16.8: Buying and Selling at Securities Exchanges
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7. Where can investors buy and sell securities, and how are securities markets regulated?
When we think of stock markets, we are typically referring to secondary markets, which handle most of the securities trading activity. The two segments of the secondary markets are broker markets and dealer markets, as Exhibit 16.6 shows. The primary difference between broker and dealer markets is the way each executes securities trades. Securities trades can also take place in alternative market systems and on non-U.S. securities exchanges.
The securities markets both in the United States and around the world are in flux and undergoing tremendous changes. We present the basics of securities exchanges in this section and discuss the latest trends in the global securities markets later in the chapter.
Broker Markets
The broker market consists of national and regional securities exchanges that bring buyers and sellers together through brokers on a centralized trading floor. In the broker market, the buyer purchases the securities directly from the seller through the broker. Broker markets account for about 60 percent of the total dollar volume of all shares traded in the U.S. securities markets.
New York Stock Exchange
The oldest and most prestigious broker market is the New York Stock Exchange (NYSE), which has existed since 1792. Often called the Big Board, it is located on Wall Street in downtown New York City. The NYSE, which lists the shares of some 2,400 corporations, had a total market capitalization (domestic and foreign companies) of $25.8 trillion at year-end 2016. On a typical day, more than 3 billion shares of stock are traded on the NYSE. 18 It represents 90 percent of the trading volume in the U.S. broker marketplace. Major companies such as IBM, Coca-Cola, AT&T, Procter & Gamble, Ford Motor Co., and Chevron list their shares on the NYSE. Companies that list on the NYSE must meet stringent listing requirements and annual maintenance requirements, which give them creditability.
The NYSE is also popular with non-U.S. companies. More than 490 foreign companies with a global market capitalization of almost $63 trillion now list their securities on the NYSE. 19
Until recently, all NYSE transactions occurred on the vast NYSE trading floor. Each of the companies traded at the NYSE is assigned to a trading post on the floor. When an exchange member receives an order to buy or sell a particular stock, the order is transmitted to a floor broker at the company’s trading post. The floor brokers then compete with other brokers on the trading floor to get the best price for their customers.
In response to competitive pressures from electronic exchanges, the NYSE created a hybrid market that combines features of the floor auction market and automated trading. Its customers now have a choice of how they execute trades. In the trends section, we’ll discuss other changes the NYSE is making to maintain a leadership position among securities exchanges.
Another national stock exchange, the American Stock Exchange (AMEX), lists the securities of more than 700 corporations but handles only 4 percent of the annual share volume of shares traded on U.S. securities exchanges. Because the AMEX’s rules are less strict than those of the NYSE, most AMEX firms are smaller and less well known than NYSE-listed corporations. Some firms move up to the NYSE once they qualify for listing there. Other companies choose to remain on the AMEX. Companies cannot be listed on both exchanges at the same time. The AMEX has become a major market, however, for exchange-traded funds and in options trading.
Regional Exchanges
The remaining 6 percent of annual share volume takes place on several regional exchanges in the United States. These exchanges list about 100 to 500 securities of firms located in their area. Regional exchange membership rules are much less strict than for the NYSE. The top regional exchanges are the Boston, Chicago, Philadelphia, and National (formerly the Cincinnati) exchanges. An electronic network linking the NYSE and many of the regional exchanges allows brokers to make securities transactions at the best prices.
The regional exchanges, which have struggled to compete, benefited from the passage of the Securities and Exchange Commission’s (SEC’s) Regulation NMS (National Market System), which became fully effective in 2007. Regulation NMS makes price the most important factor in making securities trades, and all orders must go to the trading venue with the best price. 20
Dealer Markets
Unlike broker markets, dealer markets do not operate on centralized trading floors but instead use sophisticated telecommunications networks that link dealers throughout the United States. Buyers and sellers do not trade securities directly, as they do in broker markets. They work through securities dealers called market makers , who make markets in one or more securities and offer to buy or sell securities at stated prices. A security transaction in the dealer market has two parts: the selling investor sells his or her securities to one dealer, and the buyer purchases the securities from another dealer (or in some cases, the same dealer).
