In other words, the actors function as empty signifiers in that their bodies can be read by audiences in multiple ways, and they can be placed in infinite settings without being tethered to a reality rooted in the socio-historical specifics of their racial and cultural experiences. Furthermore, racial ambiguity allows the network to claim diversity without engaging with the concept beyond superficial (physical) differences.
Strategy 3: Universal Discourse
The final strategy I want to discuss is one that, unlike the first two, is applicable to a variety of laborers in the film and television industry. It concerns distributing and marketing film with predominately Black casts that are also written, directed, and/or produced by Black creative talent. From films like The Best Man Holiday (2013) to Think Like a Man (2012), and Think Like a Man Too (2014) to the About Last Night (2014) remake, publicity and advertising largely frame these films within a universalist discourse—one designed to assure (white) mainstream audiences that the experiences onscreen are both “human” and “relatable” even though the characters may not look like them and elements remain that are, in fact, quite culturally specific. 18 Consider the promotional strategy for Think Like a Man . According to a 2012 Vulture article, while Black producer Will Packer devoted a large portion of his marketing and advertising budget to flying the cast to events with large numbers of African Americans in the audience, he deployed an alternative strategy to draw white audiences. Here he relied heavily on “crossover” comedian Steve Harvey—who hosts the daytime game show Family Feud and wrote the film’s source material—as the movie’s messenger. “Packer has deployed Steve Harvey . . . to sell the ‘everybody’s welcome!’ message to the general public, sending him out to tub-thump . . . on CBS’s This Morning and ABC’s The View , shows that Packer explains, ‘don’t necessarily over-index with African Americans.’”19 Moreover, to target white women, Packer and his team stressed the romantic comedy conventions of the film via a television campaign that, according to one former studio marketing head, looked “like classic Romantic Comedy 101. In fact, it looks like a Nancy Meyers movie, with black people. Which is fine. . . . All it has to be is funny, and make it clear that the concept has no race.” 20
Yet at what cost comes this universal rhetoric? Extra labor taken on by the actors and producers during these press junkets and promotional events to sell the film as fitting for “all” is expected from marginalized bodies if they desire to reach a mainstream audience. Films with predominately white casts are not expected to sell their films (domestically at least) as universal and relatable because they always already operate within the normative and authentic standards by which we judge the human experience. Similar to blind casting, the burden falls on the person of color to perform his or her “sameness” as a mechanism to ensure that the preferred demographic is not alienated from the production. As Packer asserts, “There is a process to get those audiences. It starts with making a film like this, which is broad, smart, and one where there’s no cultural or ethnic specificity that would not be relatable to mainstream Americans.” 21 Black cast films, then, are not a niche production; they are the benign reflection of a large, multicultured world that poses no threat to liberal sensibilities and consumption practices.
Universal discourse underscores the historical precariousness of minorities in the creative industries whose labor always existed under this double bind structure that equates success with being both similar to and different from the normative order. Such a double bind recalls how Clyde Taylor defines the mode of Black film production in early cinema as one of “unequal development,” that is, a phenomenon that exists where there is “an exploitative/dependent relationship that ultimately results in a more powerful society drawing from the less powerful selected goods and resources without regard for what the loss of those resources will mean to the exploited.” 22 According to similar logic, universal discourse insists that for creative laborers of color to participate in the film and television industries, they must embrace a rhetoric of sameness that not only elides their unequal professional footing but also encourages them to lose any sense of socio-historical specificity. Yet what must it mean to be a minority worker who, to find employment, must not only cross over to mainstream filmmaking but also disavow elements of his or her own racial identity to remain gainfully employed? Unequal development epitomizes precarity. It considers the minority worker—whose skills are utilized and borrowed, or more specifically, appropriated, for a variety of purposes—always operating on a conditional and probationary basis. Once the current diversity zeitgeist ends, so does the work.
I would extend the universal discourse to branding and would draw attention to how showrunner Shonda Rhimes, most recently described as a “revolutionary,” 23 tells the story of how when she looked at an invitation for an award ceremony at which she was to be honored and saw that she was described as the most powerful Black female showrunner, she scratched out the modifiers Black and female . 24 While I understand her desire to not be limited or constrained by those modifiers if she, in fact, is the most powerful showrunner of any description, her rationale that white men do not have to name themselves is based in structural power and an inherent specialness that allows them to be ex-nominated. They don’t have to be named because it’s common sense. Thus, though Rhimes’s refusal to take the modifier may for her be an insistence to transcend, the rhetoric ultimately reinforces the very whiteness implicit in industry. Further, while her strategy attempts to upend the “unequal development” of being considered successful only in relation to other Black female television producers—of which there is one: Rhimes—as opposed to being placed in contention with the predominately white male showrunners, by shrugging off those identity modifiers Rhimes reinscribes herself in a universalist posture. A posture that ironically makes racial difference a type of pathology one needs to be cured of, thus reinvigorating the very tenets of unequal development she hopes to quash.


2.4: Atomic Structure and Symbolism
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Learning Objectives
By the end of this section, you will be able to:
- Write and interpret symbols that depict the atomic number, mass number, and charge of an atom or ion
- Define the atomic mass unit and average atomic mass
- Calculate average atomic mass and isotopic abundance
The development of modern atomic theory revealed much about the inner structure of atoms. It was learned that an atom contains a very small nucleus composed of positively charged protons and uncharged neutrons, surrounded by a much larger volume of space containing negatively charged electrons. The nucleus contains the majority of an atom’s mass because protons and neutrons are much heavier than electrons, whereas electrons occupy almost all of an atom’s volume. The diameter of an atom is on the order of 10 −10 m, whereas the diameter of the nucleus is roughly 10 −15 m—about 100,000 times smaller. For a perspective about their relative sizes, consider this: If the nucleus were the size of a blueberry, the atom would be about the size of a football stadium (Figure \(\PageIndex{1}\)).
