3.3: Reading- Mixed Economies
A Mixed Economy exhibits characteristics of both market and planned economies, with private and state sectors providing direction.
KEY points
- The term Mixed Economy is very broadly defined and has been used to describe economies as diverse as the United States and Cuba.
- The means of production are privately owned, and markets remain the dominant form of economic coordination. However, governments wield significant influence over the economy through monetary and fiscal policy and regulation.
- Characteristics of mixed economies include welfare systems, employment standards, environmental protection, publicly owned enterprises, and antitrust policies.
- Keynesian economics advocates the presence of a mixed economy. This line of thought subsided between 1970 and 2000, but has regained considerable popularity after the financial crisis of 2008.
Terms
- Mixed economy : An economic system in which both the state and private sector direct the economy, reflecting characteristics of both market economies and planned economies.
- Welfare state: A social system in which the state takes overall responsibility for the welfare of its citizens, providing health care, education, unemployment compensation and social security.
- Mixed economies: a system in which both the state and private sector direct the way goods and services are bought and sold.
- Keynesian economics : The group of macroeconomic schools of thought based on the ideas of 20th-century economist John Maynard Keynes. Advocates of Keynesian economics argue that private sector decisions sometimes lead to inefficient macroeconomic outcomes that require active policy responses by the public sector, particularly monetary policy actions by the central bank and fiscal policy actions by the government to stabilize output over the business cycle.
Examples
- The American School (also known as the National System) is the economic philosophy that dominated United States national policies from the time of the American Civil War until the mid-twentieth century, and is an example of a mixed economy. It consisted of a three core policy initiative: protecting industry through high tariffs (1861–1932), government investment in infrastructure through internal improvements, and a national bank to promote the growth of productive enterprises. During this period the United States grew into the largest economy in the world, surpassing the UK (though not the British Empire) by 1880.
- Dirigisme is an economic policy initiated under Charles de Gaulle of France designating an economy where the government exerts strong directive influence. It involved state control of a minority of the industry, such as transportation, energy and telecommunication infrastructures, as well as various incentives for private corporations to merge or engage in certain projects. Under its influence, France experienced what is called Thirty Glorious Years of profound economic growth.
- Social market economy is the economic policy of modern Germany that steers a middle path between the goals of socialism and capitalism within the framework of a private market economy and aims at maintaining a balance between a high rate of economic growth, low inflation, low levels of unemployment, good working conditions, public welfare and public services by using state intervention.
Mixed Economies
What is a Mixed Economy?
A mixed economy is an economic system in which both the state and private sector direct the economy, reflecting characteristics of both market economies and planned economies. Most mixed economies can be described as market economies with strong regulatory oversight, in addition to having a variety of government-sponsored aspects.
While there is not one single definition for a mixed economy, the definitions always involve a degree of private economic freedom mixed with a degree of government regulation of markets.
The Plan Behind a Mixed Economy
The basic plan of the mixed economy is that:
- The means of production are mainly under private ownership;
- Markets remain the dominant form of economic coordination; and
- Profit-seeking enterprises and the accumulation of capital would remain the fundamental driving force behind economic activity. However, the government would wield considerable indirect influence over the economy through fiscal and monetary policies designed to counteract economic downturns and capitalism’s tendency toward financial crises and unemployment, along with playing a role in interventions that promote social welfare. Subsequently, some mixed economies have expanded in scope to include a role for indicative economic planning and/or large public enterprise sectors.
The relative strength or weakness of each component in the national economy can vary greatly between countries. Economies ranging from the United States to Cuba have been termed mixed economies. The term is also used to describe the economies of countries which are referred to as welfare states, such as Norway and Sweden.
What Do Governments Provide?
Governments in mixed economies often provide:
- Environmental protection,
- Maintenance of employment standards,
- A standardized welfare system, and
- Maintenance of competition.
Who Supports the Ideal of Mixed Economies?
As an economic ideal, mixed economies are supported by people of various political persuasions, typically center-left and center-right, such as social democrats or Christian democrats. Supporters view mixed economies as a compromise between state socialism and laissez-faire capitalism that is superior in net effect to either of those.
Keynesian economics advocates a mixed economy — predominantly private sector, but with a significant role of government and public sector. It also served as the economic model during the later part of the Great Depression, World War II, and the post-war economic expansion (1945–1973), though it lost some influence following the tax surcharge in 1968 and the stagflation of the 1970s. The advent of the global financial crisis in 2008 has caused a resurgence in Keynesian thought.
GLOSSARY
Capital
Money and wealth. The means to acquire goods and services, especially in a non-barter system. The uppermost part of a column. Money and wealth; the means to acquire goods and services, especially in a non-barter system. Already-produced durable goods available for use as a factor of production, such as steam shovels (equipment) and office buildings (structures).
Capitalism
A socio-economic system based on the abstraction of resources into the form of privately-owned money, wealth, and goods, with economic decisions made largely through the operation of a market unregulated by the state. A socio-economic system based on private property rights, including the private ownership of resources or capital, with economic decisions made largely through the operation of a market unregulated by the state.
Coordination
The resulting state of working together; cooperation; synchronization.
Corporation
A group of individuals, created by law or under authority of law, having a continuous existence independent of the existences of its members, and powers and liabilities distinct from those of its members.
Depression
A period of major economic contraction; officially, four consecutive quarters of negative, real GDP growth (according to NBER). In psychotherapy and psychiatry, a period of unhappiness or low morale which lasts longer than several weeks and may include ideation of self-inflicted injury or suicide.
Economic System
An economic system is the combination of the various agencies, entities (or even sectors as described by some authors) that provide the economic structure that defines the social community.
Economy
Collective focus of the study of money, currency and trade, and the efficient use of resources. The system of production and distribution and consumption. The overall measure of a currency system; as the national economy.
Employment
The work or occupation for which one is used, and often paid.
Enterprise
A company, business, organization, or other purposeful endeavor.
Fiscal Policy
Government policy that attempts to influence the direction of the economy through changes in government spending or taxes. In economics and political science, fiscal policy is the use of government revenue collection or taxation, and expenditure (spending) to influence the economy.
Framework
The identification and categorization of processes or steps that constitute a complex task or mindset in order to render explicit the tacit and implicit. A basic conceptual structure.
Goal
A desired result that one works to achieve.
Good
An object produced for market.
Incentive
Something that motivates, rouses, or encourages. It is used to motivate individuals (often, employees) for better performance by providing financial or other types of rewards. An anticipated reward or aversive event available in the environment. Something that motivates an individual to perform an action.
Industry
The sector of the economy consisting of large-scale enterprises.
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time. An increase in the general level of prices or in the cost of living. An increase in the quantity of money, leading to a devaluation of existing money.
Investment
The placement or expenditure of capital in expectation of deriving income or profit from its use.
Market
A group of potential customers for one’s product. One of the many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange.
Monetary Policy
The process by which the monetary authority of a country controls the supply of money, often targeting a rate of interest for the purpose of promoting economic growth and stability. The process by which the government, central bank, or monetary authority manages the supply of money, or trading in foreign exchange markets.
Planned Economy
An economic system in which government directly manages supply and demand for goods and services by controlling production, prices, and distribution in accordance with a long-term design and schedule of objectives.
Planning
The act of formulating a course of action, or of drawing up plans.
Private Sector
All organizations in an economy or jurisdiction that are not controlled by government, including privately owned businesses and not-for-profit organizations
Public Sector
Industries or services provided or funded by the government.
Regulation
A law or administrative rule, issued by an organization, used to guide or prescribe the conduct of members of that organization. Can specifically refer to acts in which a government or state body limits the behavior of businesses. A regulation is a legal provision that creates, limits, or constrains a right; creates or limits a duty; or allocates a responsibility.
Scope
The extent of the area or subject matter that something deals with or to which it is relevant. The breadth, depth, or reach of a subject; a domain. The breadth, depth or reach of a subject; a domain.
Services
That which is produced, then traded, bought or sold, then finally consumed and consists of an action or work.
Socialism
A political philosophy based on principles of community decision making, social equality, and the avoidance of economic and social exclusion, with preference to community goals over individual ones. Any of various economic and political philosophies that support social equality, collective decision-making, distribution of income based on contribution and public ownership of productive capital and natural resources, as advocated by socialists. The intermediate phase of social development between capitalism and full communism. This is a strategy whereby the state has control of all key resource-producing industries and manages most aspects of the economy, in contrast to laissez-faire capitalism.
Standard
Something used as a measure for comparative evaluations A level of quality or attainment.
System
A whole composed of relationships among the members. The part of the universe being studied, arbitrarily defined to any size desired.
Tariff
A system of government-imposed duties levied on imported or exported goods; a list of such duties, or the duties themselves.
- Boundless Business. Authored by : Boundless. Provided by : Boundless. Located at : https://www.boundless.com/business/ . License : CC BY-SA: Attribution-ShareAlike
- Image of USPS Mail Truck. Authored by : Coolcaesar. Located at : http://en.Wikipedia.org/wiki/Mixed_economy#mediaviewer/File:Small_USPS_Truck.jpg . License : CC BY-SA: Attribution-ShareAlike


