18: Evolution and the Origin of Species
The theory of evolution is the unifying theory of biology, meaning it is the framework within which biologists ask questions about the living world. Its power is that it provides direction for predictions about living things that are borne out in experiment after experiment. The Ukrainian-born American geneticist Theodosius Dobzhansky famously wrote that “nothing makes sense in biology except in the light of evolution. " He meant that the tenet that all life has evolved and diversified from a common ancestor is the foundation from which we approach all questions in biology.
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- 18.1: Understanding Evolution
- Evolution by natural selection describes a mechanism for how species change over time. That species change had been suggested and debated well before Darwin began to explore this idea. The view that species were static and unchanging was grounded in the writings of Plato, yet there were also ancient Greeks who expressed evolutionary ideas.
Thumbnail: A silhouette of human evolution. (CC BY-SA 3.0; Tkgd2007 via Wikimedia Commons ).


3.2: Develop an effective student-instructor connection
Now that you have developed a picture of the role that your instructors will have in your online learning journey, how might you develop an effective relationship with them. Consider the following suggestions:
- When you look at a course reading, activity, or assignment, try to put yourself inside the mind of your instructor. Why do you think they chose this particular learning experience for you? What do you think they intend for you to learn? Understanding the purpose of a learning activity can increase your motivation and help you to stay on track in your work.
- Take advantage of opportunities to connect with your instructor, either face to face or during online office hours. You can attend office hours to clarify course difficulties, but also to build your relationship with your instructor and demonstrate your interest in the course material.
- Check the course site regularly. Your instructor is likely to post announcements and other key messages for the class. This provides another point of frequent contact.
- Use email effectively to connect. You will learn more about how to construct an effective email in the section on professional communication in this workshop — but if you are curious, you may choose to view the chapter on how to email an instructor now.
- When you receive feedback on an assignment, consider it carefully. How does your instructor want to see you grow in your academic or professional skills?
For Reflection
Reflect on the questions below. If you wish, you may print out a downloadable version of this activity.
How would you define an effective relationship with an online instructor?
What is one step you will take to develop a relationship with each of your course instructors?


2.4: Reading and Writing (1) القراءة والكتابة
In this section, you will be able to:
- identify and understand words about university
- demonstrate understanding about school and subjects by answering the questions.
- talk about your university
- Write about yourself and your school.
Rana is talking with her friends about schools in the coffee shop. She is telling her friends about her cousin, Nawal, who is studying at the Lebanon University. Read the text then answer the questions
| ابنةُ عَمَّتِي نَوال |
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ابْنَةُ عَمَّتِي نَوالُ تَدْرسُ الأَدَبَ الفَرَنْسيّ في الجَامِعةِ اللُبْنَانِيَّة في بَيْرُوت. وَهِي فِي السَّنَةِ الثَّانِيَة. وَهِيَ تَدْرُسُ عَنْ بُعْد. فِي جَامِعَةِ لُبْنَانَ هناك كُلِّيَات العُلُوم وَ الهَنْدَسَة وَالتِّكْنُولُوجيا والتِّجارَة والحُقُوقِ. نَوَالُ عِندها صَدِيْقَات يَدْرُسْن فِي نَفْسِ الجَامِعَةِ عَنْ بُعْد. صَدِيقَتُها إيمان مِنَ السُّودَانِ تَدْرُسُ فِي كُلِّيَّةِ الحُقُوق. وَنَدَى مِنْ مِصر وَ سَناءُ مِنَ اليَمَنِ تَدْرُسُان فِي كُلِّيَةِ التِّجَارَة.وَالأَسَاتِذَةُ فِي الجَامِعَةِ يَتَكَلَّمُونَ اللُّغَةَ الإِنْجِلِيزِيَّةَ وَاللُغَةَ العَرَبِيَّة |
In your language class, practice introducing your paternal cousin to your classmates. Mention her name, her home country, where she's currently living in the USA, the university or college she's attending, the languages she can speak, and her field of study
You have been selected as the "New Student of the Month" at your high school. The school has asked you to write a brief introduction about yourself for the school newsletter. In your writing, include the following details: your name, nationality, information about your family, where you live, and the classes you are taking


6.11: The Third-Century Crisis, and Late Antiquity
While the second century CE was a time when the Empire flourished, the third century was a time of crisis, defined by political instability and civil wars, which ultimately demonstrated that the Empire had become too large to be effectively controlled by one ruler. Furthermore, the increasing pressures on the frontiers, which required emperors to spend much of their time on campaigns, resulted in the decline of the importance of the city of Rome. By the end of the third century, an experiment with dividing the empire showed a different model of rule, one which lasted, albeit with some interludes, until the last Western emperor, Romulus Augustulus , was deposed in 476 CE. While the political narrative of the third century and Late Antiquity could be described as a story of decline and fall of the Roman Empire (as the British historian Gibbon famously called it), nevertheless, it was a period in which culture, and especially Christian culture, flourished and replaced the traditional Roman pagan mode of thinking. Far from being culturally a time of “decline and fall,” Late Antiquity, rather, was looking forward to the world of the Middle Ages. It was also the period of Roman history that produced some of its most influential leaders, most notably, Constantine.
The Third-Century Crisis and Diocletian
Although composed during a time of prosperity in the Empire, Apuleius’ novel Metamorphoses showed tensions in the provinces, indicative of the failure of Empire to govern all portions equally effectively. While not visible in the larger urban centers until the third century CE, these tensions manifested themselves clearly during the third-century crisis, a period of almost fifty years (235 – 284 CE) that was characterized by unprecedented political, social, and economic upheaval across the Empire. In effect, the third-century crisis was the year 69 CE repeated, but this time it stretched over half a century. The same secrets of power that 69 CE revealed for the first time— that armies could make emperors and that emperors could be made outside of Rome—were now on display yet again.
235 CE, the emperor Severus Alexander was assassinated by his troops on campaign, who then proclaimed as emperor their general Maximinus Thrax. Over the subsequent half-century, twenty-six emperors were officially recognized by the Roman Senate, and a number of others were proclaimed emperors but did not live long enough to consolidate power and be officially accepted as emperors by the Senate. Most of these new emperors were military generals who were proclaimed by their troops on campaign. Most of them did not have any previous political experience and thus had no clear program for ruling the empire. The competing claims resulted in the temporary breaking away from the Roman Empire of regions to the East and the Northwest.
The political instability that resulted was not, however, the only problem with which the Empire had to contend. In addition to political upheaval and near-constant civil wars, the Empire was also dealing with increasing pressures on the frontiers, a plague that devastated the population, a famine, and rampant inflation. Roman emperors, starting with Nero, had been debasing the Roman coinage, but not until the third-century crisis did the inflation hit in full force.
The third-century crisis showed that a single emperor stationed in Rome was no longer equipped to deal with the challenges of ruling such a vast territory. And, indeed, so recognized the man who ended the crisis: the emperor Diocletian Born to a socially insignificant family in the province of Dalmatia, Diocletian had a successful military career. Proclaimed emperor by his troops in 284 CE, Diocletian promptly displayed a political acumen that none of his predecessors in the third century possessed. Realizing that, as the third-century crisis showed, a single emperor in charge of the entire empire was a “sitting duck,” whose assassination would throw the entire empire into yet another civil war, Diocletian established a new system of rule: the Tetrarchy , or the rule of four. He divided the empire into four regions, each with its own capital.
It is important to note that Rome was not the capital of its region. Diocletian clearly wanted to select as capitals cities with strategic importance, taking into account such factors as proximity to problematic frontiers. Of course, as a Dalmatian of low birth, Diocletian also lacked the emotional connection to Rome that the earliest emperors possessed. Two of the regions of the Tetrarchy were ruled by senior emperors, named Augusti (“Augustus” in the singular), and two were ruled by junior emperors, named Caesares (“Caesar” in the singular). One of the Augusti was Diocletian himself, with Maximian as the second Augustus. The two men’s sons-in-law, Galerius and Constantus Chlorus, became the two Caesares. Finally, it is important to note that in addition to reforming imperial rule, Diocletian attempted to address other major problems, such as inflation, by passing the Edict of Maximum Prices. This edict set a maximum price that could be charged on basic goods and services in the Empire. He also significantly increased the imperial bureaucracy. In a nutshell, as some modern historians have described him, Diocletian was the most significant Roman reformer since Augustus.
Diocletian’s political experiment was most clearly successful in achieving one goal: ending the third-century crisis. The four men were able to rule the empire and restore a degree of political stability. A statue column of the Tetrarchs together displays their message of unity in rule: the four men are portrayed identically, so it is impossible to tell them apart. Showing their predominantly military roles, they are dressed in military garb, rather than the toga, the garb of politicians and citizens, and each holds one hand on the hilt of his sword and hugs one of the other Tetrarchs with the other.
While it succeeded in restoring stability to the Empire, inherent within the Tetrarchy was the question of succession, which turned out to be a much greater problem than Diocletian had anticipated. Hoping to provide for a smooth transition of power, Diocletian abdicated in 305 CE and required Maximian to do the same. The two Caesares, junior emperors, were promptly promoted to Augusti, and two new Caesares were appointed. The following year, however, Constantius Chlorus, a newly minted Augustus, died. His death resulted in a series of wars for succession, which ended Diocletian’s experiment of the Tetrarchy. The wars ended with Constantius’ son, Constantine , reuniting the entire Roman Empire under his rule in 324 CE. In the process, Constantine also brought about a major religious shift in the Empire.
From Constantine to the Last Pagans of Rome
While traditional Roman religion was the ultimate melting pot, organically incorporating a broad variety of new cults and movements from the earliest periods of Roman expansion, Christianity’s monotheistic exclusivity challenged traditional Roman religion and transformed Roman ways of thinking about religion in late antiquity.By the early fourth century CE, historians estimate that about ten percent of those living in the Roman Empire were Christians. With Constantine, however, this changed, and the previously largely underground faith grew exponentially because of the emperor’s endorsement. The emperor’s conversion must have seemed nothing short of miraculous to contemporaries, and a miracle is told to explain it in contemporary sources. Before a major battle in 312 CE, Constantine reportedly had a dream or a vision in which Christ himself told Constantine to place the Greek letters X and P (Chi, Rho, the first two letters of Christ’s name in the Greek alphabet) on his soldiers’ shields in order to assure victory.