Atoms—and the protons, neutrons, and electrons that compose them—are extremely small. For example, a carbon atom weighs less than 2 × 10 −23 g, and an electron has a charge of less than 2 × 10 −19 C (coulomb). When describing the properties of tiny objects such as atoms, we use appropriately small units of measure, such as the atomic mass unit (amu) and the fundamental unit of charge (e) . The amu was originally defined based on hydrogen, the lightest element, then later in terms of oxygen. Since 1961, it has been defined with regard to the most abundant isotope of carbon, atoms of which are assigned masses of exactly 12 amu. (This isotope is known as “carbon-12” as will be discussed later in this module.) Thus, one amu is exactly \(\frac{1}{12}\) of the mass of one carbon-12 atom: 1 amu = 1.6605 × 10 −24 g. (The Dalton (Da) and the unified atomic mass unit (u) are alternative units that are equivalent to the amu.) The fundamental unit of charge (also called the elementary charge) equals the magnitude of the charge of an electron (e) with e = 1.602 × 10 −19 C.
A proton has a mass of 1.0073 amu and a charge of 1+. A neutron is a slightly heavier particle with a mass 1.0087 amu and a charge of zero; as its name suggests, it is neutral. The electron has a charge of 1− and is a much lighter particle with a mass of about 0.00055 amu (it would take about 1800 electrons to equal the mass of one proton). The properties of these fundamental particles are summarized in Table \(\PageIndex{1}\). (An observant student might notice that the sum of an atom’s subatomic particles does not equal the atom’s actual mass: The total mass of six protons, six neutrons, and six electrons is 12.0993 amu, slightly larger than 12.00 amu. This “missing” mass is known as the mass defect, and you will learn about it in the chapter on nuclear chemistry.)
| Name | Location | Charge (C) | Unit Charge | Mass (amu) | Mass (g) |
|---|---|---|---|---|---|
| electron | outside nucleus | −1.602 10 −19 | 1− | 0.00055 | 0.00091 10 −24 |
| proton | nucleus | 1.602 10 −19 | 1+ | 1.00727 | 1.67262 10 −24 |
| neutron | nucleus | 0 | 0 | 1.00866 | 1.67493 10 −24 |
The number of protons in the nucleus of an atom is its atomic number (\(Z\)) . This is the defining trait of an element: Its value determines the identity of the atom. For example, any atom that contains six protons is the element carbon and has the atomic number 6, regardless of how many neutrons or electrons it may have. A neutral atom must contain the same number of positive and negative charges, so the number of protons equals the number of electrons. Therefore, the atomic number also indicates the number of electrons in an atom. The total number of protons and neutrons in an atom is called its mass number (\(A\)) . The number of neutrons is therefore the difference between the mass number and the atomic number: A – Z = number of neutrons.
\[\begin{align*}
\text { atomic number }( Z ) & =\text { number of protons } \\[4pt]
\text { mass number }( A ) & =\text { number of protons }+ \text { number of neutrons } \\[4pt]
A - Z & =\text { number of neutrons }
\end{align*} \nonumber \]
Atoms are electrically neutral if they contain the same number of positively charged protons and negatively charged electrons. When the numbers of these subatomic particles are not equal, the atom is electrically charged and is called an ion . The charge of an atom is defined as follows:
\[\text{Atomic charge} = \text{number of protons} − \text{number of electrons} \nonumber \]
As will be discussed in more detail later in this chapter, atoms (and molecules) typically acquire charge by gaining or losing electrons. An atom that gains one or more electrons will exhibit a negative charge and is called an anion . Positively charged atoms called cations are formed when an atom loses one or more electrons. For example, a neutral sodium atom (Z = 11) has 11 electrons. If this atom loses one electron, it will become a cation with a 1+ charge (11 − 10 = 1+). A neutral oxygen atom (Z = 8) has eight electrons, and if it gains two electrons it will become an anion with a 2− charge (8 − 10 = 2−).
Example \(\PageIndex{1}\): Composition of an Atom
Iodine is an essential trace element in our diet; it is needed to produce thyroid hormone. Insufficient iodine in the diet can lead to the development of a goiter, an enlargement of the thyroid gland (Figure \(\PageIndex{2}\)).
The addition of small amounts of iodine to table salt (iodized salt) has essentially eliminated this health concern in the United States, but as much as 40% of the world’s population is still at risk of iodine deficiency. The iodine atoms are added as anions, and each has a 1− charge and a mass number of 127. Determine the numbers of protons, neutrons, and electrons in one of these iodine anions.
Solution
The atomic number of iodine (53) tells us that a neutral iodine atom contains 53 protons in its nucleus and 53 electrons outside its nucleus. Because the sum of the numbers of protons and neutrons equals the mass number, 127, the number of neutrons is 74 (127 − 53 = 74). Since the iodine is added as a 1− anion, the number of electrons is 54 [53 – (1–) = 54].
Exercise \(\PageIndex{1}\)
An ion of platinum has a mass number of 195 and contains 74 electrons. How many protons and neutrons does it contain, and what is its charge?
- Answer
-
78 protons; 117 neutrons; charge is 4+
Chemical Symbols
A chemical symbol is an abbreviation that we use to indicate an element or an atom of an element. For example, the symbol for mercury is Hg (Figure \(\PageIndex{3}\)). We use the same symbol to indicate one atom of mercury (microscopic domain) or to label a container of many atoms of the element mercury (macroscopic domain).