NASDAQ
The largest dealer market is the National Association of Securities Dealers Automated Quotation system , commonly referred to as NASDAQ. The first electronic-based stock market, the NASDAQ is a sophisticated telecommunications network that links dealers throughout the United States. Founded in 1971 with origins in the over-the-counter (OTC) market, today NASDAQ is a separate securities exchange that is no longer part of the OTC market. The NASDAQ lists more companies than the NYSE, but the NYSE still leads in total market capitalization. An average of 1.6 billion shares were exchanged daily in 2016 through NASDAQ, which is now the largest electronic stock market. 21 It provides up-to-date bid and ask prices on about 3,700 of the most active OTC securities. Its sophisticated electronic communication system provides faster transaction speeds than traditional floor markets and is the main reason for the popularity and growth of the OTC market.
In January 2006, the SEC approved NASDAQ’s application to operate as a national securities exchange. As a result, the NASDAQ Stock Market LLC began operating independently in August 2006. 22 The securities of many well-known companies, some of which could be listed on the organized exchanges, trade on the NASDAQ. Examples include Amazon, Apple, Costco, Comcast, JetBlue, Microsoft, Qualcomm, and Starbucks. The stocks of most commercial banks and insurance companies also trade in this market, as do most government and corporate bonds. More than 400 foreign companies also trade on the NASDAQ.
More than a decade ago, the NASDAQ changed its structure to a three-tier market:
- The NASDAQ Global Select Market, a new tier with “financial and liquidity requirements that are higher than those of any other market,” according to NASDAQ. More than 1,000 NASDAQ companies qualify for this group.
- The NASDAQ Global Market (formerly the NASDAQ National Market), which will list about 1,650 companies.
- The NASDAQ Capital Market will replace the NASDAQ Small Cap Market and list about 550 companies.
All three market tiers adhere to NASDAQ’s rigorous listing and corporate governance standards. 23
The Over-the-Counter Market
The over-the-counter (OTC) markets refer to those other than the organized exchanges described above. There are two OTC markets: the Over-the-Counter Bulletin Board (OTCBB) and the Pink Sheets . These markets generally list small companies and have no listing or maintenance standards, making them attractive to young companies looking for funding. OTC companies do not have to file with the SEC or follow the costly provisions of Sarbanes-Oxley. Investing in OTC companies is therefore highly risky and should be for experienced investors only.
Alternative Trading Systems
In addition to broker and dealer markets, alternative trading systems such as electronic communications networks (ECNs) make securities transactions. ECNs are private trading networks that allow institutional traders and some individuals to make direct transactions in what is called the fourth market . ECNs bypass brokers and dealers to automatically match electronic buy and sell orders. They are most effective for high-volume, actively traded stocks. Money managers and institutions such as pension funds and mutual funds with large amounts of money to invest like ECNs because they cost far less than other trading venues.
Global Trading and Foreign Exchanges
Improved communications and the elimination of many legal barriers are helping the securities markets go global. The number of securities listed on exchanges in more than one country is growing. Foreign securities are now traded in the United States. Likewise, foreign investors can easily buy U.S. securities.
Stock markets also exist in foreign countries: more than 60 countries operate their own securities exchanges. NASDAQ ranks second to the NYSE, followed by the London Stock Exchange (LSE) and the Tokyo Stock Exchange. Other important foreign stock exchanges include Euronext (which merged with the NYSE but operates separately) and those in Toronto, Frankfurt, Hong Kong, Zurich, Australia, Paris, and Taiwan. 24 The number of big U.S. corporations with listings on foreign exchanges is growing steadily, especially in Europe. For example, significant activity in NYSE-listed stocks also occurs on the LSE. The LSE also is getting a growing share of the world’s IPOs. Emerging markets such as India, whose economy has been growing 6 percent or more a year, continue to attract investor attention. The Sensex, the benchmark index of the Bombay Stock Exchange, increased close to 40 percent between 2013 and 2017 as foreign investors continue to pump billions into Indian stocks. 25
Why should U.S. investors pay attention to international stock markets? Because the world’s economies are increasingly interdependent, businesses must look beyond their own national borders to find materials to make their goods and markets for foreign goods and services. The same is true for investors, who may find that they can earn higher returns in international markets.
Regulation of Securities Markets
Both state and federal governments regulate the securities markets. The states were the first to pass laws aimed at preventing securities fraud. But most securities transactions occur across state lines, so federal securities laws are more effective. In addition to legislation, the industry has self-regulatory groups and measures.
Securities Legislation
Congress passed the Securities Act of 1933 in response to the 1929 stock market crash and subsequent problems during the Great Depression. It protects investors by requiring full disclosure of information about new securities issues. The issuer must file a registration statement with the SEC, which must be approved by the SEC before the security can be sold.