Grateful for his subsequent victory, Constantine proceeded to play a major role in the government of the church over the course of his rule, although he was not baptized himself until he was on his deathbed. Constantine, for instance, summoned the First Council of Nicaea in 325 CE, which gathered major bishops from all over the Empire. The Council settled, among other issues, the question of the relationship of God the Father and God the Son, declaring them to have been one being from the creation of the world, thus affirming the doctrine of the Trinity. The Council set a significant precedent for communication of bishops in the Empire. It ended up being merely the first of seven major ecumenical councils, the last of them being the Second Council of Nicaea in 787 CE. The councils allowed the increasingly different churches of the Eastern and Western parts of the Roman Empire to work together on key doctrines and beliefs of the church.
Last but not least, Constantine’s rule marked the end of the city of Rome as the capital of the Roman Empire. Upon reuniting the Empire in 324 CE, Constantine established his capital at the old location of the Greek city of Byzantium, but renamed it Constantinople (the location of Byzantium appears on Map \(\PageIndex{2}\)). The location had strategic advantages for the Empire at that stage. First, it had an excellent harbor. Second, it was close to the Persian frontier, as well as the Danube frontier, a trouble area that required attention from the emperor. Finally, building this new city, to which he also referred as “New Rome,” allowed Constantine to send the message that his rule was a new beginning of sorts for the Roman Empire, which was now to be a Christian empire.
With the Emperor’s backing, Christianity seems to have grown exponentially over the course of the fourth century CE, much to the chagrin of Julian the Apostate , Rome’s final pagan emperor, who tried hard to restore traditional Roman paganism during his brief rule (361 – 363 CE). Finally, the Emperor Theodosius gradually banned paganism altogether by 395 CE. Thus a mere eighty-three years after Constantine’s initial expression of support for Christianity, it became the official religion of Rome. Paganism continued to limp on for another century or so, but without state support, it slowly died out.
The Decline of the Empire—Looking Forward while Looking Back with Augustine and the Last Pagans of Rome
Imagine that you are a citizen of the greatest empire on earth. In fact, you reside in the greatest city of the greatest empire on earth. You feel protected by the pact that was made between the founders of your state and the traditional gods. The pax deorum , or peace with the gods, struck a clear bargain: as long as you and your state worshipped the gods and maintained peace with them, they would make it prosper. And prosper it did! Starting out as a tiny village on the marshes of the Tiber, the Roman Empire at its height encircled the entire Mediterranean, extending to Britain and the Rhine and Danube frontiers to the north, and including a wide strip of North Africa in its southern half. But something went so terribly wrong along the way, testing the gods’ patience with Rome. A new sect started out in Judaea in the first century CE, one which followed a crucified Messiah. Spreading outward like a wildfire to all parts of the empire, this sect challenged and gradually replaced the worship of the traditional gods, bringing even the emperors into its fold, starting with Constantine in the early fourth century CE. This outright violation of the thousand-year old pact between the Romans and their gods could have only one outcome: the ultimate punishment would come from the gods upon this rebellious state. And come it did; in 410 CE, the unthinkable happened. The city of Rome, untouched by foreign foe since the early days of the Republic, was sacked by the Goths, a Germanic tribe, led by the fearsome Alaric. How could something so terrible happen? And how could the Roman Empire recover from it? Such was the thought process of the typical Roman pagan, and especially the pagan aristocrat, as few of those as were left by 410 CE. And it was in response to these questions that Augustine, veteran theologian, philosopher, and bishop of Hippo in North Africa, wrote the final magnum opus of his career, the monumental twenty-two-book effort that he appropriately titled De Civitate Dei Contra Paganos , or On the City of God against the Pagans.
It is no coincidence that Peter Brown, the scholar credited with creating the academic field of study of Late Antiquity, began his career as a researcher by writing a biography of Augustine. Indeed, no other figure exemplifies so clearly the different culture that emerged in Late Antiquity, a culture of rethinking the Roman past, with an eye to a future in which Rome no longer existed as the capital of the Roman Empire. Born in North Africa in 354 CE, Augustine was educated in Rome and Milan, and, after a wild youth—about which he tells us in his Confessions— he rose to the post of the Bishop of Hippo in 396 CE. A famous figure by 410 CE, he was ideally suited to address the tragedy of the sack of Rome and the concerns that this event inspired in Christians and pagans alike.
In his book, Augustine presented an argument that challenged the core of Roman traditional beliefs about the state. Challenging the fundamental Roman pagan belief that Roman success was the result of the pax deorum , Augustine effectively argued that there was nothing special about Rome. It only prospered in its earlier history because God allowed it to do so. Furthermore, argued Augustine, obsession with Rome, emblematic of obsession with the earthly kingdom and way of life, was the wrong place for turning one’s attention. The City of God was the only place that mattered, and the City of God was most definitely not Rome. By turning away from this world and focusing on the next, one could find true happiness and identity as a citizen of God’s kingdom, which is the only city that is everlasting and free from threat of invasion or destruction.
Augustine’s message would have made the Republican hero Cincinnatus weep. For Cincinnatus, nothing was more valuable than Rome. For Augustine, however, nothing was less valuable than Rome.


<IP_ADDRESS>: Saving and Investing
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Find popular topics about saving and investing.
What’s on This Page
- Manage Finances and Save Money
- Steps to Make a Budget
- Saving for Retirement
- Savings Bonds
- Treasury Securities
- Trusts
- Things to Consider Before You Invest
- Tools to Research Investments
- Choose a Financial Professional
- Pyramid Schemes
Manage Finances and Save Money
To help you manage your money and reach your saving goals:
Create a Budget
A budget is your plan for how you will spend money over a set period of time. It shows how much money you make and how you spend your money. Creating a budget can help you:
- Pay your bills on time.
- Save for unplanned expenses in the future.
- Prepare for retirement.
Download a budget spreadsheet that you can use to create your own to manage your monthly income and expenses.
Consider Ways to Save
Saving money involves looking for deals and buying the quality items you need at the best price. You can save money by comparison shopping , comparing the prices and quality of products you plan to buy. MyMoney.gov offers ways to manage your spending and build your savings accounts to achieve your saving goals.
Invest in Long Term Goals
Investing is a way to make money grow, by buying shares of stocks, mutual funds, bonds, or real estate. When you invest, there is risk that you could lose the money you invest; in general the greater the earnings you can make, the greater the risk. You can save for long term goals, such as retirement (PDF,
Download Adobe Reader
) and college education, by investing. Learn how to savefor emergencies, short term and long term goals, and become an informed investor.
Steps to Make a Budget
See this video to learn how to make a budget and plan your finances.
Saving for Retirement
As you approach retirement, there are many things to think about. Experts advise that you will need about 80 percent of your pre-retirement income to continue your current quality of life. The exact amount depends on your individual needs. Some important factors to consider include:
- At what age do you plan to retire?
- Can you participate in an employer’s retirement savings plan, such as a 401(k) plan, or a traditional pension plan?
- Will your spouse or partner retire when you do?
- Where do you plan to live when you retire? Will you downsize, rent, or own your home?
- Do you expect to work part-time?
- Will you have the same medical insurance you had while working? Will your coverage change?
- Do you want to travel or pursue a new hobby that might be costly?
Resources to Help You Prepare for Retirement
To help you plan for retirement:
- Find practical tips for building retirement savings in the Top 10 Ways to Prepare for Retirement(PDF, Download Adobe Reader ).
- Use a retirement calculator to find out the best age to claim your Social Security benefits.
- myRA can help you start saving for retirement when you don’t have access to an employer-sponsored plan or lack other options to save.
- Find out the trade-offs between taking your pension in a monthly payment or in a lump sum (PDF, Download Adobe Reader ).
- Social Security pays benefits that are on average equal to about 40 percent of your pre-retirement earnings. You may be able to estimate your benefits .
- Learn how you can boost your retirement savings at Investor.gov.
- If you have a financial advisor, talk to him or her about your plans.
Savings Bonds
U.S. savings bonds are one of the safest types of investments because they are endorsed by the federal government and, therefore, are virtually risk free.
Visit TreasuryDirect, a website from the U.S. Department of the Treasury, to learn about savings bonds, treasury bonds, and securities : how to buy and redeem your investments, what to do in the event of the death of an owner, and much more. TreasuryDirect is your one-stop shopping site for government securities where you can find information about the wide range of savings options, including EE/E , HH/H , and I savings bonds .
Manage and determine the value of savings bonds using these tools:
You can give savings bonds for many occasions, such as birthdays, weddings, and graduations. Learn how to give savings bonds as gifts .
Treasury Securities
Treasury securities are debts issued by the federal government’s Bureau of Fiscal Service. When you buy a treasury security, you are lending money to the federal government for a set amount of time. In return the government promises to pay you back the entire amount, also known as the face value, when the security matures.
There are several types of treasury securities:
- Treasury Bills —Short term securities that mature between a few days and 52 weeks.
- Treasury Notes —Medium term securities that mature between one and 10 years.
- Treasury Bonds —Long term securities, with a 30 year term that pays interest every six months, until the bond matures.
- Treasury Inflation-Protected Securities (TIPS) —Securities with principle values that adjust based on inflation, but with fixed interest rates for five, 10, or 30 year maturities.
- Savings Bonds —Securities that offer a fixed interest rate over a fixed period of time.
- Floating Rate Notes (FRNs)—Securities with variable interest rates, so that as bank interest rates increase or decrease, the interest rates on the FRNs change in the same direction.
You can purchase treasury securities for yourself or as gifts. You can purchase them in several ways:
- Banks, brokers, and other financial institutions through the Commercial Book-Entry System.
- Online through Treasury Direct
- Payroll savings plans
- Public auctions
Trusts
A trust (or trust fund) is a legal entity that allows a person (the grantor, donor, or settlor) to transfer assets to another person or organization (the trustee). Once the grantor establishes the trust, the trustee controls and manages the assets for the grantor or for another beneficiary—someone who will ultimately benefit from the trust. To help you decide if a trust is right for you, first consult a licensed attorney experienced with estate planning and trust matters.
Reasons to Set Up a Trust
Some common reasons for setting up a trust include:
- Providing for minor children or family members who are inexperienced or unable to handle financial matters
- Arranging for management of personal assets, if you become unable to handle them yourself
- Avoiding probate and immediately transferring assets to beneficiaries upon death
- Reducing estate taxes and providing liquid assets to help pay for them
- The terms of a will are public while the terms of a trust are not, so privacy makes a trust an appealing option.