The symbols for several common elements and their atoms are listed in Table \(\PageIndex{2}\). Some symbols are derived from the common name of the element; others are abbreviations of the name in another language. Most symbols have one or two letters, but three-letter symbols have been used to describe some elements that have atomic numbers greater than 112. To avoid confusion with other notations, only the first letter of a symbol is capitalized. For example, Co is the symbol for the element cobalt, but CO is the notation for the compound carbon monoxide, which contains atoms of the elements carbon (C) and oxygen (O). All known elements and their symbols are in the periodic table (also found in Appendix A).
| Element | Symbol | Element | Symbol |
|---|---|---|---|
| aluminum | Al | iron | Fe (from ferrum ) |
| bromine | Br | lead | Pb (from plumbum ) |
| calcium | Ca | magnesium | Mg |
| carbon | C | mercury | Hg (from hydrargyrum ) |
| chlorine | Cl | nitrogen | N |
| chromium | Cr | oxygen | O |
| cobalt | Co | potassium | K (from kalium ) |
| copper | Cu (from cuprum ) | silicon | Si |
| fluorine | F | silver | Ag (from argentum ) |
| gold | Au (from aurum ) | sodium | Na (from natrium ) |
| helium | He | sulfur | S |
| hydrogen | H | tin | Sn (from stannum ) |
| iodine | I | zinc | Zn |
Traditionally, the discoverer (or discoverers) of a new element names the element. However, until the name is recognized by the International Union of Pure and Applied Chemistry (IUPAC), the recommended name of the new element is based on the Latin word(s) for its atomic number. For example, element 106 was called unnilhexium (Unh), element 107 was called unnilseptium (Uns), and element 108 was called unniloctium (Uno) for several years. These elements are now named after scientists (or occasionally locations); for example, element 106 is now known as seaborgium (Sg) in honor of Glenn Seaborg, a Nobel Prize winner who was active in the discovery of several heavy elements. Element 109 was named in honor of Lise Meitner, who discovered nuclear fission, a phenomenon that would have world-changing impacts; Meitner also contributed to the discovery of some major isotopes, discussed immediately below.
Link to Learning
Visit this site to learn more about IUPAC, the International Union of Pure and Applied Chemistry, and explore its periodic table.
Isotopes
The symbol for a specific isotope of any element is written by placing the mass number as a superscript to the left of the element symbol (Figure \(\PageIndex{4}\)). The atomic number is sometimes written as a subscript preceding the symbol, but since this number defines the element’s identity, as does its symbol, it is often omitted. For example, magnesium exists as a mixture of three isotopes, each with an atomic number of 12 and with mass numbers of 24, 25, and 26, respectively. These isotopes can be identified as 24 Mg, 25 Mg, and 26 Mg. These isotope symbols are read as “element, mass number” and can be symbolized consistent with this reading. For instance, 24 Mg is read as “magnesium 24,” and can be written as “magnesium-24” or “Mg-24.” 25 Mg is read as “magnesium 25,” and can be written as “magnesium-25” or “Mg-25.” All magnesium atoms have 12 protons in their nucleus. They differ only because a 24 Mg atom has 12 neutrons in its nucleus, a 25 Mg atom has 13 neutrons, and a 26 Mg has 14 neutrons.
Information about the naturally occurring isotopes of elements with atomic numbers 1 through 10 is given in Table \(\PageIndex{3}\). Note that in addition to standard names and symbols, the isotopes of hydrogen are often referred to using common names and accompanying symbols. Hydrogen-2, symbolized 2 H, is also called deuterium and sometimes symbolized D. Hydrogen-3, symbolized 3 H, is also called tritium and sometimes symbolized T.
| Element | Symbol | Atomic Number | Number of Protons | Number of Neutrons | Mass (amu) | % Natural Abundance |
|---|---|---|---|---|---|---|
| hydrogen |
\(\ce{^{1}_{1}H}\)
(protium) |
1 | 1 | 0 | 1.0078 | 99.989 |
|
(deuterium) |
1 | 1 | 1 | 2.0141 | 0.0115 | |
|
\(\ce{^{3}_{ 1} H}\)
(tritium) |
1 | 1 | 2 | 3.01605 | — (trace) | |
| helium | \(\ce{^{3}_{2}He}\) | 2 | 2 | 1 | 3.01603 | 0.00013 |
| \(\ce{^{4}_{2}He}\) | 2 | 2 | 2 | 4.0026 | 100 | |
| lithium | \(\ce{^{6}_{3 }Li}\) | 3 | 3 | 3 | 6.0151 | 7.59 |
| \(\ce{^{7}_{3 }Li}\) | 3 | 3 | 4 | 7.0160 | 92.41 | |
| beryllium | \(\ce{^{9}_{4}Be}\) | 4 | 4 | 5 | 9.0122 | 100 |
| boron | \(\ce{^{10}_{5 }B}\) | 5 | 5 | 5 | 10.0129 | 19.9 |
| \(\ce{^{11}_{5 }B}\) | 5 | 5 | 6 | 11.0093 | 80.1 | |
| carbon | \(\ce{^{12}_{ 6}C}\) | 6 | 6 | 6 | 12.0000 | 98.89 |
| \(\ce{^{13}_{ 6}C}\) | 6 | 6 | 7 | 13.0034 | 1.11 | |
| \(\ce{^{14}_{ 6}C}\) | 6 | 6 | 8 | 14.0032 | — (trace) | |
| nitrogen | \(\ce{^{14}_{ 7}N}\) | 7 | 7 | 7 | 14.0031 | 99.63 |
| \(\ce{^{15}_{ 7}N}\) | 7 | 7 | 8 | 15.0001 | 0.37 | |
| oxygen | \(\ce{^{16}_{8 1}O}\) | 8 | 8 | 8 | 15.9949 | 99.757 |
| \(\ce{^{17}_{8 1}O}\) | 8 | 8 | 9 | 16.9991 | 0.038 | |
| \(\ce{^{18}_{8 1}O}\) | 8 | 8 | 10 | 17.9992 | 0.205 | |
| fluorine | \(\ce{^{19}_{ 9}F}\) | 9 | 9 | 10 | 18.9984 | 100 |
| neon | \(\ce{^{20}_{ 10}Ne}\) | 10 | 10 | 10 | 19.9924 | 90.48 |
| \(\ce{^{21}_{ 10}Ne}\) | 10 | 10 | 11 | 20.9938 | 0.27 | |
| \(\ce{^{22}_{ 10}Ne}\) | 10 | 10 | 12 | 21.9914 | 9.25 |
Link to Learning
Use this Build an Atom simulator to build atoms of the first 10 elements, see which isotopes exist, check nuclear stability, and gain experience with isotope symbols.