9.4: Rationalize Algebraic Fractions
If the denominator of a rational expression contains sums or differences involving radicals, it is good form to always rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
The conjugate of the denominator contains the same terms, but opposite operations (addition or subtraction).
Rationalize the Denominator and Simplify:
- \(\dfrac{1}{1 − \sqrt{x}}\)
- \(\dfrac{1}{\sqrt{x} − \sqrt{y}}\)
- \(\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} − \sqrt{y}}\)
Solution
- \(\begin{array} &&\dfrac{1}{1 − \sqrt{x}} &\text{Example problem} \\ &\dfrac{(1)(1 + \sqrt{x})}{(1 − \sqrt{x})(1 + \sqrt{x})} &\text{Multiply both numerator and denominator by the conjugate, which is \((1+\sqrt{x})\)} \\ &\dfrac{1 + \sqrt{x}}{1 − \sqrt{x} + \sqrt{x} − (\sqrt{x})^2} &\text{FOIL the denominator.} \\ &\dfrac{1 + \sqrt{x}}{1 − \cancel{\sqrt{x}} + \cancel{\sqrt{x}} − (\sqrt{x})^2} &\text{Remove opposite terms that sum to zero.} \\ &\dfrac{1 + \sqrt{x}}{1 − x} &\text{The square root of \(x\), quantity squared is \(x\).} \\ &\dfrac{1 + \sqrt{x}}{1 − x} &\text{Final answer with the denominator rationalized, meaning that there are no square root terms in the denominator.} \end{array}\)
- \(\begin{array} &&\dfrac{1}{\sqrt{x} − \sqrt{y}} &\text{Example problem} \\ &\dfrac{(1)(\sqrt{x} + \sqrt{y})}{(\sqrt{x} − \sqrt{y})(\sqrt{x} + \sqrt{y})} &\text{Multiply both numerator and denominator by the conjugate, which is \((\sqrt{x} + \sqrt{y})\)} \\ &\dfrac{(\sqrt{x} + \sqrt{y})}{(\sqrt{x})^2 − \sqrt{x}\sqrt{y} + \sqrt{x}\sqrt{y} − (\sqrt{y})^2} &\text{FOIL the denominator.} \\ &\dfrac{(\sqrt{x} + \sqrt{y}) }{(\sqrt{x})^2 −\cancel{\sqrt{x}\sqrt{y}} + \cancel{\sqrt{x}\sqrt{y}} − (\sqrt{y})^2} &\text{Remove opposite terms that sum to zero.} \\ &\dfrac{\sqrt{x} + \sqrt{y}}{x − y} &\text{The square root of \(x\), quantity squared is \(x\), and the square root of \(y\), quantity squared is \(y\).} \\ &\dfrac{\sqrt{x} + \sqrt{y}}{x − y} &\text{Final answer with the denominator rationalized, meaning that there are no square root terms in the denominator.} \end{array}\)
- \(\begin{array} &&\dfrac{\sqrt{x} + \sqrt{y}}{\sqrt{x} − \sqrt{y}} &\text{Example problem} \\ &\dfrac{(\sqrt{x} + \sqrt{y})(\sqrt{x} + \sqrt{y})}{(\sqrt{x} − \sqrt{y})(\sqrt{x} + \sqrt{y})} &\text{Multiply both numerator and denominator by the conjugate, which is \((\sqrt{x} + \sqrt{y})\)} \\ &\dfrac{(\sqrt{x})^2 + 2(\sqrt{x}\sqrt{y}) + (\sqrt{y})^2}{(\sqrt{x})^2 − \sqrt{x}\sqrt{y} + \sqrt{x}\sqrt{y} − (\sqrt{y})^2} &\text{FOIL the numerator and the denominator.} \\ &\dfrac{x + 2\sqrt{x}\sqrt{y} + y}{(\sqrt{x})^2 − \cancel{\sqrt{x}\sqrt{y}} + \cancel{\sqrt{x}\sqrt{y}} − (\sqrt{y})^2} &\text{Remove opposite terms that sum to zero.} \\ &\dfrac{x + 2\sqrt{x}\sqrt{y} + y}{x − y} &\text{The square root of \(x\), quantity squared is \(x\), and the square root of \(y\), quantity squared is \(y\).} \\ &\dfrac{x + 2\sqrt{x}\sqrt{y} + y}{x − y} &\text{Final answer with the denominator rationalized, meaning that there are no square root terms in the denominator.} \end{array}\)
Rationalize the Denominator and Simplify:
- \(\dfrac{x}{1 − \sqrt{x}}\)
- \(\dfrac{1}{1 − \sqrt{x}}\)
- \(\dfrac{2 \sqrt{x}}{\sqrt{x} − 1}\)
- \(\dfrac{x − 1}{\sqrt{x} − 1}\)


2: Voice-Leading and Model Composition Last updated Save as PDF Page ID 61950 Robin Wharton and Kris Shaffer eds. Open Music Theory via Hybrid Pedagogy Publishing 2.1: Introduction to Strict Voice-Leading 2.2: Strict Two-Voice Composition (Species Counterpoint) - Composing a cantus firmus 2.3: Strict Two-Voice Composition - Composing a First-Species Counterpoint 2.4: Strict Two-Voice Composition - Composing a Second-Species Counterpoint 2.5: Strict Two-Voice Composition - Composing a Third-Species Counterpoint 2.6: Strict Two-Voice Composition - Composing a Fourth-Species Counterpoint 2.7: Strict Four-Voice Composition - Introduction to Thoroughbass 2.8: Strict Four-Voice Composition - A Brief History of basso continuo Keyboard-Style Voice-Leading 2.9: Strict Four-Voice Composition - Generating Roman Numerals from a Figured Bass Line 2.10: Strict Four-Voice Composition - Composing in basso continuo Style 2.11: Strict Four-Voice Composition - Style and Tendency 2.12: Strict Four-Voice Composition - Tendency Tones and Functional Harmonic Dissonances 2.13: Strict Four-Voice Composition - Realizing a Figured Bass in Strict basso continuo Style 2.14: Strict Four-Voice Composition - Melodic Keyboard-Style Voice-Leading 2.15: Strict Four-Voice Composition - Keyboard-Style Voice-Leading Schemata 2.16: Strict Four-Voice Composition - Realizing a Figured Bass Line in Melodic Keyboard Style 2.17: Strict Four-Voice Composition - Realizing an Unfigured Bass Line 2.18: Strict Four-Voice Composition - Embellishing Tones


2 .
Discuss the role of the nurse when the ultrasound at the first prenatal visit reveals an empty uterus.
3 .
Why is it important to continue to screen for complications throughout pregnancy?
4 .
Why are cultural considerations important when providing nutritional counseling to a person who is pregnant?
5 .
What nursing actions are expected when the pregnant person answers yes when asked if they are experiencing any pain at a prenatal visit in the second trimester?
6 .
How would the nurse explain viability to the pregnant person in the second trimester?
7 .
Explain the difference between the fetal lie and the fetal presentation.
8 .
Why is it important for the nurse to discuss the signs and symptoms of complications of pregnancy?
9 .
How does the nurse determine a family’s communication methods?
10 .
Why is a family assessment important?


17.2: Business and CSR
- Summarize the business relevance of corporate social responsibility
As Business News Daily writer Skye Schooley notes “Consumers, employees and stakeholders are beginning to prioritize CSR when choosing a brand or company. They are holding corporations accountable for [affecting] social change with their business beliefs, practices and profits.” [1] The results from public relations and marketing agency Cone Communications 2017 CSR survey illustrate how critical it is for businesses to get this right. [2] Key findings:
- 63% of Americans are hopeful businesses will take the lead to drive social and environmental change moving forward, in the absence of government regulation.
- 78% want companies to address important social justice issues.
- 87% will purchase a product because a company advocated for an issue they cared about.
- 76% will refuse to purchase a company’s products or services upon learning it supported an issue contrary to their beliefs.
As PR agency InkHouse CEO Beth Monaghan advises in an article for Forbes, “Brands must stand for something because consumers demand it, but they must also live it.” [3]
For specific examples of CSR in practice, scan the profiles on Fortune magazine’s 2018 Change the World list , recognizing companies that have made making a positive social impact a core part of their business strategy. [4] Fortune’s evaluation and ranking criteria also provide perspective reflect CSR best practices:
- Measurable social impact: We consider the reach, nature, and durability of the company’s impact on one or more specific societal problems.
- Business results: We consider the benefit the socially impactful work brings to the company. Profitability and contribution to shareholder value outweigh benefits to the company’s reputation.
- Degree of innovation: We consider how innovative the company’s effort is relative to that of others in its industry and whether other companies have followed its example.
- Corporate Integration: We consider how integral the initiative is to a company’s overall strategy, and how well that strategy is communicated through the ranks and elsewhere.
Although the majority of Change the World honorees have $1 billion in annual sales, CSR can be integrated into businesses at any scale. 1% for the Planet is a global movement that started by two businessmen who shared a love for the outdoors and committed to contributing 1% of their sales to the environment—regardless of profitability. [5] To quote Patagonia founder and 1% for the Planet co-founder Yvon Chouinard, “The intent of 1% for the Planet is to help fund these diverse environmental organizations so that collectively they can be a more powerful source in solving the world’s problems.” [6] If making an ongoing financial commitment feels too risky or you want to take action personally, individual members can participate by donating to and volunteering with local environmental nonprofits. As they put it: “everyone has a 1%.”
To get started with CSR, start local and start small. As an entrepreneur or small business, contribute expertise to events that are consistent with your mission, vision and values. Symantec’s Head of global diversity, equity and inclusion recommends involving employees and consumers in the process of deciding what causes and organizations to support, noting that these efforts can increase brand preference and employee engagement and the transparency and value of CSR initiatives. [7]
- Schooley, Skye. " What is Corporate Social Responsibility? " Business News Daily . April 22, 2019. Accessed July 18, 2019. ↵
- " 2017 Cone Communications CSR Study. " Cone. 2017. Accessed July 19, 2019. ↵
- Monaghan, Beth, " PR for a Purpose: Bringing Corporate Social Responsibility Back to Basics ." Forbes. February 13, 2018. Accessed July 19, 2019. ↵
- " Change the World ." Fortune . Accessed July 19, 2019. ↵
- " 1% for the Planet. " 1% for the Planet. Accessed July 19, 2019. ↵
- Ibid. ↵
- " Symantec 2018 Corporate Social Responsibility Report. " Symantec. 2018. Accessed July 19, 2019. ↵
- 1% For the Planet Sticker. Authored by : Ross Tossberg. Provided by : Flickr. Located at : https://www.flickr.com/photos/20209466@N06/4709903391 . License : CC BY: Attribution