Types of Trusts
Trusts can be living (inter vivos) or after-death (testamentary). A living trust is one that a grantor sets up while still alive and an after-death trust is usually established by a will after one’s death. Living trusts can be irrevocable (can’t be changed) or revocable (can be changed), although revocable trusts don’t get the same tax shelter benefits as irrevocable ones do.
The most common type of trust is the revocable living trust. If there’s a specific purpose in mind for the trust, dozens of different options exist (charitable trusts, bypass trusts, spendthrift trusts, and life insurance trusts). Two types of trusts can help pay for long-term care services:
- Charitable Remainder Trusts – This trust allows you to use your own assets to pay for long-term care services while contributing to a charity of your choice and reducing your tax burden at the same time. You can set up the trust so that you receive payments from the trust to pay for long-term care services while you are alive.
- Medicaid Disability Trusts – These trusts are limited to persons with disabilities who are under age 65 and qualify for public benefits. Parents, grandparents, and legal guardians often set up these trusts to benefit people with disabilities and a non-profit organization manages the assets. This is the only kind of trust that is exempt from rules regarding trusts and Medicaid eligibility.
Trust Scams and Fraud
If someone approaches you to set up a trust, be careful. Before signing any papers to create a living trust, will, or other kind of trust, make sure to explore all options and shop around to compare services. Some other tips to avoid trust scams and fraud include:
- Avoid high-pressure sales tactics and high-speed sales pitches.
- Stay away from salespeople who give the impression that specific organizations and recognized brands back or sell the trust.
- Research and get information about local probate laws from the Clerk or Register of Wills.
- If someone tries to sell a living trust to you, ask if they are an attorney. Some states restrict the sale of living trusts to licensed attorneys.
- If you buy a trust in your home or in another location that is not the seller’s permanent place of business, remember you have the right to take advantage of the Cooling Off Rule and cancel the transaction within three business days.
Things to Consider Before You Invest
Do you have a financial goal in mind, such as saving for retirement, paying for college, or buying a new house? If so, then you may decide to invest your money to earn enough to fund your goals. Before you invest, make sure you have answers to all of these questions:
- How quickly can you get your money back? Stocks, bonds, and shares in mutual funds usually can be sold at any time, but there is no guarantee that you will get back all the money you invested. Other investments, such as limited partnerships, certificates of deposit (CDs), or IRAs, often restrict your ability to cash out your holdings.
- What can you expect to earn on your money? While bonds generally promise a fixed return, earnings on most other securities go up and down with market changes. Keep in mind, just because an investment has done well in the past, there is no guarantee it will do well in the future.
- What type of earnings can you expect? Will you get income in the form of interest, dividends, or rent? Some investments, such as stocks and real estate, have the potential for earnings and growth in value. What is the potential for earnings over time?
- How much risk is involved? With any investment, there is always the risk that you will not get your money back or the earnings promised. There is usually a trade-off between risk and reward–the higher the potential return, the greater the risk. While the U.S. government backs U.S. Treasury securities , it does not protect against loss on any other investments.
- Are your investments diversified? Some investments perform better than others in certain situations. For example, when interest rates go up, bond prices tend to go down. One industry may struggle while another prospers. Putting your money in a variety of investment options can reduce your risk.
- Are there any tax advantages to a particular investment? U.S. savings bonds are exempt from state and local taxes. Municipal bonds are exempt from federal income tax and, sometimes, state income tax as well. Tax-deferred investments for special goals, such as paying for college and retirement, are available that let you postpone or even avoid paying income taxes.
More Information on Investing
To learn more about investing, refer to these resources:
- You can find useful tips on investing at Investor.gov, a website from the Securities and Exchange Commission (SEC).
- For all of your investment-related questions, contact the SEC’s Office of Investor Education and Advocacy .
- To get help preparing to invest, consult the Financial Industry Regulatory Authority (FINRA).
Tools to Research Investments
To help you make informed decisions when investing and avoid investment fraud, there are a variety of research tools available:
- Find investor news and alerts and a guide to investment products at Investor.gov, a website from the Securities and Exchange Commission (SEC).
- The SEC requires public companies to disclose financial and other information to help you make sound decisions. You can view the text of these files on EDGAR .
- The Financial Industry Regulatory Authority (FINRA) has up-to-date market data and information for a wide range of stocks, bonds, mutual funds, and other securities.
- If you are considering investing in collectible coins, refer to the U.S. Mint . Before you purchase coins or coin-related products, research the seller with a government agency, such as your state consumer protection office or the Federal Trade Commission (FTC) .
Choose a Financial Professional
When choosing a broker or investment adviser, research the person’s education and professional history as well as the firm the person works for. Make sure you have answers to all of these questions:
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For more information on choosing a financial professional, refer to these resources:
Pyramid Schemes
A pyramid scheme, also known as Ponzi scheme, is an illegal form of multilevel marketing. In these scams, your ability to earn profits is based on the number of new participants you recruit, instead of the amount of products or services you sell. Sometimes there actually aren’t any real products that are being sold. These types of schemes are common with investment and independent direct selling opportunities .
These schemes rely on the income from new participants in order to pay fake “profits” to people that have been part of the scheme for longer amounts of time. However, the scheme falls apart when there aren’t enough new recruits to pay into the system, so the earlier participants no longer receive earnings.
Tips to Avoid Being a Victim
Take steps to protect yourself from being a victim of a pyramid scheme:
- Be wary of “opportunities” to invest your money in franchises or investments that require you to bring in more investors to increase your profit, or recoup your initial investment.
- Be wary if the company sells non-tangible products or technical services, rather than physical items.
- Independently verify the legitimacy of any franchise or investment with the Better Business Bureau, your state Attorney General , or any licensing agencies.
- Be skeptical of success stories and testimonials of fantastic earnings.
File a Complaint
If you are aware of a pyramid scheme or have been the victim of one, file a complaint with your state consumer protection office , state Attorney General , or the Better Business Bureau (BBB). If the pyramid scheme involved securities, you should also file a complaint with your state’s securities administrator , or the Securities and Exchange Commission .
Contributors and Attributions
- Authored by : USA.gov. Located at : www.usa.gov/saving-investing. License : Public Domain: No Known Copyright


2.4: Plants for Edible Landscapes
Learning Objectives
- Describe plants suitable for edible landscapes.
As part of green infrastructure, gardening for food production offers a wide range of environmental, economic, and social benefits. Growing local food within and around communities supports:
- habitat for pollinators and biodiversity,
- regulation of local climate and water management,
- reduction of energy use and carbon footprints,
- food security and local economy, and
- physical health and social connections.
Urban agriculture is the process of growing, processing, and distributing local food and food products. There are many types of urban agriculture including community gardens, boulevard planting, green roofs, vertical farms, urban chickens, bee keeping, aquaculture, and small scale faming for farmers markets. Some other forms of food production are edible landscapes, food forests, urban orchards, gleaning (public land harvest), grow a row donations and backyard sharing, and guerrilla gardening. Figure 10.1 shows an example of products of urban agriculture available for purchase at a farmers market. Read more about the benefits and different types of urban agriculture at this link to The Urban Farmer [New Tab] [1]
Communities plan and manage urban agriculture through policies, zoning bylaws, and land use regulations that allow certain public green spaces to be used for growing food. For example, community gardens for non-commercial food production that are allowed in some or all land use designations will have guidelines for safety, accessibility, maintenance, and aesthetics. Read about an example of jurisdictional policy and regulations for community gardens available at this link to City of Victoria Community Gardens Policy [PDF][New Tab]. [2]
Food production in residential landscapes is commonly associated with vegetable plots in backyards. Annual species grown for produce are usually arranged in agricultural patterns of straight lines in designated areas. Soil is often amended with compost, heavily irrigated, and seasonally tilled over for new planting. In contrast, edible landscapes, sometimes called foodscapes, incorporate plants for food as well as ornamental value within existing and new residential and public landscape designs. In general, plants for edible landscapes are herbaceous and woody perennial species that:
- are adapted for the climate and naturally resistant to pest and disease,
- require less intensive or similar levels of maintenance and inputs as the rest of the planting area, and
- provide multiple benefits such as food, aesthetics, shading, and water management.
Plants selected for preferred foods and the attributes of form, texture, and colour are integrated with other ornamental plants to achieve a desired garden style and aesthetic appearance. For example, the fruit producing tree, Morus alba ‘Pendula’ (weeping mulberry) serves as a specimen plant with distinctive form. Shrubs with berries and vibrant autumn foliage colour like Vaccinium corymbosum (highbush blueberry) may be planted as hedging. Edible spreaders like Fragaria x ananassa (garden strawberry) and Gaultheria procumbens (wintergreen) provide ground cover while vegetables with fine texture foliage like Daucus carota ssp. sativus (carrot) contrast coarse texture plants like Rheum palmatum (rhubarb). Aromatic herbs such as Origanum laevigatum ‘Herrenhausen’ and Rosmarinus officinalis (rosemary) provide structure, scent, and visual interest alongside edible flowers like Impatiens walleriana (impatiens) and Phlox paniculata (summer phlox, border phlox). Learn more information about the origins, benefits, maintenance, and types of plants for edible landscapes available at this link to Foodscaping-Wikipedia [New Tab] [3]
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Techniques that maximize space use and vegetation cover such as interplanting larger, slow growing food plants with smaller, fast growing plants reduce soil erosion and suppress opportunistic weeds. Combining plants with different heights and structure, nutrient requirements, and rooting depths creates growing microclimates and reduces plant competition for soil nutrients. Certain companion plants such as members of the Fabaceae (pea) family that fix atmospheric nitrogen in available forms in root nodules can benefit nearby nitrogen feeders like leafy vegetables. Aromatic herbs can be used to repel pests attracted to other species by smell, and the deliberate planting of host plants distract pests from other plants and attract beneficial insects and predators that feed on pests. Read more information about the benefits of companion planting available at this link to Companion planting – Wikipedia [New Tab] [4] .
Edible landscapes that are intended to provide food products for human consumption are distinguished from planted habitats that are intended to attract wildlife. As areas of natural ecosystems are converted to residential, agricultural, industrial, and other uses, the loss of habitat negatively impacts native wildlife. However, where fragments or patches of habitat are not too small and are close together, they can be connected by corridors of vegetation that allow native species to access adequate food, water, shelter and protection. Planting regional native plants that mimic the habitat characteristics of the desired wildlife species in landscapes and gardens can provide the particular needs for food, water, shelter and protection.