Atomic Mass
Because each proton and each neutron contribute approximately one amu to the mass of an atom, and each electron contributes far less, the atomic mass of a single atom is approximately equal to its mass number (a whole number). However, the average masses of atoms of most elements are not whole numbers because most elements exist naturally as mixtures of two or more isotopes.
The mass of an element shown in a periodic table or listed in a table of atomic masses is a weighted, average mass of all the isotopes present in a naturally occurring sample of that element. This is equal to the sum of each individual isotope’s mass multiplied by its fractional abundance.
\[\text { average mass }=\sum_i(\text { fractional abundance } \times \text { isotopic mass })_i \nonumber \]
For example, the element boron is composed of two isotopes: About 19.9% of all boron atoms are \(\ce{^{10}B}\) with a mass of 10.0129 amu, and the remaining 80.1% are \(\ce{^{11}B}\) with a mass of 11.0093 amu. The average atomic mass for boron is calculated to be:
\[\begin{aligned}\text { boron average mass } & =(0.199 \times 10.0129 \,\text{amu})+(0.801 \times 11.0093 \,\text{amu}) \\[4pt]
& =1.99 \,\text{amu}+8.82 \,\text{amu}\\[4pt]
& =10.81 \,\text{amu}
\end{aligned} \nonumber \]
It is important to understand that no single boron atom weighs exactly 10.8 amu; 10.8 amu is the average mass of all boron atoms, and individual boron atoms weigh either approximately 10 amu or 11 amu.
Example \(\PageIndex{2}\): Calculation of Average Atomic Mass
A meteorite found in central Indiana contains traces of the noble gas neon picked up from the solar wind during the meteorite’s trip through the solar system. Analysis of a sample of the gas showed that it consisted of 91.84% 20 Ne (mass 19.9924 amu), 0.47% 21 Ne (mass 20.9940 amu), and 7.69% 22 Ne (mass 21.9914 amu). What is the average mass of the neon in the solar wind?
Solution
\[\begin{aligned}
\text { average mass } & =(0.9184 \times 19.9924\,\text{amu} )+(0.0047 \times 20.9940 \,\text{amu})+(0.0769 \times 21.9914 \,\text{amu}\\[4pt]
& =(18.36+0.099+1.69) \,\text{amu}\\[4pt]
& =20.15 \,\text{amu}
\end{aligned} \nonumber \]
The average mass of a neon atom in the solar wind is 20.15 amu. (The average mass of a terrestrial neon atom is 20.1796 amu. This result demonstrates that we may find slight differences in the natural abundance of isotopes, depending on their origin.)
Exercise \(\PageIndex{2}\)
A sample of magnesium is found to contain 78.70% of 24 Mg atoms (mass 23.98 amu), 10.13% of 25 Mg atoms (mass 24.99 amu), and 11.17% of 26 Mg atoms (mass 25.98 amu). Calculate the average mass of a Mg atom.
- Answer
-
24.31 amu
We can also do variations of this type of calculation, as shown in the next example.
Example \(\PageIndex{3}\): Calculation of Percent Abundance
Naturally occurring chlorine consists of 35 Cl (mass 34.96885 amu) and 37 Cl (mass 36.96590 amu), with an average mass of 35.453 amu. What is the percent composition of Cl in terms of these two isotopes?
Solution
The average mass of chlorine is the fraction that is 35 Cl times the mass of 35 Cl plus the fraction that is 37 Cl times the mass of 37 Cl.
\[\text { average mass }=\left(\text { fraction of }{ }^{35} Cl \times \text { mass of }{ }^{35} Cl \right)+\left(\text { fraction of }{ }^{37} Cl \times \text { mass of }{ }^{37} Cl \right) \nonumber \]
If we let x represent the fraction that is 35 Cl, then the fraction that is 37 Cl is represented by 1.00 − x .
(The fraction that is 35 Cl + the fraction that is 37 Cl must add up to 1, so the fraction of 37 Cl must equal 1.00 − the fraction of 35 Cl.)
Substituting this into the average mass equation, we have:
\[\begin{aligned}35.453 amu & =(x \times 34.96885 amu )+[(1.00-x) \times 36.96590 amu ] \\[4pt]
35.453 & =34.96885 x+36.96590-36.96590 x \\[4pt]
1.99705 x & =1.513 \\[4pt]
x & =\frac{1.513}{1.99705}=0.7576
\end{aligned} \nonumber \]
So solving yields: x = 0.7576, which means that 1.00 − 0.7576 = 0.2424. Therefore, chlorine consists of 75.76% 35 Cl and 24.24% 37 Cl.
Exercise \(\PageIndex{3}\)
Naturally occurring copper consists of 63 Cu (mass 62.9296 amu) and 65 Cu (mass 64.9278 amu), with an average mass of 63.546 amu. What is the percent composition of Cu in terms of these two isotopes?