2.4: Introduction to Store-Based Strategy Mixes
What you’ll learn to do: Discuss store-based strategy mixes
Retailers target specific consumers and shopping occasions. Think about consumers on their commute home from work on a Tuesday evening. Their purchase of a carton of milk at an AM/PM convenience store where they’ve also stopped for gas is very different than their purchase of two gallons of milk during a stock-up trip to a supercenter on a Saturday afternoon. As such, retailers’ store environments, layout, assortment, hours of operation, and many other factors reflect these strategic targeting decisions.
Contributors and Attributions
CC licensed content, Original
- Introduction to Store-Based Strategy Mixes. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


Questions to consider:
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How will you be able to develop your purpose?
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In what ways will you be able to create strategies for your success?
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What other resources can you use to help you succeed?
As Reginald and Madison go through their college experiences and create a balance between their academic and personal lives, their stories, no doubt, will diverge. But you can be assured that each of them will demonstrate grit, the ability to stay focused on a goal over the long-term, along the way. As Duckworth (2016) has said, it takes passion and perseverance to be gritty. It also takes resilience, or the ability to bounce back from adversity. The challenges you face will certainly stretch you, but if you have these three things—purpose, strategies, and resources—you will be more likely to bounce back, even become stronger in the process. This book has been designed with these things in mind.
Develop Your “Why”
This chapter began with the suggestion to explore why you are in college or, more simply, what your purpose is. This course—and this book—will help you continue to refine your answer and create a map for your journey to fulfill your purpose. The features in this book that help you develop your purpose include the following:
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Student Survey Questions: Each chapter opens with several questions that provide you with a snapshot on how you feel about the chapter content. How does this feature help you develop purpose? It allows you to develop better self-awareness, which will in turn help you build an awareness of your purpose.
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Analysis Questions: These questions are included throughout each chapter. Consider them “pauses” to help you reflect on what you have read and how to incorporate the information into your own journey.
Refine Your Strategies for Success
Purpose by itself may illuminate the pathway forward, but it will take strategies to help you complete your journey. Think of the strategies you will learn in this course as tools you will need along the way to completing your degree. The following features provide you with an opportunity to practice and refine strategies for success:
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Application Questions: Any time you are asked to
apply
what you are learning in the chapters, you are improving your skills. Look for them throughout and take some time to stop, think, and use the skill.
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Activities: As you read, you will also have the opportunity to interact with the content. They give you the chance to refine the strategies that will help you succeed in college.
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Career Connection: This feature allows you to consider how the skills you are developing for college connect to your future career. Making these connections will help you appreciate the deeper importance of them.
Use Your Resources
In addition to developing strategies for succeeding in your academic and future professional career, you will find that this course will point out the resources you may need to obtain more tools or refuel your desire to continue along the pathway. No one succeeds at anything by oneself. The features related to resources will certainly help you find ways to fill up your toolkit of information.
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Get Connected: Despite its ability to distract us from the work we need to do, technology can help you accomplish your day-to-day tasks with relative ease. This feature offers suggestions for apps and websites that can help you build skills or just keep track of due dates!
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Where Do You Go from Here?: The skills and habits you are building now will serve you well in your future endeavors. This feature is designed to help you dig deeper into the chapter content and refine your research skills. It also asks that you find ways to connect what you are learning now to your life and career.
All of these features, in addition to the content, will help you see yourself for who you are and provide opportunities to develop in ways that will make reaching your goal a little easier. Will it be challenging at times? Yes, it will. Will it take time to reflect on those challenges and find better ways to learn and reach your goals? Most definitely. But the effort you put into completing your college degree will result in the confidence you will gain from knowing that anything you set your mind to do—and you work hard for—can be accomplished.
Chapter Takeaways
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The first year of college is the most critical. Make the commitment to overcome any obstacles to a successful transition and stay committed and motivated to succeed.
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Although college students differ in many ways, all successful students share certain common traits, including a positive attitude, effective critical thinking skills, good time management skills, effective study skills, interactions with instructors and other students, and good habits for personal health and financial stability.
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You can learn to maximize your learning by attending to each step of the learning process: preparing, absorbing, capturing, and reviewing.
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It is important to understand your personal learning style and use it well in classes, while also making the effort to learn in new ways and work with other students for a more effective overall learning experience.
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Working with your academic advisor and taking advantage of the many resources available at your college are key actions to ensure success.
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Understanding the larger characteristics of college success leads to a richer college experience, supplementing the value of good grades.
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While it may take a few weeks to develop all the skills needed for success in college, there are many steps you can begin taking today to get moving in the right direction.
This chapter provides an introduction to the transition to college by first asking “Why?”  These transitional experiences are part of being in college, and this chapter provides you with information about what to expect and how to handle the changes you will go through. Next, the chapter discusses college culture and how to understand the customs and language of higher education. The chapter ends with resources throughout the text that can help you practice skills and dive deeper into the topics.
Where do you go from here?
Making the transition into college smoother for you can have long-term benefits. What have you learned about in this chapter that you want to know more about that could help you? Review the questions below. The answer to these questions will be further discussed in future chapters.
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What is the long-term value of a college degree?
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What is the “hidden curriculum,” and how can knowing about it help you succeed in college?
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What learning strategies are the most effective?
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What kinds of resources and services do colleges now offer that help students’ personal development?
Chapter Takeaways
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An active social life and social interaction with a variety of people on campus contribute to college students’ well-being and overall academic success.
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Successfully interacting with diverse others requires effective communication skills, including both listening skills and assertive communication rather than passive or aggressive communication.
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Social interaction can be heightened by productive and moderate online networking.
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Time management and study skills help one avoid problems when balancing social life and academic studies.
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To prevent or resolve conflicts that may occur in any social interaction, maintain an attitude of respect for others, be open minded and willing to compromise, and know how to work together calmly to resolve conflicts.
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Diversity on campus is beneficial for all students, not just those from ethnic or minority groups. The wider perspectives of students from different backgrounds and the greater variety of teaching methods help everyone gain more fully in educational experiences. Socially, students develop a more mature worldview and are better prepared for interacting with a diverse world in the future.
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Multiculturalism involves an attitude of respect for the ideas, feelings, behaviors, and experiences of others who differ from oneself in any way. Colleges promote both diversity in the student body and multiculturalism among all students.
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To gain a multicultural perspective, challenge your own learned stereotypes while you learn more about other cultural groups. Understanding what can be learned from others leads to celebrating the diversity found on most campuses.
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Take a personal responsibility both for broadening your own social world and for speaking out against prejudice and discrimination wherever encountered.
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Take advantage of campus opportunities to increase your cultural awareness and to form social relationships with diverse others. Organized campus groups and events can help you broaden your horizons in many beneficial ways.
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Participation in campus clubs and other organizations is not only fun and a good way to reduce stress but also helps develop social, intellectual, and technical skills that may serve you well in your future career or other endeavors.
-
Visit your college’s Web site and look for a section on student activities and organizations. Try to identify two or three groups you might be interested to learn more about.
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Next time you walk across campus or through the student center, stop to look at bulletin boards and posters. Look for upcoming events that celebrate cultural diversity in some way. Read the information in detail and imagine how much fun the event might be while you also learn something new. Then ask a friend to go with you.
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Go to www.understandingrace.org/liv...rts/index.html—a Web site of the American Anthropological Association—and take the short online sports quiz. Many things have been said about why certain races or people from certain geographic areas excel at certain sports. People often talk about differences in biology and other differences among ethnic groups as related to sports. How much is true, partly true, or blatantly false? How much do you know about what are real or not real differences?
Rethinking
Revisit the questions you answered at the beginning of the chapter, and consider one option you learned in this chapter that might change your answer to them.
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I am fully aware of the expectations of college and how to meet them.
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I know why I am in college and have clear goals that I want to achieve.
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Most of the time, I take responsibility for my learning new and challenging concepts.
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I feel comfortable working with faculty, advisors, and classmates to accomplish my goals.