Creating connections between native and ornamental vegetation and water sources in urban forests, parks, gardens, boulevards, and other plantings allows wildlife to move safely among habitat patches in urban areas. For example, evergreen trees like Cryptomeria japonica (Japanese cedar) with branches close to the ground and deciduous trees with open canopies and multiple branches such as Frangula purshiana (cascara), and Prunus padus var. commutata (European bird cherry) offer shelter and protection, as well as nesting sites and food. Interplanting layers of shrubs like Ribes alpinum (alpine currant), Ribes sanguineum (flowering currant, winter currant), and Rubus spectabilis (salmonberry) with herbaceous species like Andropogon gerardii (big bluestem), Asarum caudatum (western wild ginger), and Polystichum munitum (western sword fern) provides a range of wildlife species with food, shelter, and protection. Review images of plant examples at this link to KPU Plant Database [New Tab] [5] w Tab]. Learn more about gardening for wildlife habitat available at this link to Fraser Valley Conservancy Native Plants Guide [PDF][New Tab] [6]
Habitat loss and invasive species are major threats to wildlife habitats, particularly in wetlands and forests. Selecting ornamental plants for habitat planting includes examining the potential for species to escape, establish, and overtake natural ecosystems. Non-invasive ornamentals and regional native plants are the responsible alternative to invasive plants. For example, an introduced horticultural plant that has become invasive in wetlands is Butomus umbellatus (flowering rush). Alternate choices for this plant include the native species Scirpus microcarpus (s mall-flowered bulrush), Carex spp. (sedges), and Sagittaria latifolia (wapato, arrowhead). Alternate choices for another invasive, Euphorbia esula (green spurge, leafy spurge) include species in the genera Delosperma (ice plant) and Helianthemum (rock rose). Species in the genera Salvia (sage), and Penstemon (beardtongue) provide alternate choices for the invasive species, Echium vulgare (blueweed). Another invasive species, Linaria vulgaris (toadflax) can be replaced with selections from the genera Penstemon (beardtongue), Hemerocallis (daylily), Antirrhinum (snapdragon), and Kniphofia (torch lily). Learn more about the threat of invasive horticultural plants and alternative plant choices at this link to Invasive Species Council of BC Grow Me Instead [PDF][New Tab] [7]
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10.4: Relationship Stages
Process and Stages Common to Relationships
We have many relationships throughout our lives, and every one of those relationships starts somewhere. The relationship begins and develops, and then sooner or later, it eventually ends. Take into consideration one of your closest friends. Do you remember when you first met them or how your relationship even started? Think even further as to how you two became closer?
Mark Knapp and Anita Vangelisti (2009) looked at how relationships come together, how they are maintained, and how they come apart. They divided relationships into 10 stages, to help us better identify and understand how interpersonal relationships form and dissolve. In this section, we will discuss each of the stages in greater detail. Although the model shown in Table \(\PageIndex{1}\) lists the stages in a specific order, they are not always experienced in such a linear way.
Moreover, relationships are dynamic and constantly changing. Therefore, we may not experience all of the stages. Alternately, we may experience the stages multiple times, or more than one stage may occur at any given time. In short, this model is a starting point. As you read through the various stages, ask yourself what stages you experienced in a recent relationship. Which did you skip? You might consider how culture and identity may influence the relationship stages.
| Process | Stage |
|---|---|
| Coming together |
|
| Coming apart |
|
Coming Together
Knapp and Vangelisti's (2009) model separates relationship stages into two distinct phases: coming together and coming apart. We will start by going through the first five stages, grouped under the coming together phase in the model: initiating, experimenting, intensifying, integrating, and bonding.
Initiating
When we first encounter a new person, we decide if we want to put in the energy and effort to make contact or start a conversation. This first contact happens in the initiating stage. There are different ways to initiate. Someone may approach you in a store to say “hello, what’s up?” In a different initiation example, a person could smile at you. At the initiation stage, we are primarily interested in making contact with the other person.
Experimenting
After the initiation stage, we move forward to the experimenting stage. At this stage, you are figuring out if you want to continue the relationship and want to get to know the other person. This is where you begin to start to peel back the layers of the onion to learn more about the person on a deeper, more personal level.
Let’s say two people meet on the first day of work. On the second day, during lunch, they talk about their favorite movies and notice they share some interests. In the experimenting stage, interactions tend to be casual. We ask reciprocal questions to look for common ground or similarities, to further build the relationship. Just like the name of the stage, we are experimenting and trying to figure out if we should move towards the next stage.
Intensifying
If after the experimenting stage, we decide to continue the relationship, we typically move to the intensifying stage. During this stage, we share more intimate and personal information about ourselves with the other person. For example, the co-workers who previously discussed common interests may dive deeper to talk about their childhoods, long-term goals, dreams, fears, and desires. Conversations at the intensifying stage become more serious, and the interactions are more meaningful and deep, as the layers of the onion peel away.
Integrating
The integrating stage is where two people truly become a couple and are viewed as a unit. In this stage, they let others know they are dating exclusively. Once a couple enters the integrating stage, friends are more likely to invite both people over for any dinners or events, rather than one or the other.
Bonding
The final stage of the coming together phase is called bonding , where people reveal their relationship to the world. This could be as simple as an Instagram or TikTok post declaring the relationship officially or sending a holiday card together. The bonding stage may include a formal engagement announcement and/or a wedding ceremony. In every case, the couple makes a public announcement of their relationship, to let others know their relationship is real and meaningful.
Coming Apart
The second distinct phase in Knapp and Vangelisti's (2009) model is called coming apart . This phase details the stages that occur as a relationship becomes increasingly distant. In terms of communication, there may be increased conflict or decreased use of positive words in the couple's relationship. The five relationship stages in the coming apart phase include differentiating, circumscribing, stagnating, avoiding, and terminating.
Differentiating
Differentiating is a process of disengaging or uncoupling. In other words, differentiating is the opposite of integrating. This is where we use communication to purposely divide the relationship or bond. Part of this may involve reclaiming or splitting up friendships that were once mutual. For example, Mark may say to Tarek, “My friends and I are going out for beers tonight, and we would like to have privacy for the evening once we come back to the apartment.”
Circumscribing
To circumscribe means to limit something (Cambridge Dictionary Online, 2021). When people circumscribe in a relationship, communication decreases and certain areas or subjects become restricted as individuals verbally close themselves off from each other. This could be in the form of silence or passive-aggressive behavior. For example, Tarek used to walk with Mark to work each morning, but lately Mark has been leaving the apartment without him.
Stagnating
When a relationship has stagnated , it feels like it is not moving forward any longer or has come to a halt. In some cases, verbal communication slows or may be avoided and the relationship feels strained. For example, Mark and Tarek now have limited communication around the apartment, and barely talk at work anymore.
Avoiding
In the avoidance stage, people stay out of each other’s physical space. For example, you may decide not to go to a specific social gathering when you know that other person will be there. However, when actual physical avoidance cannot take place, people will simply avoid each other while they’re together and treat the other as if they don’t exist. When avoiding, the individuals in the relationship become separate from one another physically, emotionally, and mentally.
Terminating
There are many reasons why a relationship may terminate. Termination can result from outside circumstances, such as geographic separation or internal factors, such as changing values or personalities that lead to a weakening of the bond. Termination may occur through the exchange of verbal or nonverbal messages. One example of nonverbal termination is called ghosting , where one person disappears unexpectedly and never communicates with the other again. In this final stage, Mark and Tarek both move out of the apartment and into separate homes. They stop making plans to see each other. Of course, if one person dies, the relationship, by default, also terminates.


17.5: Normal Modes of a Standing Sound Wave
- Explain the mechanism behind sound-reducing headphones
- Describe resonance in a tube closed at one end and open at the other end
- Describe resonance in a tube open at both ends
Interference is the hallmark of waves, all of which exhibit constructive and destructive interference exactly analogous to that seen for water waves. In fact, one way to prove something “is a wave” is to observe interference effects. Since sound is a wave, we expect it to exhibit interference.
Interference of Sound Waves
In waves , we discussed the interference of wave functions that differ only in a phase shift. We found that the wave function resulting from the superposition of \(y_{1}(x, t)=A \sin (k x-\omega t+\phi)\) and \(y_{2}(x, t)=A \sin (k x-\omega t)\) is
\[ y(x, t)=\left[2 A \cos \left(\frac{\phi}{2}\right)\right] \sin \left(k x-\omega t+\frac{\phi}{2}\right). \nonumber \]
One way for two identical waves that are initially in phase to become out of phase with one another is to have the waves travel different distances; that is, they have different path lengths. Sound waves provide an excellent example of a phase shift due to a path difference. As we have discussed, sound waves can basically be modeled as longitudinal waves, where the molecules of the medium oscillate around an equilibrium position, or as pressure waves.
When the waves leave the speakers, they move out as spherical waves (Figure \(\PageIndex{1}\)). The waves interfere; constructive inference is produced by the combination of two crests or two troughs, as shown. Destructive interference is produced by the combination of a trough and a crest.
The phase difference at each point is due to the different path lengths traveled by each wave. When the difference in the path lengths is an integer multiple of a wavelength,
\[ \Delta r=\left|r_{2}-r_{1}\right|=n \lambda, \text { where } n=0,1,2,3, \ldots \nonumber \]
the waves are in phase and there is constructive interference. When the difference in path lengths is an odd multiple of a half wavelength,
\[ \Delta r=\left|r_{2}-r_{1}\right|=n \frac{\lambda}{2}, \text { where } n=1,3,5, \ldots \nonumber \]
the waves are 180°(\(\pi\) rad) out of phase and the result is destructive interference. These points can be located with a sound-level intensity meter.
Two speakers are separated by 5.00 m and are being driven by a signal generator at an unknown frequency. A student with a sound-level meter walks out 6.00 m and down 2.00 m, and finds the first minimum intensity, as shown below. What is the frequency supplied by the signal generator? Assume the wave speed of sound is v = 343.00m/s.
Strategy
The wave velocity is equal to \(v = \frac{\lambda}{T} = \lambda f\). The frequency is then \(f = \frac{v}{\lambda}\) . A minimum intensity indicates destructive interference and the first such point occurs where there is path difference of \(\Delta r = \lambda / 2\), which can be found from the geometry.