- Answer
-
69.15% Cu-63 and 30.85% Cu-65
Link to Learning
Visit this site to make mixtures of the main isotopes of the first 18 elements, gain experience with average atomic mass, and check naturally occurring isotope ratios using the Isotopes and Atomic Mass simulation.
As you will learn, isotopes are important in nature and especially in human understanding of science and medicine. Let's consider just one natural, stable isotope: Oxygen-18, which is noted in the table above and is referred to as one of the environmental isotopes. It is important in paleoclimatology, for example, because scientists can use the ratio between Oxygen-18 and Oxygen-16 in an ice core to determine the temperature of precipitation over time. Oxygen-18 was also critical to the discovery of metabolic pathways and the mechanisms of enzymes. Mildred Cohn pioneered the usage of these isotopes to act as tracers, so that researchers could follow their path through reactions and gain a better understanding of what is happening. One of her first discoveries provided insight into the phosphorylation of glucose that takes place in mitochondria. And the methods of using isotopes for this research contributed to entire fields of study.
The occurrence and natural abundances of isotopes can be experimentally determined using an instrument called a mass spectrometer. Mass spectrometry (MS) is widely used in chemistry, forensics, medicine, environmental science, and many other fields to analyze and help identify the substances in a sample of material. In a typical mass spectrometer (Figure \(\PageIndex{5}\)), the sample is vaporized and exposed to a high-energy electron beam that causes the sample’s atoms (or molecules) to become electrically charged, typically by losing one or more electrons. These cations then pass through a (variable) electric or magnetic field that deflects each cation’s path to an extent that depends on both its mass and charge (similar to how the path of a large steel ball rolling past a magnet is deflected to a lesser extent that that of a small steel ball). The ions are detected, and a plot of the relative number of ions generated versus their mass-to-charge ratios (a mass spectrum ) is made. The height of each vertical feature or peak in a mass spectrum is proportional to the fraction of cations with the specified mass-to-charge ratio. Since its initial use during the development of modern atomic theory, MS has evolved to become a powerful tool for chemical analysis in a wide range of applications.


10.11: References
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American Academy of Family Physicians. (2023). AAFP-ACOG Joint Statement on Cooperative Practice and Hospital Privileges . https://www.aafp.org/about/policies/...statement.html
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American College of Obstetricians and Gynecologists. (2020a). Physical activity and exercise during pregnancy and the postpartum period. (ACOG Committee Opinion 804). https://www.acog.org/clinical/clinic...tpartum-period
American College of Obstetricians and Gynecologists. (2020b). Moderate caffeine consumption during pregnancy. (ACOG Committee Opinion 462). https://www.acog.org/-/media/project...-pregnancy.pdf
American College of Obstetricians and Gynecologists. (2022, July). Skin conditions during pregnancy. https://www.acog.org/womens-health/f...ring-pregnancy
American College of Obstetrics and Gynecology. (2018). Screening for perinatal depression . (ACOG Committee Opinion 757). journals.lww.com/greenjourna...ng_for.42.aspx
American College of Obstetrics and Gynecology. (2019, August). Levels of maternal care. (Obstetric Care Consensus 9). https://www.acog.org/clinical/clinic...-maternal-care
American Medical Association. (n.d.). Prenatal Screening Questionnaire . https://www.ama-assn.org/media/8706/download
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Atif, M., Farooq, M., Shafiq, M., Ayub, G., & Ilyas, M. (2023). The impact of partner’s behaviour on pregnancy related outcomes and safe child-birth in Pakistan. BMC Pregnancy and Childbirth , 23 (1), 516. https://doi.org/10.1186/s12884-023-05814-z
Bjelica, A., Cetkovic, N., Trninic-Pjevic, A., & Mladenovic-Segedi, L. (2018). The phenomenon of pregnancy: A psychological view. Ginekologia Polska, 89 (2), 102–106. https://doi.org/10.5603/GP.a2018.0017
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Eick, S. M., Goin, D. E., Izano, M. A., Cushing, L., DeMicco, E., Padula, A. M., Woodruff, T. J., & Morello-Frosch, R. (2020, June 12). Relationships between psychosocial stressors among pregnant women in San Francisco: A path analysis. Plos One . https://doi.org/10.1371/journal.pone.0234579
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Pantelis, A., Sotiriadis , A., Chatzistamatiou, K., Pratilas, G., & Dinas, K. (2018). Serum relaxin and cervical length for prediction of spontaneous preterm birth in second-trimester symptomatic women. Ultrasound in Obstetrics and Gynecology, 52 , 763–768. doi.org/10.1002/uog.18972
Parsons, A. M., & Bouma, G. J. (2021). A potential role and contribution of androgens in placental development and pregnancy. Life , 11 (7), 644. https://doi.org/10.3390/life11070644
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Sapkota, D., Dennison, S., Allen, J., Gamble, J., Williams, C., Malope-Rwodzi, N., Baar, L., Ransley, J., & McGee, T. R. (2022). Navigating pregnancy and early motherhood in prison: A thematic analysis of mothers’ experiences. Health & Justice, 10 (32). https://doi.org/10.1186/s40352-022-00196-4
Scrimshaw, S. C., & Backes, E. P. (2020). Birth settings in America: Outcomes, quality, access, and choice . (A Consensus Study Report of the National Academies of Health, Science, and Engineering). National Academies Press. https://nap.nationalacademies.org/re...5636/chapter/1
Seiermann, A. U., Al-Mufti, H., Waid, J. L., Wendt, A. S., Sobhan, S., & Gabrysch, S. (2021). Women’s fasting habits and dietary diversity during Ramadan in rural Bangladesh. Maternal & Child Nutrition , 17 (3), e13135. doi.org/10.1111/mcn.13135
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4.5: Division of Fractions
- be able to determine the reciprocal of a number
- be able to divide one fraction by another
Reciprocals
Two numbers whose product is 1 are called reciprocals of each other.