The Securities Exchange Act of 1934 formally gave the SEC power to regulate securities exchanges. The act was amended in 1964 to give the SEC authority over the dealer markets as well. The amendment included rules for operating the stock exchanges and granted the SEC control over all participants (exchange members, brokers, dealers) and the securities traded in these markets.
The 1934 act also banned insider trading , the use of information that is not available to the general public to make profits on securities transactions. Because of lax enforcement, however, several big insider trading scandals occurred during the late 1980s. The Insider Trading and Fraud Act of 1988 greatly increased the penalties for illegal insider trading and gave the SEC more power to investigate and prosecute claims of illegal actions. The meaning of insider was expanded beyond a company’s directors, employees, and their relatives to include anyone who gets private information about a company.
Other important legislation includes the Investment Company Act of 1940, which gives the SEC the right to regulate the practices of investment companies (such as mutual funds managed by financial institutions), and the Investment Advisers Act of 1940, which requires investment advisers to disclose information about their background. The Securities Investor Protection Corporation (SIPC) was established in 1970 to protect customers if a brokerage firm fails, by insuring each customer’s account for up to $500,000.
In response to corporate scandals that hurt thousands of investors, the SEC passed new regulations designed to restore public trust in the securities industry. It issued Regulation FD (for “fair disclosure”) in October 2000. Regulation FD requires public companies to share information with all investors at the same time, leveling the information playing field. The Sarbanes-Oxley Act of 2002 has given the SEC more power when it comes to regulating how securities are offered, sold, and marketed.
Self-Regulation
The investment community also regulates itself, developing and enforcing ethical standards to reduce the potential for abuses in the financial marketplace. The Financial Industry Regulatory Authority (FINRA) oversees the nation’s more than 3,700 brokerage firms and more 600,000 registered brokers. It develops rules and regulations, provides a dispute resolution forum, and conducts regulatory reviews of member activities for the protection and benefit of investors.
In response to “Black Monday”—October 19, 1987, when the Dow Jones Industrial Average plunged 508 points and the trading activity severely overloaded the exchange’s computers—the securities markets instituted corrective measures to prevent a repeat of the crisis. Now, under certain conditions, circuit breakers stop trading for a 15-minute cooling-off period to limit the amount the market can drop in one day. Under revised rules approved in 2012 by the SEC, market-wide circuit breakers kick in when the S&P 500 Index drops 7 percent (level 1), 13 percent (level 2), and 20 percent (level 3) from the prior day’s closing numbers. 26
ETHICS IN PRACTICE
Blowing the Whistle on Financial Fraud
As part of the 2010 Dodd-Frank legislation passed by Congress in response to the 2008 financial crisis, the Securities and Exchange Commission (SEC) established a whistleblower-rewards program to provide employees and other individuals with the opportunity to report financial securities misconduct. More than seven years after starting the Office of the Whistleblower, the SEC reports that the rewards program has recovered almost $1 billion in financial penalties from companies that have done things to damage their own reputation as well as those of employees and other stakeholders.
According to a recent SEC report, 2016 was a banner year for individuals reporting financial wrongdoings and whistleblowers being rewarded for what they discovered. In 2016 alone, more than $57 million was awarded to whistleblowers—an amount greater than the total amount of rewards issued since the program’s inception in 2011.
The whistleblower program is based on three key components: monetary awards, prohibition of employer retaliation, and protection of the whistleblower’s identity. The program requires the SEC to pay out monetary awards to eligible individuals who voluntarily provide original information about a violation of federal securities laws that has occurred, is ongoing, or is about to take place. The information supplied must lead to a successful enforcement action or monetary sanctions exceeding $1 million. No awards are paid out until the sanctions are collected from the offending firm.
A whistleblower must be an individual (not a company), and that individual does not need to be employed by a company to submit information about that specific organization. A typical award to a whistleblower is between 10 and 30 percent of the monetary sanctions the SEC and others (for example, the U.S. attorney general) are able to collect from the company in question.
Through September 2016, the whistleblower program received more than 18,000 tips, with more than 4,200 tips reported in 2016 alone. The program is not limited to U.S. citizens or residents; foreign persons living abroad may submit tips and are eligible to receive a monetary award. In fact, the SEC gave the largest monetary award to date of $30 million to a foreign national living abroad for original information relating to an ongoing fraud.
Despite criticisms from some financial institutions, the whistleblower-rewards program continues to be a success—reinforcing the point that financial fraud will not go unnoticed by the SEC, employees, and others individuals.
Critical Thinking Questions
- Despite assurances that companies involved in financial fraud are not allowed to retaliate against their accusers, would you blow the whistle on your employer? Why or why not?