Solution
1. Find the path length to the minimum point from each speaker
\[ r_{1}=\sqrt{(6.00 \: \mathrm{m})^{2}+(2.00 \: \mathrm{m})^{2}}=6.32 \: \mathrm{m}, r_{2}=\sqrt{(6.00 \: \mathrm{m})^{2}+(3.00 \: \mathrm{m})^{2}}=6.71 \: \mathrm{m} \nonumber \]
2. Use the difference in the path length to find the wavelength.
\begin{array}{c}
\Delta r=\left|r_{2}-r_{1}\right|=|6.71 \: \mathrm{m}-6.32 \: \mathrm{m}|=0.39 \: \mathrm{m} \nonumber \\
\lambda=2 \Delta r=2(0.39 \: \mathrm{m})=0.78 \: \mathrm{m} \nonumber
\end{array}
3. Find the frequency.
\[ f=\frac{v}{\lambda}=\frac{343.00 \: \mathrm{m} / \mathrm{s}}{0.78 \: \mathrm{m}}=439.74 \: \mathrm{Hz} \nonumber \]
Significance
If point P were a point of maximum intensity, then the path length would be an integer multiple of the wavelength.
If you walk around two speakers playing music, how come you do not notice places where the music is very loud or very soft, that is, where there is constructive and destructive interference?
The concept of a phase shift due to a difference in path length is very important. You will use this concept again in Interference and Photons and Matter Waves, where we discuss how Thomas Young used this method in his famous double-slit experiment to provide evidence that light has wavelike properties.
Noise Reduction Through Destructive Interference
Figure \(\PageIndex{3}\) shows a clever use of sound interference to cancel noise. Larger-scale applications of active noise reduction by destructive interference have been proposed for entire passenger compartments in commercial aircraft. To obtain destructive interference, a fast electronic analysis is performed, and a second sound is introduced 180° out of phase with the original sound, with its maxima and minima exactly reversed from the incoming noise. Sound waves in fluids are pressure waves and are consistent with Pascal’s principle; that is, pressures from two different sources add and subtract like simple numbers. Therefore, positive and negative gauge pressures add to a much smaller pressure, producing a lower-intensity sound. Although completely destructive interference is possible only under the simplest conditions, it is possible to reduce noise levels by 30 dB or more using this technique.
Describe how noise-canceling headphones differ from standard headphones used to block outside sounds.
Where else can we observe sound interference? All sound resonances, such as in musical instruments, are due to constructive and destructive interference. Only the resonant frequencies interfere constructively to form standing waves, whereas others interfere destructively and are absent.
Resonance in a Tube Closed at One End
As we discussed in Waves , standing waves are formed by two waves moving in opposite directions. When two identical sinusoidal waves move in opposite directions, the waves may be modeled as
\[ y_{1}(x, t)=A \sin (k x-\omega t) \text { and } y_{2}(x, t)=A \sin (k x+\omega t) \nonumber .\]
When these two waves interference, the resultant wave is a standing wave:
\[ y_{\mathrm{R}}(x, t)=[2 A \sin (k x)] \cos (\omega t) . \nonumber \]
Resonance can be produced due to the boundary conditions imposed on a wave. In Waves , we showed that resonance could be produced in a string under tension that had symmetrical boundary conditions, specifically, a node at each end. We defined a node as a fixed point where the string did not move. We found that the symmetrical boundary conditions resulted in some frequencies resonating and producing standing waves, while other frequencies interfere destructively. Sound waves can resonate in a hollow tube, and the frequencies of the sound waves that resonate depend on the boundary conditions.
Suppose we have a tube that is closed at one end and open at the other. If we hold a vibrating tuning fork near the open end of the tube, an incident sound wave travels through the tube and reflects off the closed end. The reflected sound has the same frequency and wavelength as the incident sound wave, but is traveling in the opposite direction. At the closed end of the tube, the molecules of air have very little freedom to oscillate, and a node arises. At the open end, the molecules are free to move, and at the right frequency, an antinode occurs. Unlike the symmetrical boundary conditions for the standing waves on the string, the boundary conditions for a tube open at one end and closed at the other end are anti-symmetrical: a node at the closed end and an antinode at the open end.
If the tuning fork has just the right frequency, the air column in the tube resonates loudly, but at most frequencies it vibrates very little. This observation just means that the air column has only certain natural frequencies. Consider the lowest frequency that will cause the tube to resonate, producing a loud sound. There will be a node at the closed end and an antinode at the open end, as shown in Figure \(\PageIndex{4}\).
The standing wave formed in the tube has an antinode at the open end and a node at the closed end. The distance from a node to an antinode is one-fourth of a wavelength, and this equals the length of the tube; thus, \(\lambda_1 = 4L\). This same resonance can be produced by a vibration introduced at or near the closed end of the tube (Figure \(\PageIndex{5}\)). It is best to consider this a natural vibration of the air column, independently of how it is induced.
Given that maximum air displacements are possible at the open end and none at the closed end, other shorter wavelengths can resonate in the tube, such as the one shown in Figure \(\PageIndex{6}\). Here the standing wave has three-fourths of its wavelength in the tube, or \(\frac{3}{4} \lambda_3 = L\), so that \(\lambda_3 = \frac{4}{3} L\). Continuing this process reveals a whole series of shorter-wavelength and higher-frequency sounds that resonate in the tube. We use specific terms for the resonances in any system. The lowest resonant frequency is called the fundamental , while all higher resonant frequencies are called overtones . The resonant frequencies that are integral multiples of the fundamental are collectively called harmonics . The fundamental is the first harmonic, the second harmonic is twice the frequency of the first harmonic, and so on. Some of these harmonics may not exist for a given scenario. Figure \(\PageIndex{7}\) shows the fundamental and the first three overtones (or the first, third, fifth, and seventh harmonics) in a tube closed at one end.
The relationship for the resonant wavelengths of a tube closed at one end is
\[\lambda_{n}=\frac{4}{n} L \quad n=1,3,5, \ldots \label{17.13} \]
Now let us look for a pattern in the resonant frequencies for a simple tube that is closed at one end. The fundamental has \(\lambda = 4L\), and frequency is related to wavelength and the speed of sound as given by
\[ v = f \lambda \nonumber .\]
Solving for \(f\) in this equation gives
\[ f=\frac{v}{\lambda}=\frac{v}{4 L}, \nonumber \]
where v is the speed of sound in air. Similarly, the first overtone has \(\lambda = 4L/3\) (see Figure \(\PageIndex{7}\)), so that
\[ f_{3}=3 \frac{v}{4 L}=3 f_{1} . \nonumber \]
Because \(f_3 = 3 f_1\), we call the first overtone the third harmonic. Continuing this process, we see a pattern that can be generalized in a single expression. The resonant frequencies of a tube closed at one end are
\[ f_{n}=n \frac{v}{4 L}, \quad n=1,3,5, \dots \label{17.14} \]
where \(f_1\) is the fundamental, \(f_3\) is the first overtone, and so on. It is interesting that the resonant frequencies depend on the speed of sound and, hence, on temperature. This dependence poses a noticeable problem for organs in old unheated cathedrals, and it is also the reason why musicians commonly bring their wind instruments to room temperature before playing them.
Resonance in a Tube Open at Both Ends
Another source of standing waves is a tube that is open at both ends. In this case, the boundary conditions are symmetrical: an antinode at each end. The resonances of tubes open at both ends can be analyzed in a very similar fashion to those for tubes closed at one end. The air columns in tubes open at both ends have maximum air displacements at both ends (Figure \(\PageIndex{8}\)). Standing waves form as shown.
The relationship for the resonant wavelengths of a tube open at both ends is
\[ \lambda_{n}=\frac{2}{n} L, \quad n=1,2,3, \ldots \label{17.15} \]
Based on the fact that a tube open at both ends has maximum air displacements at both ends, and using Figure \(\PageIndex{8}\) as a guide, we can see that the resonant frequencies of a tube open at both ends are
\[ f_{n}=n \frac{v}{2 L}, \quad n=1,2,3 \dots, \label{17.16} \]
where \(f_1\) is the fundamental, \(f_2\) is the first overtone, \(f_3\) is the second overtone, and so on. Note that a tube open at both ends has a fundamental frequency twice what it would have if closed at one end. It also has a different spectrum of overtones than a tube closed at one end.
Note that a tube open at both ends has symmetrical boundary conditions, similar to the string fixed at both ends discussed in Waves . The relationships for the wavelengths and frequencies of a stringed instrument are the same as given in Equation \ref{17.15} and Equation \ref{17.16}. The speed of the wave on the string (from Waves ) is \(v = \sqrt{\frac{F_{T}}{\mu}}\). The air around the string vibrates at the same frequency as the string, producing sound of the same frequency. The sound wave moves at the speed of sound and the wavelength can be found using \(v = \lambda f\).
How is it possible to use a standing wave’s node and antinode to determine the length of a closed-end tube
You observe two musical instruments that you cannot identify. One plays high-pitched sounds and the other plays low-pitched sounds. How could you determine which is which without hearing either of them play?


3.6: Handel’s Messiah
In the previous chapter you were introduced to the life and music of G. F. Handel and the way he and others have drawn on the Bible to provide entertainment.
In this chapter you will
- explore Handel’s most famous work, The Messiah, in detail
- learn why Handel’s most famous oratorio is not in fact an oratorio
- compare the most famous part of The Messiah with another work by Handel that you may remember from television
- learn about the impact of The Messiah on one of the performers and some of Handel’s contemporaries
Handel’s Magnum Opus: The Messiah
For many people, if they know the musical term oratorio , it is because of George Frideric Handel’s most famous work , The Messiah . Ironically, however, while Handel composed a great many works that fall into the category of “biblical oratorio,” The Messiah is not one of them, technically speaking. If an oratorio may be defined as an opera without the costumes and acting, The Messiah does not have characters in the way operas and oratorios normally would. Listen to the entire thing if you did not do so in the previous chapter. [1]
The words of the libretto by Charles Jennens are all taken from biblical texts . They are not always taken directly from the King James Version (KJV) or the Book of Common Prayer (although most of them are). Jennens seems to have drawn on Henry Hammond’s paraphrase and annotations on the books of the Bible, [2] which included criticism of the KJV.