The following pairs of numbers are reciprocals.
\(\underbrace{\dfrac{3}{4} \text{and} \dfrac{4}{3}}_{\dfrac{3}{4} \cdot \dfrac{4}{3} = 1}\)
\(\underbrace{\dfrac{7}{16} \text{and} \dfrac{16}{7}}_{\dfrac{7}{16} \cdot \dfrac{16}{7} = 1}\)
\(\underbrace{\dfrac{1}{6} \text{and} \dfrac{6}{1}}_{\dfrac{1}{6} \cdot \dfrac{6}{1} = 1}\)
Notice that we can find the reciprocal of a nonzero number in fractional form by inverting it (exchanging positions of the numerator and denominator).
Practice Set A
Find the reciprocal of each number.
\(\dfrac{3}{10}\)
- Answer
-
\(\dfrac{10}{3}\)
Practice Set A
\(\dfrac{2}{3}\)
- Answer
-
\(\dfrac{3}{2}\)
Practice Set A
\(\dfrac{7}{8}\)
- Answer
-
\(\dfrac{8}{7}\)
Practice Set A
\(\dfrac{1}{5}\)
- Answer
-
5
Practice Set A
\(2 \dfrac{2}{7}\)
Write this number as an improper fraction first.
- Answer
-
\(\dfrac{7}{16}\)
Practice Set A
\(5 \dfrac{1}{4}\)
- Answer
-
\(\dfrac{4}{21}\)
Practice Set A
\(10 \dfrac{3}{16}\)
- Answer
-
\(\dfrac{16}{163}\)
Dividing Fractions
Our concept of division is that it indicates how many times one quantity is contained in another quantity. For example, using the diagram we can see that there are 6 one-thirds in 2.
There are 6 one-thirds in 2.
Since 2 contains six \(\dfrac{1}{3}\)'s we express this as
Using these observations, we can suggest the following method for dividing a number by a fraction.
Dividing One Fraction by Another
Fraction
To divide a first fraction by a second, nonzero fraction, multiply the first traction by the reciprocal of the second fraction.
Invert and
Multiply
This method is commonly referred to as
"invert the divisor and
multiply."
Perform the following divisions.
\(\dfrac{1}{2} \div \dfrac{3}{4}\). The divisor is \(\dfrac{3}{4}\). Its reciprocal is \(\dfrac{4}{3}\). Multiply \(\dfrac{1}{3}\) by \(\dfrac{4}{3}\).
\(\dfrac{1}{3} \cdot \dfrac{4}{3} = \dfrac{1 \cdot 4}{3 \cdot 3} = \dfrac{4}{9}\)
\(\dfrac{1}{2} \div \dfrac{3}{4} = \dfrac{4}{9}\)
\(\dfrac{3}{8} \div \dfrac{5}{4}\). The divisor is \(\dfrac{5}{4}\). Its reciprocal is \(\dfrac{4}{5}\). Multiply \(\dfrac{3}{8}\) by \(\dfrac{4}{5}\).
\(\dfrac{3}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{4}} \end{array}}{5} = \dfrac{3 \cdot 1}{2 \cdot 5} = \dfrac{3}{10}\)
\(\dfrac{3}{8} \div \dfrac{5}{4} = \dfrac{3}{10}\)
\(\dfrac{5}{6} \div \dfrac{5}{12}\). The divisor is \(\dfrac{5}{12}\). Its reciprocal is \(\dfrac{12}{5}\). Multiply \(\dfrac{5}{6}\) by \(\dfrac{12}{5}\).
\(\dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{\begin{array} {c} {\cancel{6}} \\ {^1} \end{array}} \cdot \dfrac{\begin{array} {c} {^2} \\ {\cancel{12}} \end{array}}{\begin{array} {c} {\cancel{5}} \\ {^1} \end{array}} = \dfrac{1 \cdot 2}{1 \cdot 1} = \dfrac{2}{1} = 2\)
\(\dfrac{5}{6} \div \dfrac{5}{12} = 2\)
\(2 \dfrac{2}{9} \div 3 \dfrac{1}{3}\). Convert each mixed number to an improper fraction.
\(2 \dfrac{2}{9} = \dfrac{9 \cdot 2 + 2}{9} = \dfrac{20}{9}\).
\(3 \dfrac{1}{3} = \dfrac{3 \cdot 3 + 1}{3} = \dfrac{10}{3}\).
\(\dfrac{20}{9} \div \dfrac{10}{3}\) The divisor is \(\dfrac{10}{3}\). Its reciprocal is \(\dfrac{3}{10}\). Multiply \(\dfrac{20}{9}\) by \(\dfrac{3}{10}\).
\(\dfrac{\begin{array} {c} {^2} \\ {\cancel{20}} \end{array}}{\begin{array} {c} {\cancel{9}} \\ {^3} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{10}} \\ {^1} \end{array}} = \dfrac{2 \cdot 1}{3 \cdot 1} = \dfrac{2}{3}\)
\(2 \dfrac{2}{9} \div 3 \dfrac{1}{3} = \dfrac{2}{3}\)
\(\dfrac{12}{11} \div 8\). First conveniently write 8 as \(\dfrac{8}{1}\).
\(\dfrac{12}{11} \div \dfrac{8}{1}\). The divisor is \(\dfrac{8}{1}\). Its reciprocal is \(\dfrac{1}{8}\). Multiply \(\dfrac{12}{11}\) by \(\dfrac{1}{8}\).