10.10: Surveying the Stars
Learning Objectives
By the end of this section, you will be able to:
- Understand the concept of triangulating distances to distant objects, including stars
- Explain why space-based satellites deliver more precise distances than ground-based methods
- Discuss astronomers’ efforts to study the stars closest to the Sun
It is an enormous step to go from the planets to the stars. For example, our Voyager 1 probe, which was launched in 1977, has now traveled farther from Earth than any other spacecraft. As this is written in 2016, Voyager 1 is 134 AU from the Sun. 1 The nearest star, however, is hundreds of thousands of AU from Earth. Even so, we can, in principle, survey distances to the stars using the same technique that a civil engineer employs to survey the distance to an inaccessible mountain or tree—the method of triangulation .
Triangulation in Space
A practical example of triangulation is your own depth perception. As you are pleased to discover every morning when you look in the mirror, your two eyes are located some distance apart. You therefore view the world from two different vantage points, and it is this dual perspective that allows you to get a general sense of how far away objects are.
 Look at it first with one eye (closing the other) and then switch eyes. Note how the pen seems to shift relative to objects across the room. Now hold the pen at arm’s length: the shift is less. If you play with moving the pen for a while, you will notice that the farther away you hold it, the less it seems to shift.
If your arms were made of rubber, you could stretch the pen far enough away from your eyes that the shift would become imperceptible. This is because our depth perception fails for objects more than a few tens of meters away. In order to see the shift of an object a city block or more from you, your eyes would need to be spread apart a lot farther.
Let’s see how surveyors take advantage of the same idea. Suppose you are trying to measure the distance to a tree across a deep river (Figure \(\PageIndex{1}\)). You set up two observing stations some distance apart. That distance (line AB in Figure \(\PageIndex{1}\)) is called the baseline . Now the direction to the tree (C in the figure) in relation to the baseline is observed from each station. Note that C appears in different directions from the two stations. This apparent change in direction of the remote object due to a change in vantage point of the observer is called parallax .
The parallax is also the angle that lines AC and BC make—in mathematical terms, the angle subtended by the baseline. A knowledge of the angles at A and B and the length of the baseline, AB, allows the triangle ABC to be solved for any of its dimensions—say, the distance AC or BC. The solution could be reached by constructing a scale drawing or by using trigonometry to make a numerical calculation. If the tree were farther away, the whole triangle would be longer and skinnier, and the parallax angle would be smaller. Thus, we have the general rule that the smaller the parallax, the more distant the object we are measuring must be.
In practice, the kinds of baselines surveyors use for measuring distances on Earth are completely useless when we try to gauge distances in space. The farther away an astronomical object lies, the longer the baseline has to be to give us a reasonable chance of making a measurement. Unfortunately, nearly all astronomical objects are very far away. To measure their distances requires a very large baseline and highly precise angular measurements. The Moon is the only object near enough that its distance can be found fairly accurately with measurements made without a telescope. Ptolemy determined the distance to the Moon correctly to within a few percent. He used the turning Earth itself as a baseline, measuring the position of the Moon relative to the stars at two different times of night.
With the aid of telescopes, later astronomers were able to measure the distances to the nearer planets and asteroids using Earth’s diameter as a baseline. This is how the AU was first established. To reach for the stars, however, requires a much longer baseline for triangulation and extremely sensitive measurements. Such a baseline is provided by Earth’s annual trip around the Sun.
Distances to Stars
As Earth travels from one side of its orbit to the other, it graciously provides us with a baseline of 2 AU, or about 300 million kilometers. Although this is a much bigger baseline than the diameter of Earth, the stars are so far away that the resulting parallax shift is still not visible to the naked eye—not even for the closest stars.
In the chapter on Observing the Sky: The Birth of Astronomy, we discussed how this dilemma perplexed the ancient Greeks, some of whom had actually suggested that the Sun might be the center of the solar system, with Earth in motion around it. Aristotle and others argued, however, that Earth could not be revolving about the Sun. If it were, they said, we would surely observe the parallax of the nearer stars against the background of more distant objects as we viewed the sky from different parts of Earth’s orbit (Figure \(\PageIndex{3}\)). Tycho Brahe (1546–1601) advanced the same faulty argument nearly 2000 years later, when his careful measurements of stellar positions with the unaided eye revealed no such shift.
These early observers did not realize how truly distant the stars were and how small the change in their positions therefore was, even with the entire orbit of Earth as a baseline. The problem was that they did not have tools to measure parallax shifts too small to be seen with the human eye. By the eighteenth century, when there was no longer serious doubt about Earth’s revolution, it became clear that the stars must be extremely distant. Astronomers equipped with telescopes began to devise instruments capable of measuring the tiny shifts of nearby stars relative to the background of more distant (and thus unshifting) celestial objects.
This was a significant technical challenge, since, even for the nearest stars, parallax angles are usually only a fraction of a second of arc. Recall that one second of arc (arcsec) is an angle of only 1/3600 of a degree. A coin the size of a US quarter would appear to have a diameter of 1 arcsecond if you were viewing it from a distance of about 5 kilometers (3 miles). Think about how small an angle that is. No wonder it took astronomers a long time before they could measure such tiny shifts.
The first successful detections of stellar parallax were in the year 1838, when Friedrich Bessel in Germany (Figure \(\PageIndex{2}\)), Thomas Henderson, a Scottish astronomer working at the Cape of Good Hope, and Friedrich Struve in Russia independently measured the parallaxes of the stars 61 Cygni, Alpha Centauri, and Vega, respectively. Even the closest star, Alpha Centauri, showed a total displacement of only about 1.5 arcseconds during the course of a year.
Figure \(\PageIndex{3}\) shows how such measurements work. Seen from opposite sides of Earth’s orbit, a nearby star shifts position when compared to a pattern of more distant stars. Astronomers actually define parallax to be one-half the angle that a star shifts when seen from opposite sides of Earth’s orbit (the angle labeled P in Figure \(\PageIndex{3}\)). The reason for this definition is just that they prefer to deal with a baseline of 1 AU instead of 2 AU.
Units of Stellar Distance
With a baseline of one AU, how far away would a star have to be to have a parallax of 1 arcsecond? The answer turns out to be 206,265 AU, or 3.26 light-years. This is equal to 3.1 × 10 13 kilometers (in other words, 31 trillion kilometers). We give this unit a special name, the parsec (pc)—derived from “the distance at which we have a par allax of one sec ond.” The distance (\(D\)) of a star in parsecs is just the reciprocal of its parallax ( p ) in arcseconds; that is,
\[D=\frac{1}{p} \nonumber\]
Thus, a star with a parallax of 0.1 arcsecond would be found at a distance of 10 parsecs, and one with a parallax of 0.05 arcsecond would be 20 parsecs away.
Back in the days when most of our distances came from parallax measurements, a parsec was a useful unit of distance, but it is not as intuitive as the light-year. One advantage of the light-year as a unit is that it emphasizes the fact that, as we look out into space, we are also looking back into time. The light that we see from a star 100 light-years away left that star 100 years ago. What we study is not the star as it is now, but rather as it was in the past. The light that reaches our telescopes today from distant galaxies left them before Earth even existed.
In this text, we will use light-years as our unit of distance, but many astronomers still use parsecs when they write technical papers or talk with each other at meetings. To convert between the two distance units, just bear in mind: 1 parsec = 3.26 light-year, and 1 light-year = 0.31 parsec.
Example \(\PageIndex{1}\): How Far Is a Light-Year?
A light-year is the distance light travels in 1 year. Given that light travels at a speed of 300,000 km/s, how many kilometers are there in a light-year?
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We learned earlier that speed = distance/time. We can rearrange this equation so that \(\text{distance } = \text{ velocity } \times \text{ time }\). Now, we need to determine the number of seconds in a year.
There are approximately 365 days in 1 year. To determine the number of seconds, we must estimate the number of seconds in 1 day.
We can change units as follows (notice how the units of time cancel out):
\[ 1 \text{ day } \times 24 \text{ hr/day } \times 60 \text{ min/hr } \times 60 \text{ s/min } = 86,400 \text{ s/day} \nonumber\]
Next, to get the number of seconds per year:
\[ 365 \text{ days/year } \times 86,400 \text{ s/day } = 31,536,000 \text{ s/year} \nonumber\]
Now we can multiply the speed of light by the number of seconds per year to get the distance traveled by light in 1 year:
\[ \begin{aligned} \text{distance } & = \text{ velocity } \times \text{ time} \\ & = 300,000 \text{ km/s } \times 31,536,000 \text{ s} \\ & =9.46 \times 10^{12} \text{ km} \end{aligned} \nonumber\]
That’s almost 10,000,000,000,000 km that light covers in a year. To help you imagine how long this distance is, we’ll mention that a string 1 light-year long could fit around the circumference of Earth 236 million times.
Exercise \(\PageIndex{1}\)
The number above is really large. What happens if we put it in terms that might be a little more understandable, like the diameter of Earth? Earth’s diameter is about 12,700 km.
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\[ \begin{aligned} 1 \text{ light-year } & = 9.46 \times 10^{12} \text{ km} \\ & =9.46 \times 10^{12} \text{ km } \times \frac{1 \text{ Earth diameter}}{12,700 \text{ km}} \\ & =7.45 \times 10^8 \text{ Earth diameters} \end{aligned} \nonumber\]
That means that 1 light-year is about 745 million times the diameter of Earth.
NAMING STARS
You may be wondering why stars have such a confusing assortment of names. Just look at the first three stars to have their parallaxes measured: 61 Cygni, Alpha Centauri, and Vega. Each of these names comes from a different tradition of designating stars.
The brightest stars have names that derive from the ancients. Some are from the Greek, such as Sirius, which means “the scorched one”—a reference to its brilliance. A few are from Latin, but many of the best-known names are from Arabic because, as discussed in Observing the Sky: The Birth of Astronomy, much of Greek and Roman astronomy was “rediscovered” in Europe after the Dark Ages by means of Arabic translations. Vega, for example, means “swooping Eagle,” and Betelgeuse (pronounced “Beetle-juice”) means “right hand of the central one.”
In 1603, German astronomer Johann Bayer (1572–1625) introduced a more systematic approach to naming stars. For each constellation, he assigned a Greek letter to the brightest stars, roughly in order of brightness. In the constellation of Orion, for example, Betelgeuse is the brightest star, so it got the first letter in the Greek alphabet—alpha—and is known as Alpha Orionis. (“Orionis” is the possessive form of Orion, so Alpha Orionis means “the first of Orion.”) A star called Rigel, being the second brightest in that constellation, is called Beta Orionis (Figure \(\PageIndex{4}\)). Since there are 24 letters in the Greek alphabet, this system allows the labeling of 24 stars in each constellation, but constellations have many more stars than that.