As we said, Messiah is not, strictly speaking, an oratorio as typically defined in that era, since there is no drama. Oratorio meant something that was much the same as opera, without the acting but with different individuals nonetheless voicing different roles. Some appreciated the difference in the case of Messiah . Dr. Edward Synge, bishop of Elphin, commented that the lack of dialogue was a plus. [3]
The text on its own may not, at first glance, seem like the sort of thing that might inspire music, much less memorable, great music. Read it on its own and you will see what I mean. Any biblical text can be set to music, of course, and there are a great many different instances of texts being brought together and set to music. Is there anything special about this particular group of texts, combined in this particular way, that might account for the work’s popularity?
A textual study of the libretto by Martin Dicke notes, “His primary source was the King James Version of the Bible. For all but one of the Psalm texts, however, he used the 1662 Book of Common Prayer. Altogether, the libretto is taken from 81 Bible verses that come from 14 different books of the Bible. Of these books, Isaiah is quoted the most frequently (21 verses) followed by the Book of Psalms (15 verses) and 1 Corinthians (10 verses). It is interesting and significant to note that all of the passages from 1 Corinthians come from 1 Corinthians 15.” [4] For those not familiar with it, chapter 15 of Paul’s first letter to the church in Corinth focuses on resurrection—both the resurrection of Jesus and resurrection as the form of afterlife for all. Many of the texts incorporated into the libretto are well known among the biblically literate—they are not, for the most part, obscure.
Musically, hints of elements that are to come later are woven in very early. For instance, specific harmonic progressions and cadences in “And the glory of the Lord shall be revealed” will be fleshed out more fully in the “Hallelujah” chorus. The latter is about the revelation of the glory of the Lord in its fullest eschatological (end of time/history) sense, as the kingdoms of the world become the kingdom of God and his Christ forever more. If one listens to Handel’s coronation anthem “Zadok the Priest,” composed for the 1727 coronation of King George II of England, one hears similarities with the famous “Hallelujah Chorus.” That work is a setting of a biblical text and may be known to some today from its appearance in the television show The Crown in the scene of Elizabeth’s coronation . It is interesting to think about whether the musical and lyrical similarity between the two works suggests that the famous “Hallelujah Chorus” ought to be thought of as a coronation anthem for Jesus. [5] The scene from Revelation that is depicted (see Revelation 11:15 )—and indeed, in a sense, the book as a whole—could be interpreted in that way. Listen from around 2:43 in this recording of the work, and see if the similarity strikes you in the same way. [6]
We see from these few examples, among many others that could be provided, that Handel was theologically as well as musically astute, as he would need to be in order to find in Jennens’s libretto not a mere catena of texts but a combination that has profound depth and significance. And this helps us recognize the key elements in the reception of the Bible in Messiah. The biblical text is interpreted whenever one makes omissions or takes verses and passages from their original context and places them side by side. [7] Setting them to music then interprets them again. (And when cuts are made to Messiah , the meaning changes once more.) The combination of ancient Israelite prophecy, Gospel narrative, and New Testament prophecy that we encounter in Messiah creates new meanings and interpretations of the biblical text even apart from the music. A great example of the change of meaning when texts are removed from their original context and juxtaposed can be found in Part II of Messiah , “Lift Up Your Heads,” which sets Psalm 24 in the context of Jesus’s descent into hell and ascension into heaven so that the gates become those of hell or those of heaven, but in either case, they are no longer those of the earthly city of Jerusalem.
The music begins softly and builds from there—although the total ensemble is sparser than in other oratorios Handel wrote. The work includes musical “word painting” that anyone can appreciate—listen to the leaps on the word “exalted,” the jagged up and down on “crooked,” and so on. [8] Yet the music is also rich from the perspective of performers. Were this not so, the piece would not have become such a global phenomenon. Music that people enjoy listening to but no one relishes performing is not going to be performed and recorded as frequently as Messiah has been. And so one can analyze the work musically—noting, for instance, that in Part I of Messiah , the first of each group of three sections tends to be in a key that is subsequently revealed to be the dominant key of the next, musically depicting the sense of expectation that the words also convey.
Thus, while some take the easy route of attributing this masterpiece (as any other) to “inspiration,” it is a work that only someone with exceptional musical and theological ability could produce. [9] Handel made improvements to the music subsequent to the first performance (as well as other adjustments for performances in specific locations where the musicians and vocalists available differed). [10] Noting these points is not to remove the element of inspiration but to emphasize that great words, great music, and the greater combination of the two do not come to those who lack the training to channel the ideas. Moreover, the fact that great writers and composers did not simply have ideas come to them fully formed in a final and perfect version but created drafts that they subsequently revised and improved is crucial knowledge for would-be writers and composers in our time.
It is truly the interplay of the words and music that makes Messiah so powerful, so memorable, so meaningful, and so popular. The words on their own might or might not be deemed an interesting selection of biblical texts. The music on its own would undoubtedly be enjoyable if you like music from this era. When the two are combined, however, something emerges that is more than the sum of their parts. Try simply reading the libretto or listening to the music of Messiah without the words and you’ll see what I mean. This is not to suggest that the music on its own does not have merit, much less to suggest something of that sort about the lyrics! The point is that the combination becomes something greater—in this case, something much greater—than either is on its own.
Let me end this brief consideration of the interplay of words and music in Messiah by noting Jennens’s own reaction to the work, which Handel composed in only three weeks after receiving the libretto. In a January 1743 letter to Edward Holdsworth, Jennens wrote, “His Messiah has disappointed me, being set in great haste, tho’ he said he would be a year about it, & make it the best of all his compositions. I shall put no more sacred words into his hands to be thus abus’d.” Jennens’s impression of the piece improved once he heard it, but for years to follow, he would complain that the piece could have been better still had Handel made the improvements Jennens suggested to him.
Context of The Messiah
The historical context in which Messiah was composed was an era of increasing secularism, in which the use of sacred words for entertainment was controversial. David Greene writes, “It is a piece all of whose texts are drawn directly from the Bible, yet it is written for the concert hall, and uses a musical style that had been developed for the opera house, not the church.” [11] It would be too simple, however, to say only that Messiah exemplifies the outcome of a secularizing impulse. For it also shows another process at work, a process that is the reverse of secularization. It transforms the secular format and style, and they become the bearer of renewed and refreshed theological reflection. This process characterized Handel’s other oratorios, which—like seventeenth- and eighteenth-century oratorios in general—show the sacred attempting to reach into the profane and resanctify it, turning what had become profane into an aspect of the sacred. Indeed, the explicit strategy of those who commissioned the early Italian oratorios was to use a style that would reach out to people who were at risk of becoming indifferent to the life of the church and quicken their spirituality. Bach’s Passions are only works of art by accident, as it were, whereas Handel’s Messiah is art first and edification second. In an era that also witnessed the spread of Deism (the belief that God created the world to run according to natural laws so that divine interventions and miracles are unnecessary and do not occur), the question of how Messiah related to that is worth asking. Its outlook is orthodox, not Deistic—it emphasizes the activity of God in history and the person and work of Jesus in fairly traditional terms.
The first performance took place in Dublin as a benefit concert to raise money for two hospitals as well as prisoners in jails. Clergy objected to the work’s use of Scripture for entertainment purposes when the work was then performed in London. But interestingly enough, when it was offered for a benefit concert there, that marked a turning point in the work’s reception. And so it is important to reflect on the fact that just as words and music must be considered together, so too must both be in relation to their historical setting and their economic implications.
As is typical in Christian theology (in that time as often in our own), there is little focus on Jesus’s public activity and teaching, with just a hint of miracle (“Then shall the eyes of the blind be opened” [ Isaiah 35:5 ]). Instead, Jesus’s fulfilment of prophecy, birth, suffering, and death takes center stage, with the most focus on his suffering and death. [12]
Reception of The Messiah
By One of Its Earliest Performers
The alto Susannah Cibber had been involved in a messy divorce for which she was widely criticized and, as a result, had moved from England to Dublin in Ireland in the early 1740s. Denis Stevens writes , “In the Dublin performance, it was reputedly Mrs. Cibber’s deeply emotional and theatrical performance of ‘I know that my Redeemer liveth’ that brought Rev. Dr. Patrick Delaney to his feet with ‘Woman, for this be all thy sins forgiven!’ If she had redistributed the words, lessening the stress on ‘my’ and strengthening the key-word ‘Redeemer,’ Delaney might have said nothing at all. English may not have been Handel’s native language, but he knew how to set it with an assurance only bettered by Purcell.” [13] (Henry Purcell was one of Britain’s leading composers in the Baroque era.)
By Preachers
John Newton preached on the texts of Handel’s oratorio in the Parish Church of St. Mary Woolnoth in London from 1784 to 1785 (published in 1786). This was not because he was a fan. He said/wrote, “It is probable that those of my hearers who admire this oratorio, and are often present when it is performed, may think me harsh and singular in my opinion that of all our musical compositions this is the most improper for a public entertainment. But while it continues to be equally acceptable, whether performed in a church or in the theatre, and while the greater part of the performers and of the audience, are the same at both places, I can rate it no higher than as one of the many fashionable amusements which mark the character of this age of dissipation. I am afraid it is no better than a profanation of the name and truths of God, a crucifying the Son of God afresh. You may judge for yourselves.” [14]
Yet from our time, we can safely say that some people know these texts from the Bible solely as a result of Handel’s Messiah —and no one knows them today directly as a result of Newton’s preaching. On the initial objections to Handel’s depiction of this sacred oratorio in a secular setting, Ben Finane writes, “Messiah did not fare as well, owing to outcry over the singing of Scripture in the theatre rather than the church; this panicked Handel to the extant that, without changing its title, he advertised Messiah in vague terms as ‘A New Sacred Oratorio.’” [15] It will strike many today as odd that oratorios on sacred themes were not considered inappropriate, while actual use of Scripture was.
By Other Composers
There is a new setting of the same words by Sven-David Sandström, who, like Handel, is also not a native speaker of English. It is interesting to listen to this, if only because it shows that the combination of Jennens’s libretto with music does not automatically become what Handel’s Messiah has become. Here is an interview with the composer about the project. [16]
For Further Reading
Barrett, Wayne. “G. F. Handel’s ‘Messiah’: Drama Theologicus: A Discussion of ‘Messiah’s’ Text with Implications for Its Performance.” Choral Journal 46, no. 6 (2005): 8–14.
Bullard, Roger A. Messiah: The Gospel According to Handel’s Oratorio . Grand Rapids: Eerdmans, 1993.
Burrows, Donald. Handel, Messiah . Cambridge Music Handbooks. New York: Cambridge University Press, 1991.