- What can companies do to make sure their employees are aware of the consequences of financial securities fraud? Provide several examples.
Sources: “Office of the Whistleblower,” https://www.sec.gov , accessed November 1, 2017; Erika A. Kelton, “Four Important Dodd-Frank Whistleblower Program Developments to Watch for in 2017,” https://wp.nyu.edu , accessed November 1, 2017; Jason Zuckerman and Matt Stock, “One Billion Reasons Why the SEC Whistleblower-Reward Program Is Effective,” Forbes, http://www.forbes.com , July 18, 2017; John Maxfield, “The Dodd-Frank Act Explained,” USA Today, https://www.usatoday.com , February 3, 2017; Eduardo Singerman and Paul Hugel, “The Tremendous Impact of the Dodd-Frank Whistleblower Program in 2016,” Accounting Today, https://www.accountingtoday.com , December 28, 2016; Samuel Rubenfeld, “Dodd-Frank Rollback to Spare SEC Whistleblower Program, Experts Say,” The Wall Street Journal, www.blogs.wsj.com, November 15, 2016.
CONCEPT CHECK
- How do the broker markets differ from dealer markets, and what organizations compose each of these two markets?
- Why is the globalization of the securities markets important to U.S. investors? What are some of the other exchanges where U.S companies can list their securities?
- Briefly describe the key provisions of the main federal laws designed to protect securities investors. What is insider trading, and how can it be harmful? How does the securities industry regulate itself?


1.8: Relapse Scenario- Naela
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Relapse Scenario-Naela
Naela is a 35 year old African American woman who has been in intensive outpatient treatment (IOP) for two months. During the first month she had two relapses where she had drinks at home when feeling down which led to more throughout the evening. She has been able to attain 48 days clean since her last lapse. During this time she felt she was doing well at not drinking but she still felt miserable. However, she was able to begin establishing contact with her daughters by phone. This was going well until her daughters disclosed that their father was seeing someone and was “getting serious.” Naela began to question them about how they felt about the woman and they said they were looking forward to having her as a step-mom, that she was fun to be around and good to them. Naela had been successfully negotiating not going out with her old co-workers for drinks when they invited her by going to meetings and treatment groups instead. She told Nancy, one of her old co-workers, who called on occasion to invite her out that she was concentrating on getting healthy and had a new fitness routine. In fact, her brother did add her to his gym membership and she went a few times recently. She did not disclose that she was not drinking to Nancy or that she had a problem with alcohol.
The evening after Naela talked with her daughters she was very upset the next day. She was going to go to a meeting but couldn’t stop thinking that another woman was going to take her place in her daughter’s lives. She started to look on their Facebook pages and when she did she saw a picture of them with an adult woman biking and laughing in a park. She went to her ex-husband’s page and saw this same woman arm in arm with him. She felt like she was going to be sick and started sweating. It was Friday and she knew Nancy would most likely be going out tonight. She called her, telling herself that she needed someone to talk to and Nancy was always a good listener. In fact, Nancy was going out and said she would pick her up on the way. Naela started to get ready and her mood started to lift. She told herself that she deserved to have fun, after all they were all having fun biking and laughing. She was tired of feeling sad all the time with no energy to do anything.
Finally, Nancy arrived and they went to the bar. She started drinking and told herself she would only have a few. However, by midnight she was sloppy drunk and crying to Nancy and anyone else that would listen about her daughters and the new woman. The next morning she woke up at Nancy’s apartment. She didn’t remember how she got there. Nancy woke up and told her that she really tied one on. Naela asked if she did anything crazy.  Nancy said, “You were so upset and kept saying you were always rejected and it wasn’t fair.” Naela was holding her head in her hands and Nancy tried to console her by saying it was his loss!
Naela spent the rest of the weekend in bed telling her brother she had the flu. He didn’t suspect anything because honestly she spent a lot of time sleeping so it wasn’t out of the ordinary anyway. On Monday she returned to IOP and told the group that she had relapsed. She said, “This was a bad one.” “I don’t know how to stop it when it gets started.” She disclosed to the group that she had just wanted to feel good and was so tired of feeling so sad all the time and tired. She said,”I’m not sure I can stay sober if I feel like this all the time.”