In 1725, the English Astronomer Royal John Flamsteed introduced yet another system, in which the brighter stars eventually got a number in each constellation in order of their location in the sky or, more precisely, their right ascension. (The system of sky coordinates that includes right ascension was discussed in Earth, Moon, and Sky.) In this system, Betelgeuse is called 58 Orionis and 61 Cygni is the 61st star in the constellation of Cygnus, the swan.
It gets worse. As astronomers began to understand more and more about stars, they drew up a series of specialized star catalogs, and fans of those catalogs began calling stars by their catalog numbers. If you look at Appendix I—our list of the nearest stars (many of which are much too faint to get an ancient name, Bayer letter, or Flamsteed number)—you will see references to some of these catalogs. An example is a set of stars labeled with a BD number, for “Bonner Durchmusterung.” This was a mammoth catalog of over 324,000 stars in a series of zones in the sky, organized at the Bonn Observatory in the 1850s and 1860s. Keep in mind that this catalog was made before photography or computers came into use, so the position of each star had to be measured (at least twice) by eye, a daunting undertaking.
There is also a completely different system for keeping track of stars whose luminosity varies, and another for stars that brighten explosively at unpredictable times. Astronomers have gotten used to the many different star-naming systems, but students often find them bewildering and wish astronomers would settle on one. Don’t hold your breath: in astronomy, as in many fields of human thought, tradition holds a powerful attraction. Still, with high-speed computer databases to aid human memory, names may become less and less necessary. Today’s astronomers often refer to stars by their precise locations in the sky rather than by their names or various catalog numbers.
The Nearest Stars
No known star (other than the Sun) is within 1 light-year or even 1 parsec of Earth. The stellar neighbors nearest the Sun are three stars in the constellation of Centaurus. To the unaided eye, the brightest of these three stars is Alpha Centauri, which is only 30○from the south celestial pole and hence not visible from the mainland United States. Alpha Centauri itself is a binary star—two stars in mutual revolution—too close together to be distinguished without a telescope. These two stars are 4.4 light-years from us. Nearby is a third faint star, known as Proxima Centauri. Proxima, with a distance of 4.3 light-years, is slightly closer to us than the other two stars. If Proxima Centauri is part of a triple star system with the binary Alpha Centauri, as seems likely, then its orbital period may be longer than 500,000 years.
Proxima Centauri is an example of the most common type of star, and our most common type of stellar neighbor (as we saw in Stars: A Celestial Census.) Low-mass red M dwarfs make up about 70% of all stars and dominate the census of stars within 10 parsecs of the Sun. The latest survey of the solar neighborhood has counted 357 stars and brown dwarfs within 10 parsecs, and 248 of these are red dwarfs. Yet, if you wanted to see an M dwarf with your naked eye, you would be out of luck. These stars only produce a fraction of the Sun’s light, and nearly all of them require a telescope to be detected.
The nearest star visible without a telescope from most of the United States is the brightest appearing of all the stars, Sirius, which has a distance of a little more than 8 light-years. It too is a binary system, composed of a faint white dwarf orbiting a bluish-white, main-sequence star. It is an interesting coincidence of numbers that light reaches us from the Sun in about 8 minutes and from the next brightest star in the sky in about 8 years.
Example \(\PageIndex{2}\): Calculating the Diameter of the Sun
For nearby stars, we can measure the apparent shift in their positions as Earth orbits the Sun. We wrote earlier that an object must be 206,265 AU distant to have a parallax of one second of arc. This must seem like a very strange number, but you can figure out why this is the right value. We will start by estimating the diameter of the Sun and then apply the same idea to a star with a parallax of 1 arcsecond. Make a sketch that has a round circle to represent the Sun, place Earth some distance away, and put an observer on it. Draw two lines from the point where the observer is standing, one to each side of the Sun. Sketch a circle centered at Earth with its circumference passing through the center of the Sun. Now think about proportions. The Sun spans about half a degree on the sky. A full circle has 360○. The circumference of the circle centered on Earth and passing through the Sun is given by:
\[\text{circumference } =2 \pi \times 93,000,000 \text{ miles} \nonumber\]
Then, the following two ratios are equal:
\[\frac{0.5^{\circ}}{360^{\circ}} = \frac{\text{diameter of Sun}}{2 \pi \times 93,000,000} \nonumber\]
Calculate the diameter of the Sun. How does your answer compare to the actual diameter?
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To solve for the diameter of the Sun, we can evaluate the expression above.
\[\begin{aligned} \text{diameter of the sun } & = \frac{0.5^{\circ}}{360^{\circ}}
\times 2 \pi \times 93,000,000 \text{ miles} \\ =811,577 \text{ miles} \end{aligned} \nonumber\]This is very close to the true value of about 848,000 miles.
Exercise \(\PageIndex{2}\)
Now apply this idea to calculating the distance to a star that has a parallax of 1 arcsec. Draw a picture similar to the one we suggested above and calculate the distance in AU. (Hint: Remember that the parallax angle is defined by 1 AU, not 2 AU, and that 3600 arcseconds = 1 degree.)
- Answer
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206,265 AU
Measuring Parallaxes in Space
The measurements of stellar parallax were revolutionized by the launch of the spacecraft Hipparcos in 1989, which measured distances for thousands of stars out to about 300 light-years with an accuracy of 10 to 20% (see Figure \(\PageIndex{5}\) and the feature on Parallax and Space Astronomy below). However, even 300 light-years are less than 1% the size of our Galaxy’s main disk.
In December 2013, the successor to Hipparcos, named Gaia , was launched by the European Space Agency. Gaia is measuring the position and distances to almost one billion stars with an accuracy of a few millionths of an arcsecond. Gaia’s distance limit will extend well beyond Hipparcos, studying stars out to 30,000 light-years (100 times farther than Hipparcos, covering nearly 1/3 of the galactic disk). Gaia will also be able to measure proper motions 2 for thousands of stars in the halo of the Milky Way—something that can only be done for the brightest stars right now. At the end of Gaia’s mission, we will not only have a three-dimensional map of a large fraction of our own Milky Way Galaxy, but we will also have a strong link in the chain of cosmic distances that we are discussing in this chapter. Yet, to extend this chain beyond Gaia’s reach and explore distances to nearby galaxies, we need some completely new techniques.
Parallax and space astronomy
One of the most difficult things about precisely measuring the tiny angles of parallax shifts from Earth is that you have to observe the stars through our planet’s atmosphere. As we saw in Astronomical Instruments, the effect of the atmosphere is to spread out the points of starlight into fuzzy disks, making exact measurements of their positions more difficult. Astronomers had long dreamed of being able to measure parallaxes from space, and two orbiting observatories have now turned this dream into reality.
The name of the Hipparcos satellite, launched in 1989 by the European Space Agency, is both an abbreviation for High Precision Parallax Collecting Satellite and a tribute to Hipparchus, the pioneering Greek astronomer whose work we discussed in the Observing the Sky: The Birth of Astronomy. The satellite was designed to make the most accurate parallax measurements in history, from 36,000 kilometers above Earth. However, its onboard rocket motor failed to fire, which meant it did not get the needed boost to reach the desired altitude. Hipparcos ended up spending its 4-year life in an elliptical orbit that varied from 500 to 36,000 kilometers high. In this orbit, the satellite plunged into Earth’s radiation belts every 5 hours or so, which finally took its toll on the solar panels that provided energy to power the instruments.
Nevertheless, the mission was successful, resulting in two catalogs. One gives positions of 120,000 stars to an accuracy of one-thousandth of an arcsecond—about the diameter of a golf ball in New York as viewed from Europe. The second catalog contains information for more than a million stars, whose positions have been measured to thirty-thousandths of an arcsecond. We now have accurate parallax measurements of stars out to distances of about 300 light-years. (With ground-based telescopes, accurate measurements were feasible out to only about 60 light-years.)
In order to build on the success of Hipparcos, in 2013, the European Space Agency launched a new satellite called Gaia . The Gaia mission is scheduled to last for 5 years. Because Gaia carries larger telescopes than Hipparcos, it can observe fainter stars and measure their positions 200 times more accurately. The main goal of the Gaia mission is to make an accurate three-dimensional map of that portion of the Galaxy within about 30,000 light-years by observing 1 billion stars 70 times each, measuring their positions and hence their parallaxes as well as their brightnesses.
For a long time, the measurement of parallaxes and accurate stellar positions was a backwater of astronomical research—mainly because the accuracy of measurements did not improve much for about 100 years. However, the ability to make measurements from space has revolutionized this field of astronomy and will continue to provide a critical link in our chain of cosmic distances.
The European Space Agency (ESA) maintains a Gaia mission website where you can learn more about the Gaia mission and to get the latest news on Gaia observations.
To learn more about Hipparcos, explore this European Space Agency webpage with an ESA vodcast Charting the Galaxy—from Hipparcos to Gaia .
Summary
For stars that are relatively nearby, we can “triangulate” the distances from a baseline created by Earth’s annual motion around the Sun. Half the shift in a nearby star’s position relative to very distant background stars, as viewed from opposite sides of Earth’s orbit, is called the parallax of that star and is a measure of its distance. The units used to measure stellar distance are the light-year, the distance light travels in 1 year, and the parsec (pc), the distance of a star with a parallax of 1 arcsecond (1 parsec = 3.26 light-years). The closest star, a red dwarf, is over 1 parsec away. The first successful measurements of stellar parallaxes were reported in 1838. Parallax measurements are a fundamental link in the chain of cosmic distances. The Hipparcos satellite has allowed us to measure accurate parallaxes for stars out to about 300 light-years, and the Gaia mission will result in parallaxes out to 30,000 light-years.
Footnotes
1 To have some basis for comparison, the dwarf planet Pluto orbits at an average distance of 40 AU from the Sun, and the dwarf planet Eris is currently roughly 96 AU from the Sun.
2 Proper motion (as discussed in Analyzing Starlight , is the motion of a star across the sky (perpendicular to our line of sight.)
Glossary
- parallax
- an apparent displacement of a nearby star that results from the motion of Earth around the Sun
- parsec
- a unit of distance in astronomy, equal to 3.26 light-years; at a distance of 1 parsec, a star has a parallax of 1 arcsecond
Contributors and Attributions
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Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