Davies, Andrew. “Oratorio as Exegesis: The Use of the Book of Isaiah in Handel’s Messiah.” In Retellings—the Bible in Literature, Music, Art and Film , edited by J. Cheryl Exum, 464–84. Leiden: E. J. Brill, 2007.
Greene, David B. “Handel’s ‘Messiah’: Music, Theology, and Ritual.” Soundings 75, no. 1 (1992): 43–59.
Reno, Timothy John. “A ‘Messiah’ for Our Time: An Analysis of Sven-David Sandström’s New Oratorio.” MA thesis, University of Maryland, 2010.
Roberts, John H. “False Messiah.” Journal of the American Musicological Society 63, no. 1 (2010): 45–97.
- Martin Pearlman conducts Boston Baroque. Licensed to YouTube by UMG. ↵
- The full title is Deuterai phrontides, or, A review of the paraphrase & annotations on all the books of the New Testament with some additions & alterations . ↵
- See Donald Burrows, Handel, Messiah (New York: Cambridge University Press, 1991), 20. ↵
- There are, in fact, seventeen quotations from the Psalms if one includes New Testament quotations from that book. ↵
- Calvin R. Stapert, Handel’s Messiah: Comfort for God’s People (Grand Rapids: Eerdmans, 2010), 135. ↵
- Performed by the Academy of Ancient Music and the Choir of the Academy of Ancient Music, conducted by Richard Egarr. Recorded live at Symphony Hall in Birmingham. Shared on YouTube by the Academy of Ancient Music. ↵
- For example, you may know what Paul had in view when he wrote to the Philippians, “I can do all things through Christ who strengthens me,” but if you wear it on a T-shirt, you are probably already applying it beyond Paul’s intended scope. Even if not, some who see your T-shirt will not know the original context and are likely to misunderstand. ↵
- This is known as “madrigalism,” on which see further Stapert, Handel’s Messiah , 83. ↵
- On this, see, for example, Michael Steinberg, Choral Masterworks: A Listener’s Guide (New York: Oxford University Press, 2008), 148–50; and Stapert, Handel’s Messiah , 42. ↵
- Revisions were also made in later times for performances— for instance, by none other than Mozart ! See Steinberg, Choral Masterworks , 150–54. ↵
- David B. Greene, The Theology of Handel’s “Messiah,” Beethoven’s “Credo,” and Verdi’s “Dies Irae”: How Listening to Sung Theology Leads to the Contemplation of God (Lewiston: Edwin Mellen, 2012), 14. See also p. 18, where he writes, “The harmonically generated relations among the texts give each text a particular force and nuance. In short, musical relations supplement or replace the role of grammar in relating the words to one another.” ↵
- There has been extensive discussion in recent years about whether Messiah is anti-Jewish. It is interesting that the work can be construed both as reflecting and as challenging points of view that prevailed in its time. See Michael Marissen, Tainted Glory in Handel’s Messiah (New Haven, CT: Yale University Press, 2014); the response to an earlier version of one of Marissen’s chapters by John H. Roberts, “False Messiah,” Journal of the American Musicological Society 63, no. 1 (2010): 45–97; and Benjamin Ivry’s article “Is Handel’s ‘Messiah’ an Anti-Semitic Screed?,” Forward, April 12, 2014, https://forward.com/news/196156/is-handel-s-messiah-an-anti-semitic-screed/ . ↵
- Denis Stevens, “Messiah: An Oratorio Written by George Frideric Handel,” Baroque Music, accessed September 12, 2022, http://www.baroquemusic.org/HandelMessiah.html . ↵
- John Newton, Messiah. Fifty Expository Discourses, on the Series of Scriptural Passages, Which Form the Subject of the Celebrated Oratorio of Handel (London: Printed for the author and sold by J. Buckland and J. Johnson, 1786), https://www.loc.gov/item/10000700/ ; John Newton, “Sermon L. The Universal Chorus,” in Messiah , vol. 2, Christian Classics Ethereal Library, accessed September 30, 2022, https://ccel.org/ccel/newton/messiah2/messiah2.xxvi.html . ↵
- Ben Finane, Handel’s Messiah and His English Oratorios: A Closer Look (New York: Continuum, 2009), 22. See also pp. 38–40 on Handel’s Israel in Egypt and the reaction to it. ↵
- This interview with Soli Deo Gloria is shared on their YouTube channel. ↵


1.6: The Six Dimensions of Health
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Body size and functioning; recognizing the need for physical activity, healthy foods, and adequate sleep; avoiding unhealthy habits
Developing a sense of connection, belonging, and sustained support system; having positive relationships
Recognizing creative abilities and finding ways to expand knowledge and skills; being open-minded
Coping effectively with life and expressing emotions in an appropriate manner
Having a sense of purpose and meaning in life; establishing peace, harmony, and balance in our lives
Occupying pleasant, healthy, and safe environments that support wellbeing; positively impacting the quality of our surroundings (including protecting and preserving nature)
Learning about the Six Dimensions of Health can help a person choose how to make wellness a part of everyday life.


4.11: Piecewise-Definition Functions
Piecewise-Defined Functions are functions that are defined using different equations for different parts of the domain.
Evaluate the following piecewise-defined function for the given values of \(x\), and graph the function:
\(f(x) = \left\{\begin{array}{cc}−2x + 1 & −1 \leq x < 0 \\ x^2 + 2 &0 \leq x \leq 2\end{array} \right.\)
Solution
To graph this function, make a table of solutions:
|
Table of Solutions for \(f(x) = −2x + 1 \) Domain \(−1 \leq x < 0\) |
|
| \(x\) | \(f(x)\) |
| -1 | 3 |
| 0 | 1 (open circle here, 0 not in the domain) |
|
Table of Solutions for \(f(x) = x^2 + 2\) Domain \(0 \leq x \leq 2\) |
|
| \(x\) | \(f(x)\) |
| 0 | 2 |
| 1 | 3 |
| 2 | 6 |
Evaluate the following piecewise-defined function for the given values of \(x\), and graph the function:
\(f(x) = \left\{\begin{array}{cc} −x + 1 &x \leq −1 \\ 2 & −1 < x \leq 1 \\ −x + 3 &x > 1 \end{array}\right.\)
Solution
To graph this function, once again make a table of solutions:
|
Table of Solutions for \(f(x) = −x + 1\) Domain \(x \leq −1\) |
|
| \(x\) | \(f(x)\) |
| -3 | 4 |
| -2 | 3 |
| -1 | 2 (closed circle here, -1 is in the domain) |
|
Table of Solutions for \(f(x) = 2\) Domain \(−1 < x \leq 1\) |
|
| \(x\) | \(f(x)\) |
| -1 | 2 (open circle filled in by the previous function, -1 not in the domain) |
| 0 | 2 |
| 1 | 2 (closed circle here, 1 is in the domain) |
|
Table of Solutions for \(f(x) = −x + 3\) Domain \(x > 1\) |
|
| \(x\) | \(f(x)\) |
| 1 | 2 (open circle filled in by the previous function, 1 not in the domain) |
| 2 | 1 |
| 3 | 0 |
Evaluate the following piecewise-defined functions for the given values of x, and graph the functions:.
-
\(f(x)=\left\{\begin{array}{cc}
x & x<0\\
2 x+1 &x\geq 0
\end{array}\right.\) - \(g(x) = \left\{\begin{array}{cc} 4 − x& x < 2\\ 2x − 2 &x \geq 2 \end{array} \right.\)
- \(h(x) = \left\{\begin{array}{cc} −x − 1 & x < −1 \\ 0& −1 \leq x \leq 1 \\ x + 1 & x > 1 \end{array} \right.\)
- \(g(x) = \left\{\begin{array}{cc} 6 & −8 \leq x < −4 \\ 3 &−4 \leq x \leq 5 \end{array}\right.\)
- \(f(x) = \left\{\begin{array}{cc} −x + 1 & −1 \leq x < 1 \\ \sqrt{x − 1 } &1 \leq x \leq 5\end{array}\right.\)


2.6.E: Problems on Roots, Powers, and Irrationals (Exercises)
Let \(n \in N\) in \(E^{1} ;\) let \(p>0\) and \(a>0\) be elements of an ordered field \(F\).
Prove that
(i) if \(p^{n}>a,\) then \((\exists x \in F) p>x>0\) and \(x^{n}>a\);
(ii) if \(p^{n}<a,\) then \((\exists x \in F) x>p\) and \(x^{n}<a\).
[Hint: For (i), put
\[
x=p-d, \text { with } 0<d<p.
\]
Use the Bernoulli inequality (Problem 5 (ii) in §§5-6) to find \(d\) such that
\[
x^{n}=(p-d)^{n}>a,
\]
i.e.,
\[
\left(1-\frac{d}{p}\right)^{n}>\frac{a}{p^{n}}.
\]
Solving for \(d,\) show that this holds if
\[
0<d<\frac{p^{n}-a}{n p^{n-1}}<p . \quad \text { (Why does such a } d \text { exist? } )
\]
For \((\mathrm{ii}),\) if \(p^{n}<a,\) then
\[
\frac{1}{p^{n}}>\frac{1}{a}.
\]
Use (i) with \(a\) and \(p\) replaced by 1\(/ a\) and 1\(/ p . ]\)
Prove Theorem 1 assuming that
(i) \(a>1\);
(ii) \(0<a<1\) (the cases \(a=0\) and \(a=1\) are trivial).
\([\text { Hints: }(\text { i) Let }\)
\[
A=\left\{x \in F | x \geq 1, x^{n}>a\right\}.
\]
Show that \(A\) is bounded below (by 1 ) and \(A \neq \emptyset\) (e.g., \(a+1 \in A-\)why?).
By completeness, put \(p=\) inf \(A .\)
Then show that \(p^{n}=a\) (i.e., \(p\) is the required \(\sqrt[n]{a} )\).
Indeed, if \(p^{n}>a,\) then Problem 1 would yield an \(x \in A\) with
\[
x<p=\inf A . \text { (Contradiction!) }
\]
Similarly, use Problem 1 to exclude \(p^{n}<a\).
To prove uniqueness, use Problem 4(ii) of §§5-6.
Case (ii) reduces to (i) by considering 1\(/ a\) instead of \(a . ]\)
Prove Note 1.