\(\dfrac{\begin{array} {c} {^3} \\ {\cancel{12}} \end{array}}{11} \cdot \dfrac{1}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} = \dfrac{3 \cdot 1}{11 \cdot 2} = \dfrac{3}{22}\)
\(\dfrac{12}{11} \div 8 = \dfrac{3}{22}\)
\(\dfrac{7}{8} \div \dfrac{21}{20} \cdot \dfrac{3}{35}\). The divisor is \(\dfrac{21}{20}\). Its reciprocal is \(\dfrac{20}{21}\).
\(\dfrac{\begin{array} {c} {^1} \\ {\cancel{7}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^{^1}} \\ {^{\cancel{5}}} \\ {\cancel{20}} \end{array}}{\begin{array} {c} {\cancel{21}} \\ {^{\cancel{3}}} \\ {^{^1}} \end{array}} \dfrac{\begin{array} {c} {^1} \\ {\cancel{3}} \end{array}}{\begin{array} {c} {\cancel{35}} \\ {^7} \end{array}} = \dfrac{1 \cdot 1 \cdot 1}{2 \cdot 1 \cdot 7} = \dfrac{1}{14}\)
\(\dfrac{7}{8} \div \dfrac{21}{20} \cdot \dfrac{3}{25} = \dfrac{1}{14}\)
How many \(2 \dfrac{3}{8}\) inch-wide packages can be placed in a box 19 inches wide?
The problem is to determine how many two and three eighths are contained in 19, that is, what is \(19 \div 2 \dfrac{3}{8}\)?
\(2\dfrac{3}{8} = \dfrac{19}{8}\) Convert the divisor \(2 \dfrac{3}{8}\) to an improper fraction.
\(19 = \dfrac{19}{1}\) Write the dividend 19 as \(\dfrac{19}{1}\).
\(\dfrac{19}{1} \div \dfrac{19}{8}\) The divisor is \(\dfrac{19}{8}\). Its reciprocal is \(\dfrac{8}{19}\).
\(\dfrac{\begin{array} {c} {^1} \\ {\cancel{19}} \end{array}}{1} \cdot \dfrac{8}{\begin{array} {c} {\cancel{19}} \\ {^1} \end{array}} = \dfrac{1 \cdot 8}{1 \cdot 1} = \dfrac{8}{1} = 8\)
Thus, 8 packages will fit into the box.
Practice Set B
Perform the following divisions.
\(\dfrac{1}{2} \div \dfrac{9}{8}\)
- Answer
-
\(\dfrac{4}{9}\)
Practice Set B
\(\dfrac{3}{8} \div \dfrac{9}{24}\)
- Answer
-
1
Practice Set B
\(\dfrac{7}{15} \div \dfrac{14}{15}\)
- Answer
-
\(\dfrac{1}{2}\)
Practice Set B
\(8 \div \dfrac{8}{15}\)
- Answer
-
15
Practice Set B
\(6 \dfrac{1}{4} \div \dfrac{5}{12}\)
- Answer
-
15
Practice Set B
\(3 \dfrac{1}{3} \div 1 \dfrac{2}{3}\)
- Answer
-
2
Practice Set B
\(\dfrac{5}{6} \div \dfrac{2}{3} \cdot \dfrac{8}{25}\)
- Answer
-
\(\dfrac{2}{5}\)
Practice Set B
A container will hold 106 ounces of grape juice. How many \(6 \dfrac{5}{8}\) -ounce glasses of grape juice can be served from this container?
- Answer
-
16 glasses
Determine each of the following quotients and then write a rule for this type of division.
Practice Set B
\(1 \div \dfrac{2}{3}\)
- Answer
-
\(\dfrac{3}{2}\)
Practice Set B
\(1 \div \dfrac{3}{8}\)
- Answer
-
\(\dfrac{8}{3}\)
Practice Set B
\(1 \div \dfrac{3}{4}\)
- Answer
-
\(\dfrac{4}{3}\)
Practice Set B
\(1 \div \dfrac{5}{2}\)
- Answer
-
\(\dfrac{2}{5}\)
Practice Set B
When dividing 1 by a fraction, the quotient is the .
- Answer
-
is the reciprocal of the fraction.
Exercises
For the following problems, find the reciprocal of each number.
Exercise \(\PageIndex{1}\)
\(\dfrac{4}{5}\)
- Answer
-
\(\dfrac{5}{4}\) or \(1 \dfrac{1}{4}\)
Exercise \(\PageIndex{2}\)
\(\dfrac{8}{11}\)
Exercise \(\PageIndex{3}\)
\(\dfrac{2}{9}\)
- Answer
-
\(\dfrac{9}{2}\) or \(4 \dfrac{1}{2}\)
Exercise \(\PageIndex{4}\)
\(\dfrac{1}{5}\)
Exercise \(\PageIndex{5}\)
\(3\dfrac{1}{4}\)
- Answer
-
\(\dfrac{4}{13}\)
Exercise \(\PageIndex{6}\)
\(8 \dfrac{1}{4}\)
Exercise \(\PageIndex{7}\)
\(3\dfrac{2}{7}\)
- Answer
-
\(\dfrac{7}{23}\)
Exercise \(\PageIndex{8}\)
\(5 \dfrac{3}{4}\)
Exercise \(\PageIndex{9}\)
1
- Answer
-
1
Exercise \(\PageIndex{10}\)
4
For the following problems, find each value.