12.7: Discussion Questions
- What has the Supreme Court’s 2022 overturning of Roe V. Wade changed about reproductive rights in the United States?
- Higher rates of STIs are seen in communities of people of color, particularly highest in the Black population in the United States (Park, 2021). How might racism contribute to this? What can be done to reduce this disparity?
- How might we reduce shame and stigma around STIs? If we were able to reduce stigma, how might this affect STI rates?
- Have you ever talked with a medical professional about STI testing? If so, how do you feel it was handled? How might we help medical professionals feel more comfortable and be more competent at addressing STI concerns?
- STI testing is offered more to people with vulvas since they may be receiving routine gynecological care, as opposed to people with penises. Why do you think this is and does it make sense or should we be doing this differently? Explain your reasoning.


10.9: Returning to Rehabilitation in the Contemporary Juvenile Justice System
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- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
Empirical research drives recent reform efforts. The past decade has witnessed the identification of key developmental processes associated with delinquent behavior, such as brain development research. Ergo, evidence-based practices , which utilize the scientific method to assess the effectiveness of interventions, policies, and programs. In looking at what works, what doesn’t, and what is promising, researchers and policymakers alike assess the implementation of interventions to best meet the needs of the individual youth.
Additionally, several noteworthy Supreme Court cases exemplify society’s evolving standards of decency and the treatment of youth. These key cases demonstrate a move back to rehabilitation and acknowledge the fundamental differences between children and adults.
Key Supreme Court Cases
Roper v Simmons (2005)
In 2005, a landmark decision by the Supreme Court ruled it unconstitutional to impose a death penalty sentence on any youth who was under the age of 18 when they committed their offense ( Roper v. Simmons ). Although Christopher Simmons planned and committed a capital offense (he murdered his neighbor, Shirley Cook), the court ruled that 18 years of age is where criminal responsibility should rest. That is to say, if a child is too young to vote, sign contracts, or do a number of other things (because society deems them not responsible enough), then they are too young to receive the death penalty. The court stated,“the evolving standards of decency that mark the progress of a maturing society” to determine which punishments are so disproportionate as to be “cruel and unusual.” Simmons received life in prison. It was ruled that imposing the death penalty on a person who was under the age of 18 at the time of the crime constituted cruel and unusual punishment. At the time of the Roper v Simmons verdict, the U.S. was only one out of a handful of countries that still imposed the death penalty on juveniles (among other countries were Yemen, Saudi Arabia, and Iran).
Graham v Florida (2010)
While the death penalty was taken off the table for youth under the age of 18, they were instead being sentenced to life in prison without the possibility of parole (LWOP ). This was until the 2010 case of Graham v. Florida . Terrance Graham received life in prison for a felony offense (armed burglary) when he was only 16 years old. Since Florida does not have parole, his sentence de facto became a life without the possibility of parole. The Supreme Court heard his case and ruled that it was unconstitutional to sentence a minor to life without the possibility of parole for a non-homicide offense.
Miller v Alabama (2012)
Two years later, juvenile law again rested in the hands of the Supreme Court. Even though Graham v. Florida abolished life without the possibility of parole for non-homicide offenses, youth under the age of 18 were still receiving that sentence for crimes of murder. In 2012, Evan Miller was 14 years old when he killed his neighbor by severely beating him with a baseball bat while attempting to rob him.  In the majority opinion, Justice Elena Kagan wrote that “mandatory life without parole for those under age of 18 at the time of their crime violates the 8th Amendment ’s prohibition on cruel and unusual punishments. ” “Mandatory life without parole for a juvenile precludes consideration of his chronological age and its hallmark features – among them, immaturity, impetuosity, and failure to appreciate risks and consequences,” Justice Kagan said. “It prevents taking into account the family and home environment that surrounds him – and from which he cannot usually extricate himself – no matter how brutal or dysfunctional.”
This seemed like a huge win for juvenile justice reformers. Juveniles could no longer receive the death penalty, life without parole for non-homicide, nor mandatory life without parole for homicide. However, there were still so many people serving LWOP sentences who were juveniles when they committed their crime.
Montgomery v Louisiana (2016)
In 2016, the Supreme Court heard the case of Henry Montgomery, who was 17 years old in 1963 when killed a sheriff’s deputy. He initially received a death sentence, but this was overturned because of the racial tension of the time (Montgomery was black youth who killed a white law enforcement officer.) He instead received a life sentence and appealed this sentence after the Miller v. Alabama ruling. Montgomery v. Alabama barred mandatory life without parole sentences retroactively. This meant that all youth sentenced prior to 2012 with LWOP sentences needed to be retried.
These four major Supreme Court cases identify the differences between adults and juveniles. They recognize the difference in brain formation and culpability, owning the ability for rehabilitation of youth and moving step by step away from a retribution/punishment model for youth.


3.4: Inner Product Spaces
We have discussed the inner product (or dot product) between two column matrices. Recall that the inner product between, say, two three-by-one column matrices
\[\text{u}=\left(\begin{array}{c}u_1\\u_2\\u_3\end{array}\right),\quad\text{v}=\left(\begin{array}{c}v_1\\v_2\\v_3\end{array}\right)\nonumber \]
is given by
\[\text{u}^{\text{T}}\text{v}=u_1v_1+u_2v_2+u_3v_3.\nonumber \]
We now generalize the inner product so that it is applicable to any vector space, including those containing functions.
We will denote the inner product between any two vectors \(\text{u}\) and \(\text{v}\) as \((\text{u}, \text{v})\), and require the inner product to satisfy the same arithmetic rules that are satisfied by the dot product. With \(\text{u}\), \(\text{v}\), \(\text{w}\) vectors and \(c\) a scalar, these rules can be written as
\[(\text{u},\text{v})=(\text{v},\text{u}),\quad (\text{u}+\text{v},\text{w})=(\text{u},\text{w})+(\text{v},\text{w}),\quad (\text{cu},\text{v})=\text{c}(\text{u},\text{v})=(\text{u},\text{cv});\nonumber \]
and \((\text{u},\text{u})\geq 0\), where the equality holds if and only if \(\text{u}=0\).
Generalizing our definitions for column matrices, the norm of a vector \(\text{u}\) is defined as
\[||\text{u}||=(\text{u},\text{u})^{1/2}.\nonumber \]
A unit vector is a vector whose norm is one. Unit vectors are said to be normalized to unity , though sometimes we just say that they are normalized . We say two vectors are orthogonal if their inner product is zero. We also say that a basis is orthonormal (as in an orthonormal basis) if all the vectors are mutually orthogonal and are normalized to unity. For an orthonormal basis consisting of the vectors \(v_1,\: v_2,\cdots , v_n\), we write
\[(v_i,v_j)=\delta_{ij},\nonumber \]
where \(\delta_{ij}\) is called the Kronecker delta, defined as
\[\delta_{ij}=\left\{\begin{array}{ll}1,&\text{if }i=j; \\ 0,&\text{if }i\neq j.\end{array}\right.\nonumber \]
Oftentimes, basis vectors are used that are orthogonal but are normalized to other values besides unity.
Define an inner product for \(\mathbb{P}_n\).
Solution
Let \(p(x)\) and \(q(x)\) be two polynomials in \(\mathbb{P}_n\). One possible definition of an inner product is given by
\[(p,q)=\int_{-1}^1 p(x)q(x)dx.\nonumber \]
You can check that all the conditions of an inner product are satisfied.
Show that the first four Legendre polynomials form an orthogonal basis for \(\mathbb{P}_3\) using the inner product defined above.
Solution
The first four Legendre polynomials are given by
\[P_0(x)=1,\quad P_1(x)=x,\quad P_2(x)=\frac{1}{2}(3x^2-1),\quad P_3(x)=\frac{1}{2}(5x^3-3x),\nonumber \]
and these four polynomials form a basis for \(\mathbb{P}_3\). With an inner product defined on \(\mathbb{P}_n\) as
\[(p,q)=\int_{-1}^1p(x)q(x)dx,\nonumber \]
it can be shown by explicit integration that
\[(P_m,P_n)=\frac{2}{2n+1}\delta_{m,n},\nonumber \]
so that the first four Legendre polynomials are mutually orthogonal. They are normalized so that \(P_n(1) = 1\).
Define an inner product on \(\mathbb{P}_n\) such that the Hermite polynomials are orthogonal.
Solution
For instance, the first four Hermite polynomials are given by
\[H_0(x)=1,\quad H_1(x)=2x,\quad H_2(x)=4x^2-2,\quad H_3(x)=8x^3-12x,\nonumber \]
which also form a basis for \(\mathbb{P}_3\). Here, define an inner product on \(\mathbb{P}_n\) as
\[(p,q)=\int_{-\infty}^\infty p(x)q(x)e^{-x^2}dx.\nonumber \]
It can be shown that
\[(H_m,H_n)=2^n\pi^{1/2}n!\delta_{m,n},\nonumber \]
so that the Hermite polynomials are orthogonal with this definition of the inner product. These Hermite polynomials are normalized so that the leading coefficient of \(H_n\) is given by \(2^n\).