[Hint: Suppose first that \(a\) is not divisible by any square of a prime, i.e.,
\[
a=p_{1} p_{2} \cdots p_{m},
\]
where the \(p_{k}\) are distinct primes. (We assume it known that each \(a \in N\) is the product of [possibly repeating] primes.) Then proceed as in the proof of Theorem 2, replacing "even" by "divisible by \(p_{k}\)."
The general case, \(a=p^{2} b,\) reduces to the previous case since \(\sqrt{a}=p \sqrt{b} . ]\)
Prove that if \(r\) is rational and \(q\) is not, then \(r \pm q\) is irrational; so also are \(r q, q / r,\) and \(r / q\) if \(r \neq 0\).
[Hint: Assume the opposite and find a contradiction.]
\(\Rightarrow 5 .\) Prove the density of irrationals in a complete field \(F :\) If \(a<b(a, b \in F)\), there is an irrational \(x \in F\) with
\[
a<x<b
\]
(hence infinitely many such irrationals \(x ) .\) See also Chapter 1, §9, Problem \(4 .\)
[Hint: By Theorem 3 of §10,
\[
(\exists r \in R) \quad a \sqrt{2}<r<b \sqrt{2}, r \neq 0 .(\mathrm{Why} ?)
\]
Put \(x=r / \sqrt{2} ;\) see Problem 4].
Prove that the rational subfield \(R\) of any ordered field is Archimedean.
[Hint: If
\[
x=\frac{k}{m} \text { and } y=\frac{p}{q} \quad(k, m, p, q \in N),
\]
then \(n x>y\) for \(n=m p+1 ]\).
Verify the formulas in \((1)\) for powers with positive rational exponents \(r, s .\)
Prove that
(i) \(a^{r+s}=a^{r} a^{s}\) and
(ii) \(a^{r-s}=a^{r} / a^{s}\) for \(r, s \in E^{1}\) and \(a \in F(a>0)\).
[Hints: For \((\mathrm{i}),\) if \(r, s>0\) and \(a>1,\) use Problem 9 in §§8-9 to get
Verify that
\[
\begin{aligned} A_{a r} A_{a s} &=\left\{a^{x} a^{y} | x, y \in R, 0<x \leq r, 0<y \leq s\right\} \\ &=\left\{a^{z} | z \in R, 0<z \leq r+s\right\}=A_{a, r+s} \end{aligned}
\]
Hence deduce that
\[
a^{r} a^{s}=\sup \left(A_{a, r+s}\right)=a^{r+s}
\]
by Definition 2.
For \((\mathrm{ii}),\) if \(r>s>0\) and \(a>1,\) then by \((\mathrm{i})\),
\[
a^{r-s} a^{s}=a^{r}
\]
so
\[
a^{r-s}=\frac{a^{r}}{a^{s}}.
\]
For the cases \(r<0\) or \(s<0,\) or \(0<a<1,\) use the above results and Definition 2\((\mathrm{ii})(\mathrm{iii}) . ]\)
From Definition 2 prove that if \(r>0\left(r \in E^{1}\right),\) then
\[
a>1 \Longleftrightarrow a^{r}>1
\]
for \(a \in F(a>0)\).
Prove for \(r, s \in E^{1}\) that
(i) \(r<s \Leftrightarrow a^{r}<a^{s}\) if \(a>1\);
(ii) \(r<s \Leftrightarrow a^{r}>a^{s}\) if \(0<a<1\).
[Hints: (i) By Problems 8 and 9,
\[
a^{s}=a^{r+(s-r)}=a^{r} a^{s-r}>a^{r}
\]
since \(a^{s-r}>1\) if \(a>1\) and \(s-r>0\).
(ii) For the case \(0<a<1,\) use Definition 2(ii).]
Prove that
\[
(a \cdot b)^{r}=a^{r} b^{r} \text { and }\left(\frac{a}{b}\right)^{r}=\frac{a^{r}}{b^{r}}
\]
for \(r \in E^{1}\) and positive \(a, b \in F\).
[Hint: Proceed as in Problem \(8 . ]\)
Given \(a, b>0\) in \(F\) and \(r \in E^{1},\) prove that
(i) \(a>b \Leftrightarrow a^{r}>b^{r}\) if \(r>0,\) and
(ii) \(a>b \Leftrightarrow a^{r}<b^{r}\) if \(r<0\).
[Hint:
\[
a>b \Longleftrightarrow \frac{a}{b}>1 \Longleftrightarrow\left(\frac{a}{b}\right)^{r}>1
\]
if \(r>0\) by Problems 9 and 11].
Prove that
\[
\left(a^{r}\right)^{s}=a^{r s}
\]
for \(r, s \in E^{1}\) and \(a \in F(a>0)\).
[Hint: First let \(r, s>0\) and \(a>1 .\) To show that
\[
\left(a^{r}\right)^{s}=a^{r s}=\sup A_{a, r s}=\sup \left\{a^{x y} | x, y \in R, 0<x y \leq r s\right\},
\]
use Problem 13 in §§8-9.
Fix any \(d>1\) and put \(b=a^{r} .\) Then
\[
\left(a^{r}\right)^{s}=b^{s}=\sup A_{b s}=\sup \left\{b^{y} | y \in R, 0<y \leq s\right\}.
\]
Fix that \(y .\) Now
\[
a^{r}=\sup A_{a r}=\sup \left\{a^{x} | x \in R, 0<x \leq r\right\};
\]
so
\[
(\exists x \in R | 0<x \leq r) \quad a^{r}<d^{\frac{1}{2 y}} a^{x} . \quad(\mathrm{Why} ?)
\]
Combining all and using the formulas in \((1)\) for rationals \(x, y,\) obtain
\[
\left(a^{r}\right)^{s}<d^{\frac{1}{2}}\left(a^{r}\right)^{y}<d^{\frac{1}{2}}\left(d^{\frac{1}{2 y}} a^{x}\right)^{y}=d a^{x y},
\]
thus proving (ii)].


1.21: Assignment- College Reflection
STEP 1 : To practice reflection, try this activity writing about your first week at college and how college is similar and/or different from what you expected. You can fill it out in the chart shown below, or just write down your thoughts in paragraphs, but complete it in three steps:
First, spend five minutes brainstorming what you expected college to be like. Think about these kinds of things:
| Things to Consider | Expectations | Reality | Observations |
| What did you expect the students to be like? | |||
| What did you expect the teachers to be like? | |||
| What did you expect the workload to be like? | |||
| What did you expect the classroom environment to be like? | |||
| What did you expect the campus climate/community to be like? | |||
| What did you expect your living environment to be like (for those living on campus)? | |||
| What did you expect your commute to be like (for those commuting to campus)? | |||
| What did you expect it to feel like to be a college student? | |||
| Anything else? |
Next, spend five minutes brainstorming what college has actually been like for you so far.
Finally, reflect on how your expectations for college differ from your actual experiences. How off base were you? How accurate were your expectations? What areas of college have met your expectations?
STEP 2 : Pick one or two rows from the chart and discuss the similarities and/or differences between your expectations for college and the reality of your experience. Think also about what equipped you to be accurate in your expectations. For example, maybe you knew what to expect because “I have an older sister who attends this institution and she told me exactly what to expect” or maybe you were stepping into the unknown—”I am the first in my family to attend college and I had no idea what to expect.”
This response doesn’t need to be formatted in any particular way—just write, think, and reflect. Your response should be at least 200 words (about two single-spaced paragraphs).
STEP 3 : Write a paragraph reflecting on this activity (at least 75 words, about one short paragraph). Was it easy to think about and discuss your expectations about college? Were you surprised in any way by your own response? What did you enjoy or dislike about this activity? Why? What lessons can you learn about yourself as a writer from participating in this activity?
Rubric
| Criteria | Proficient | Developing | Not Evident | Points |
| College expectations and reality | Response includes a freewrite chart with at least five items | Response includes a partial list of | Incomplete list | __/3 |
| Freewrite | Writes at least 200 words about the differences/similarities between expectations for and reality of college. The writing is easy to read and follow. | Response is too short, hard to follow, or off-topic. | Missing or incomplete freewrite | __/8 |
| Reflection | Includes a reflection paragraph about the writing assignment that reflects on the freewrite activity and offers some thoughts about lessons learned. At least 75 words. | Reflection is too short or does not dive into enough detail about the assignment. | Incomplete or missing reflection | __/4 |
| Total | __/15 |
Sample Response
Here’s one sample response below. Note that he starts by writing about the first item on the food list, but then goes on to write about his mom and his relationship with his dad. There is no right way to write your response —you too should feel free to let your reflective writing guide you.
My College Experience
One thing that surprised me about college was the high cost of books. I had prepared to pay for my books by working over the summer as a lifeguard. It was a hot, boring job, but I knew I needed to be responsible. My family is invested in my education, but they also expect me to pay for some of my college expenses (like books) myself, and I thought I was prepared.
Imagine my surprise when I got to the bookstore and saw that the cost of my math textbook was more than $150. I knew from friends that students used and rental books were often available and cheaper, so I sought out that option. I was even more shocked to discover that a rental of the math textbook cost $120. The savings for a rental was only $30, and if I lost or damaged the book, I would be out the full cost. Now, I know myself and I can get distracted. The chances that I lose the book aren’t so small.
In high school, books were supplied to us by the school. Now that I’m in college, I’m responsible for paying tuition and for covering the cost of textbooks. It’s a lot more than I expected. I guess I may need to do a lot more lifeguarding!
Reflection
It was pretty easy to write this response. I knew right away that I wanted to write about the textbook issue. I mean, I had heard about the cost of books, but that really surprised me.  Maybe that’s why this was pretty easy for me to write, actually. I was kind of annoyed about how much the book cost! Maybe one takeaway is that when I care about something, it’s easier to write about it.
Contributors and Attributions
- Authored by : Audrey Fisch for Lumen Learning. Provided by : Lumen Learning. License : CC BY: Attribution


5.17: Putting It Together
SUMMARY
In this section, you learned the following:
- The different types of plate boundaries
- What type of motion is associated with each boundary
- The mechanisms that cause the plate movements
- The geologic features associated with each boundary
- What evidence supported Continental Drift
- Why certain areas are prone to mountain building, earthquakes and volcanoes
- The different types of deformation caused by plate tectonics
- What hotspots are and how they can create new land
Contributors and Attributions
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Original content from Kimberly Schulte (Columbia Basin College) and supplemented by Lumen Learning . The content on this page is copyrighted under a Creative Commons Attribution 4.0 International license.