Exercise \(\PageIndex{11}\)
\(\dfrac{3}{8} \div \dfrac{3}{5}\)
- Answer
-
\(\dfrac{5}{8}\)
Exercise \(\PageIndex{12}\)
\(\dfrac{5}{9} \div \dfrac{5}{6}\)
Exercise \(\PageIndex{13}\)
\(\dfrac{9}{16} \div \dfrac{15}{8}\)
- Answer
-
\(\dfrac{3}{10}\)
Exercise \(\PageIndex{14}\)
\(\dfrac{4}{9} \div \dfrac{6}{15}\)
Exercise \(\PageIndex{15}\)
\(\dfrac{25}{49} \div \dfrac{4}{9}\)
- Answer
-
\(\dfrac{225}{196}\) or \(1 \dfrac{29}{196}\)
Exercise \(\PageIndex{16}\)
\(\dfrac{15}{4} \div \dfrac{27}{8}\)
Exercise \(\PageIndex{17}\)
\(\dfrac{24}{75} \div \dfrac{8}{15}\)
- Answer
-
\(\dfrac{3}{5}\)
Exercise \(\PageIndex{18}\)
\(\dfrac{5}{7} \div 0\)
Exercise \(\PageIndex{19}\)
\(\dfrac{7}{8} \div \dfrac{7}{8}\)
- Answer
-
1
Exercise \(\PageIndex{20}\)
\(0 \div \dfrac{3}{5}\)
Exercise \(\PageIndex{21}\)
\(\dfrac{4}{11} \div \dfrac{4}{11}\)
- Answer
-
1
Exercise \(\PageIndex{22}\)
\(\dfrac{2}{3} \div \dfrac{2}{3}\)
Exercise \(\PageIndex{23}\)
\(\dfrac{7}{10} \div \dfrac{10}{7}\)
- Answer
-
\(\dfrac{49}{100}\)
Exercise \(\PageIndex{24}\)
\(\dfrac{3}{4} \div 6\)
Exercise \(\PageIndex{25}\)
\(\dfrac{9}{5} \div 3\)
- Answer
-
\(\dfrac{3}{5}\)
Exercise \(\PageIndex{26}\)
\(4 \dfrac{1}{6} \div 3 \dfrac{1}{3}\)
Exercise \(\PageIndex{27}\)
\(7 \dfrac{1}{7} \div 8 \dfrac{1}{3}\)
- Answer
-
\(\dfrac{6}{7}\)
Exercise \(\PageIndex{28}\)
\(1 \dfrac{1}{2} \div 1 \dfrac{1}{5}\)
Exercise \(\PageIndex{29}\)
\(3 \dfrac{2}{5} \div \dfrac{6}{25}\)
- Answer
-
\(\dfrac{85}{6}\) or \(14 \dfrac{1}{6}\)
Exercise \(\PageIndex{30}\)
\(5 \dfrac{1}{6} \div \dfrac{31}{6}\)
Exercise \(\PageIndex{31}\)
\(\dfrac{35}{6} \div 3 \dfrac{3}{4}\)
- Answer
-
\(\dfrac{28}{18} = \dfrac{14}{9}\) or \(1 \dfrac{5}{9}\)
Exercise \(\PageIndex{32}\)
\(5 \dfrac{1}{9} \div \dfrac{1}{18}\)
Exercise \(\PageIndex{33}\)
\(8 \dfrac{3}{4} \div \dfrac{7}{8}\)
- Answer
-
10
Exercise \(\PageIndex{34}\)
\(\dfrac{12}{8} \div 1 \dfrac{1}{2}\)
Exercise \(\PageIndex{35}\)
\(3 \dfrac{1}{8} \div \dfrac{15}{16}\)
- Answer
-
\(\dfrac{10}{3}\) or \(3 \dfrac{1}{3}\)
Exercise \(\PageIndex{36}\)
\(11 \dfrac{11}{12} \div 9 \dfrac{5}{8}\)
Exercise \(\PageIndex{37}\)
\(2 \dfrac{2}{9} \div 11 \dfrac{2}{3}\)
- Answer
-
\(\dfrac{4}{21}\)
Exercise \(\PageIndex{38}\)
\(\dfrac{16}{3} \div 6 \dfrac{2}{5}\)
Exercise \(\PageIndex{39}\)
\(4 \dfrac{3}{25} \div 2 \dfrac{56}{75}\)
- Answer
-
\(\dfrac{3}{2}\) or \(1 \dfrac{1}{2}\)
Exercise \(\PageIndex{40}\)
\(\dfrac{1}{1000} \div \dfrac{1}{100}\)
Exercise \(\PageIndex{41}\)
\(\dfrac{3}{8} \div \dfrac{9}{16} \cdot \dfrac{6}{5}\)
- Answer
-
\(\dfrac{4}{5}\)
Exercise \(\PageIndex{42}\)
\(\dfrac{3}{16} \cdot \dfrac{9}{8} \cdot \dfrac{6}{5}\)
Exercise \(\PageIndex{43}\)
\(\dfrac{4}{15} \div \dfrac{2}{25} \cdot \dfrac{9}{10}\)
- Answer
-
3
Exercise \(\PageIndex{44}\)
\(\dfrac{21}{30} \cdot 1 \dfrac{1}{4} \div \dfrac{9}{10}\)
Exercise \(\PageIndex{45}\)
\(8 \dfrac{1}{3} \cdot \dfrac{36}{75} \div 4\)
- Answer
-
1
Exercises for Review
Exercise \(\PageIndex{46}\)
What is the value of 5 in the number 504,216?
Exercise \(\PageIndex{47}\)
Find the product of 2,010 and 160.
- Answer
-
321,600
Exercise \(\PageIndex{48}\)
Use the numbers 8 and 5 to illustrate the commutative property of multiplication.
Exercise \(\PageIndex{49}\)
Find the least common multiple of 6, 16, and 72.
- Answer
-
144
Exercise \(\PageIndex{50}\)
Find \(\dfrac{8}{9}\) of \(6 \dfrac{3}{4}\)