4.3.2: Types of Communications in Organizations
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Learning Objectives
- Know the types of communications that occur in organizations.
In the communication model described above, three types of communication can be used by either the communicator in the initial transmission phase or the receiver in the feedback phase. These three types are discussed next.
Oral Communication
This consists of all messages or exchanges of information that are spoken, and it’s the most prevalent type of communication.
Written Communication
This includes e-mail, texts, letters, reports, manuals, and annotations on sticky notes. Although managers prefer oral communication for its efficiency and immediacy, the increase in electronic communication is undeniable. As well, some managers prefer written communication for important messages, such as a change in a company policy, where precision of language and documentation of the message are important.
Managerial Leadership
Dealing with Information Overload
One of the challenges in many organizations is dealing with a deluge of emails, texts, voicemails, and other communication. Organizations have become flatter, outsourced many functions, and layered technology to speed communication with an integrated communication programs such as Slack, which allows users to manage all their communication and access shared resources in one place. This can lead to information overload, and crucial messages may be drowned out by the volume in your inbox.
Add the practice of “reply to all,” which can add to the volume of communication, that many coworkers use, and that means that you may get five or six versions of an initial e-mail and need to understand all of the responses as well as the initial communication before responding or deciding that the issue is resolved and no response is needed. Here are suggestions to dealing with e-mail overload upward, horizontally, and downward within your organization and externally to stakeholders and customers.
One way to reduce the volume and the time you spend on e-mail is to turn off the spigot of incoming messages.  Also, consider whether your colleagues or direct reports are copying you on too many emails as an FYI. If yes, explain that you only need to be updated at certain times or when a final decision is made.
You will also want to set up a system that will organize your inbox into “folders” that will allow you to manage the flow of messages into groups that will allow you to address them appropriately. Your system might look something like this:
- Inbox: Treat this as a holding pen. E-mails shouldn’t stay here any longer than it takes for you to file them into another folder. The exception is when you respond immediately and are waiting for an immediate response.
- Today: This is for items that need a response today.
- This week: This is for messages that require a response before the end of the week.
- This month/quarter: This is for everything that needs a longer-term response. Depending on your role, you may need a monthly or quarterly folder.
- FYI: This is for any items that are for information only and that you may want to refer back to in the future.
This system prioritizes e-mails based on timescales rather than the e-mails’ senders, enabling you to better schedule work and set deadlines.
Another thing to consider is your outgoing e-mail. If your outgoing messages are not specific, too long, unclear, or are copied too widely, your colleagues are likely to follow the same practice when communicating with you. Keep your communication clear and to the point, and managing your outbox will help make your inbound e-mails manageable.
critical thinking questions
- How are you managing your e-mails now? Are you mixing personal and school and work-related e-mails in the same account?
- How would you communicate to a colleague that is sending too many FYI e-mails, sending too may unclear e-mails, or copying too many people on her messages?
sources
Amy Gallo, Stop Email Overload, Harvard Business Review , February 21, 2012, https://hbr.org/2012/02/stop-email-overload-1 ;
Barry Chingel, “How to beat email Overload in 2018”, CIPHER , January 16, 2018, https://www.ciphr.com/advice/email-overload/ ;
Monica Seely, “At the Mercy of Your Inbox? How to Cope With Email Overload”, The Guardian , November 6, 2017, https://www.theguardian.com/small-bu...email-overload .
Nonverbal Communication
There is also the transformation of information without speaking or writing. Some examples of this are things such as traffic lights and sirens as well as things such as office size and placement, which connote something or someone of importance. As well, things such as body language and facial expression can convey either conscious or unconscious messages to others.
Major Influences on Interpersonal Communication
Regardless of the type of communication involved, the nature, direction, and quality of interpersonal communication processes can be influenced by several factors. 3
Social Influences
Communication is a social process, as it takes at least two people to have a communication episode. There is a variety of social influences that can affect the accuracy of the intended message. For example, status barriers between employees at different levels of the organization can influence things such as addressing a colleague as at a director level as “Ms. Jones” or a coworker at the same level as “Mike.” Prevailing norms and roles can dictate who speaks to whom and how someone responds. Figure 16.2.2 illustrates a variety of communications that illustrate social influences in the workplace.
Perception
In addition, the communication process is heavily influenced by perceptual processes. The extent to which an employee accurately receives job instructions from a manager may be influences by her perception of the manager, especially if the job instructions conflict with her interest in the job or if they are controversial. If an employee has stereotyped the manager as incompetent, chances are that little that the manager says will be taken seriously. If the boss is well regarded or seen as influential in the company, everything that she says may be interpreted as important.
Interaction Involvement
Communication effectiveness can be influenced by the extent to which one or both parties are involved in conversation. This attentiveness is called interaction attentiveness or interaction involvement. 4 If the intended receiver of the message is preoccupied with other issues, the effectiveness of the message may be diminished. Interaction involvement consists of three interrelated dimensions: responsiveness, perceptiveness, and attentiveness.
Organizational Design
The communication process can also be influenced by the design of the organization. It has often been argued to decentralize an organization because that will lead to a more participative structure and lead to improved communication in the organization. When messages must travel through multiple levels of an organization, the possibility of distortion can also occur, which would be diminished with more face-to-face communication.
concept check
- What are the three major types of communication?
- How can you manage the inflow of electronic communication?
- What are the major influences on organizational communication, and how can organizational design affect communication?


2.2: Naming Animals
Of course we know what a cat, a dog and a whale are but, in some situations using the common names for animals can be confusing. Problems arise because people in different countries, and even sometimes in the same country, have different common names for the same animals. For example a cat can be a chat, a Katze, gato, katt, or a moggie, depending on which language you use. To add to the confusion sometimes the same name is used for different animals. For example, the name ‘gopher’ is used for ground squirrels, rodents (pocket gophers), for moles and in the south-eastern United States for a turtle. This is the reason why all animals have been given an official scientific or binomial name . Unfortunately these names are always in Latin. For example:
- Common rat: Rattus rattus
- Human: Homo sapiens
- Domestic cat: Felis domesticus
- Domestic dog: Canis familiaris
As you can see from the above there are certain rules about writing scientific names:
- They always have 2 parts to them.
- The first part is the genus name and is always written with a capital first letter.
- The second name is the species name and is always written in lower case .
- The name is always underlined or printed in italics .
The first time you refer to an organism you should write the whole name in full. If you need to keep referring to the same organism you can then abbreviate the genus name to just the initial. Thus “ Canis familiaris ” becomes “ C. familiaris ” the second and subsequent times you refer to it.


11.10: Short Responses to Criticism
- Write a short message responding to internal or external criticism.
In today’s business world, your professional demeanor is not limited to your physical workplace. Your online presence is an extension of you and also must convey all of the characteristics we have discussed in the last section. Given the amount of cyber-bullying and just plain rudeness we all see online, it is even more challenging to maintain your professionalism when responding to an external critical message on social media.
Here are seven tips from Caron Beesley in her article “7 Tips for Dealing With Criticism of Your Business on Social Media“: [1]
- Get Listening
- Should You Respond?
- Don’t Let Negative Comments Linger
- Always Acknowledge, Never Deny
- How to Apologize
- Take the Conversation Offline
- The Bottom Line
Using these guidelines, you can easily craft a short response like the following examples:
Thanks for your feedback! We’ll take a look at the issue and get back to you within a week. We are so sorry for your negative experience with our product. We’ve sent this issue to our customer support team, and they should be reaching out to you within the next day to resolve the problem.Of course, if you promise to get back to someone, you need to remember to do so; however, these responses both show your willingness to hear complaints and follow tip number 6: take the conversation offline. By doing this, you can have a much more interesting and productive conversation with the person who is making the critique.
In addition to these guidelines, keeping things light and somewhat tongue-in-cheek can also be effective for a short message response.
Wendy’s Twitter has a bit of a reputation for being irreverent and “savage” (just check out this article from Market Watch). While this brand of humor has gained Wendy’s quite a bit of positive attention, some people argue that Wendy’s is making outdated jokes, and simply trying to appeal to its audience by trying to be current. In response to this broad criticism, Wendy’s tweeted the following:
buy our cheeseburgers pic.twitter.com/tohfFsyeR4
— Wendy's (@Wendys) June 28, 2018
In this tweet, Wendy’s largely agreed with the comments that their Twitter account is a marketing move, but they were able to make a joke out of the interaction and maintain the tone of their Twitter presence.
- Beesley, Caron. "7 Tips for Dealing With Criticism of Your Business on Social Media." SBA Blogs: Managing a Business . 11 Oct 2012. Web. 10 July 2018.
Contributors and Attributions
- Short Responses to Criticism. Authored by : Robert Danielson. Provided by : Lumen Learning. License : CC BY: Attribution


12: Introduction to Calculus
Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. Calculus has two basic applications: differential calculus and integral calculus.
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- 12.2: Finding Limits - Properties of Limits
- Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits. Knowing the properties of limits allows us to compute limits directly.
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- 12.3: Continuity
- A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example. A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function. We can check three different conditions to decide if a function is continuous at a particular number.


17.2: An Introduction to Human Resource Management
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What has been the evolution of human resource management (HRM) over the years, and what is the current value it provides to an organization?
Human resource management over the years has served many purposes within an organization. From its earliest inception as a primarily compliance-type function, it has further expanded and evolved into its current state as a key driver of human capital development. In the book HR From the Outside In (Ulrich, Younger, Brockbank, Younger, 2012), the authors describe the evolution of HR work in “waves”. Wave 1 focused on the administrative work of HR personnel, such as the terms and conditions of work, delivery of HR services, and regulatory compliance. This administrative side still exists in HR today, but it is often accomplished differently via technology and outsourcing solutions. The quality of HR services and HR’s credibility came from the ability to run administrative processes and solve administrative issues effectively. Wave 2 focused on the design of innovative HR practice areas such as compensation, learning, communication, and sourcing. The HR professionals in these practice areas began to interact and share with each other to build a consistent approach to human resource management. The HR credibility in Wave 2 came from the delivery of best-practice HR solutions.
Wave 3 HR, over the last 15–20 years or so, has focused on the integration of HR strategy with the overall business strategy. Human resources appropriately began to look at the business strategy to determine what HR priorities to work on and how to best use resources. HR began to be a true partner to the business, and the credibility of HR was dependent upon HR having a seat at the table when the business was having strategic discussions. In Wave 4, HR continues to be a partner to the business, but has also become a competitive practice for responding to external business conditions. HR looks outside their organizations to customers, investors, and communities to define success—in the form of customer share, investor confidence, and community reputation. HR’s credibility is thus defined in terms of its ability to support and drive these external metrics. Although each “wave” of HR’s evolution is important and must be managed effectively, it is the “outside in” perspective that allows the human resource management function to shine via the external reputation and successes of the organization.
catching the entrepreneurial spirit
Human Resources Outsourcing—Entrepreneurial Ventures
Human resources is a key function within any company, but not all companies are able to afford or justify full-time HR staff. Over the last decade, HR outsourcing has become a good business decision for many small companies whose current staff doesn’t have the bandwidth or expertise to take on the risks of employee relations issues, benefits and payroll, or HR compliance responsibilities. This has led many HR practitioners to try out their entrepreneurial skills in the areas of HR outsourcing and “fractional HR.”
Human resources outsourcing is very commonly used by smaller companies (and often large companies too) to cover such tasks as benefits and payroll management. This is an area that has been outsourced to third parties for many years. More recent is the trend to have “fractional HR” resources to help with the daily/weekly/monthly HR compliance, employee relations, and talent management issues that companies need to address. Fractional HR is a growing industry, and it has become the service offering of many entrepreneurial HR ventures. Fractional HR is essentially as it sounds—it is the offering of HR services to a company on a part-time or intermittent basis when the company may not be able to justify the cost of a full-time HR resource. An HR professional can be available onsite for a specified number of hours or days weekly or monthly, depending on the company’s needs and budget. The HR professional handles everything from HR compliance issues and training to employee issues support. Also, for companies that are keen on development of employees, the HR resource can drive the talent management processes—such as performance management, succession planning, training, and development—for companies who require more than just basic HR compliance services.
How does a business leader decide whether HR outsourcing is needed? There are generally two factors that drive a leader to consider fractional HR or HR outsourcing—time and risk. If a leader is spending too much time on HR issues and employee relations, he may decide that it is a smart tradeoff to outsource these tasks to a professional. In addition, the risk inherent in some HR issues can be very great, so the threat of having a lawsuit or feeling that the company is exposed can lead the company to seek help from a fractional HR professional.
HR entrepreneurs have taken full advantage of this important trend, which many say will likely continue as small companies grow and large companies decide to off-load HR work to third parties. Some HR companies offer fractional HR as part of their stated HR services, in addition to payroll and benefits support, compensation, and other HR programmatic support. Having a fractional HR resource in place will often illuminate the need for other HR services and program builds, which are generally supported by those same companies. Whether you are an individual HR practitioner or have a small company of HR practitioners and consultants, fractional HR and HR outsourcing can be a very viable and financially rewarding business model. It can also be very personally rewarding, as the HR professional enables smaller companies to grow and thrive, knowing that its HR compliance and processes are covered.
Discussion Questions
- At what point should a company consider bringing on a full-time HR resource instead of using a fractional HR resource? What questions should the company ask itself?
Human resource management provides value to an organization, to a large extent, via its management of the overall employee life cycle that employees follow—from hiring and onboarding, to performance management and talent development, all the way through to transitions such as job change and promotion, to retirement and exit. Human capital is a key competitive advantage to companies, and those who utilize their human resource partners effectively to drive their human capital strategy will reap the benefits.
Human resource management includes the leadership and facilitation of the following key life cycle process areas:
- Human resources compliance
- Employee selection, hiring, and onboarding
- Performance management
- Compensation rewards and benefits
- Talent development and succession planning
Human resources is responsible for driving the strategy and policies in these areas to be in accordance with and in support of the overall business strategy. Each of these areas provides a key benefit to the organization and impacts the organization’s value proposition to its employees.
concept check
- In what way do you usually interact with human resources?


10.4: The Matrix Exponential via the Laplace Transform
You may recall from the Laplace Transform module that may achieve \(e^{at}\) via
\[e^{at} = \mathscr{L}^{-1}(\frac{1}{s-a}) \nonumber\]
The natural matrix definition is therefore
\[e^{At} = \mathscr{L}^{-1} ((sI-A)^{-1}) \nonumber\]
where \(I\) is the n-by-n identity matrix.
The easiest case is the diagonal case, e.g.,
\[A = \begin{pmatrix} {1}&{0}\\ {0}&{2} \end{pmatrix} \nonumber\]
for then
\[(sI-A)^{-1} = \begin{pmatrix} {\frac{1}{s-1}}&{0}\\ {0}&{\frac{1}{s-2}} \end{pmatrix} \nonumber\]
and so
\[e^{At} = \begin{pmatrix} {\mathscr{L}^{-1} (\frac{1}{s-1})}&{0}\\ {0}&{\mathscr{L}^{-1} (\frac{1}{s-2})} \end{pmatrix} \nonumber\]
As a second example let us suppose
\[A = \begin{pmatrix} {0}&{1}\\ {-1}&{0} \end{pmatrix} \nonumber\]
and compute, in matlab,
>> inv(s*eye(2)-A) ans = [ s/(s^2+1), 1/(s^2+1)] [-1/(s^2+1), s/(s^2+1)] >> ilaplace(ans) ans = [ cos(t), sin(t)] [-sin(t), cos(t)]
If
\[A = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \nonumber\]
then
>> inv(s*eye(2)-A) ans = [ 1/s, 1/s^2] [ 0, 1/s] >> ilaplace(ans) ans = [ 1, t] [ 0, 1]


16.3: Floating-Point Data Type
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Overview
A floating-point data type uses a formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. For this reason, floating-point computation is often found in systems which include very small and very large real numbers, which require fast processing times. A number is, in general, represented approximately to a fixed number of significant digits and scaled using an exponent in some fixed base. [1]
Discussion
The floating-point data type is a family of data types that act alike and differ only in the size of their domains (the allowable values). The floating-point family of data types represents number values with fractional parts. They are technically stored as two integer values: a mantissa and an exponent . The floating-point family has the same attributes and acts or behaves similarly in all programming languages. They can always store negative or positive values thus they always are signed; unlike the integer data type that could be unsigned. The domain for floating-point data types varies because they could represent very large numbers or very small numbers. Rather than talk about the actual values, we mention the precision . The more bytes of storage the larger the mantissa and exponent, thus more precision.
| Language | Reserved Word | Size | Precision | Range |
|---|---|---|---|---|
| C++ |
float
|
32 bits / 4 bytes | 7 decimal digits | ±3.40282347E+38 |
| C++ |
double
|
64 bits / 8 bytes | 15 decimal digits | ±1.79769313486231570E+308 |
| C# |
float
|
32 bits / 4 bytes | 7 decimal digits | ±3.40282347E+38 |
| C# |
double
|
32 bits / 4 bytes | 15 decimal digits | ±1.79769313486231570E+308 |
| Java |
float
|
32 bits / 4 bytes | 7 decimal digits | ±3.40282347E+38 |
| Java |
double
|
32 bits / 4 bytes | 15 decimal digits | ±1.79769313486231570E+308 |
| JavaScript |
Number
|
64 bits / 8 bytes | 15 decimal digits | ±1.79769313486231570E+308 |
| Python |
float()
|
64 bits / 8 bytes | 15 decimal digits | ±1.79769313486231570E+308 |
| Swift |
Float
|
32 bits / 4 bytes | 7 decimal digits | ±3.40282347E+38 |
| Swift |
Double
|
64 bits / 8 bytes | 15 decimal digits | ±1.79769313486231570E+308 |
Key Terms
- double
- The most often used floating-point family data type used.
- mantissa exponent
- The two integer parts of a floating-point value.
- precision
- The effect on the domain of floating-point values given a larger or smaller storage area in bytes.