10: Conception, Pregnancy and Childbirth Last updated Save as PDF Page ID 167215 Susan Rahman with Nathan Bowman, Dahmitra Jackson, Lushtak, Remi Newman, & Prateek Sunder California Consortium for Equitable Change in Hispanic Serving Institutions Open Educational Resources (CC ECHO) 10.1: Introduction to Conception, Pregnancy and Childbirth 10.2: Fertilization 10.3: Pre-Embryonic Period 10.4: Overview of Developmental Milestones 10.5: Embryonic Development at Specific Stages 10.6: Pregnancy from the mother's perspective 10.7: Prenatal Care 10.8: Labor and Birth 10.9: Childbirth Disparities 10.10: Postpartum Care 10.11: Risks in Pregnancy 10.12: Staying Healthy 10.13: In Vitro Fertilization and Artificial Implantation 10.14: Pregnancy and Lactation 10.15: Postpartum Depression 10.16: Conclusion 10.17: Discussion Questions 10.18: For Further Exploration 10.19: Glossary 10.20: References


15.3: 1863- The Changing Nature of the War
Wars have their own logic; they last far longer than anyone anticipates at the beginning of hostilities. As they drag on, the energy and zeal that marked the entry into warfare often wane, as losses increase and people on both sides suffer the tolls of war. The American Civil War is a case study of this characteristic of modern war.
Although Northerners and Southerners both anticipated that the battle between the Confederacy and the Union would be settled quickly, it soon became clear to all that there was no resolution in sight. The longer the war continued, the more it began to affect life in both the North and the South. Increased need for manpower, the issue of slavery, and the ongoing challenges of keeping the war effort going changed the way life on both sides as the conflict progressed.
MASS MOBILIZATION
By late 1862, the course of the war had changed to take on the characteristics oftotal war, in which armies attempt to demoralize the enemy by both striking military targets and disrupting their opponent’s ability to wage war through destruction of their resources. In this type of war, armies often make no distinction between civilian and military targets. Both the Union and Confederate forces moved toward total war, although neither side ever entirely abolished the distinction between military and civilian. Total war also requires governments to mobilize all resources, extending their reach into their citizens’ lives as never before. Another reality of war that became apparent in 1862 and beyond was the influence of combat on the size and scope of government. Both the Confederacy and the Union governments had to continue to grow in order to manage the logistics of recruiting men and maintaining, feeding, and equipping an army.
Confederate Mobilization
The Confederate government in Richmond, Virginia, exercised sweeping powers to ensure victory, in stark contradiction to the states’ rights sentiments held by many Southern leaders. The initial emotional outburst of enthusiasm for war in the Confederacy waned, and the Confederate government instituted a military draft in April 1862. Under the terms of the draft, all men between the ages of eighteen and thirty-five would serve three years. The draft had a different effect on men of different socioeconomic classes. One loophole permitted men to hire substitutes instead of serving in the Confederate army. This provision favored the wealthy over the poor, and led to much resentment and resistance. Exercising its power over the states, the Confederate Congress denied state efforts to circumvent the draft.
In order to fund the war, the Confederate government also took over the South’s economy. The government ran Southern industry and built substantial transportation and industrial infrastructure to make the weapons of war. Over the objections of slaveholders, it impressed slaves, seizing these workers from their owners and forcing them to work on fortifications and rail lines. Concerned about the resistance to and unhappiness with the government measures, in 1862, the Confederate Congress gave President Davis the power to suspend the writ of habeas corpus, the right of those arrested to be brought before a judge or court to determine whether there is cause to hold the prisoner. With a stated goal of bolstering national security in the fledgling republic, this change meant that the Confederacy could arrest and detain indefinitely any suspected enemy without giving a reason. This growth of the Confederate central government stood as a glaring contradiction to the earlier states’ rights argument of pro-Confederate advocates.
The war efforts were costing the new nation dearly. Nevertheless, the Confederate Congress heeded the pleas of wealthy plantation owners and refused to place a tax on slaves or cotton, despite the Confederacy’s desperate need for the revenue that such a tax would have raised. Instead, the Confederacy drafted a taxation plan that kept the Southern elite happy but in no way met the needs of the war. The government also resorted to printing immense amounts of paper money, which quickly led to runaway inflation. Food prices soared, and poor, white Southerners faced starvation. In April 1863, thousands of hungry people rioted in Richmond, Virginia (Figure 15.3.1). Many of the rioters were mothers who could not feed their children. The riot ended when President Davis threatened to have Confederate forces open fire on the crowds.
One of the reasons that the Confederacy was so economically devastated was its ill-advised gamble that cotton sales would continue during the war. The government had high hopes that Great Britain and France, which both used cotton as the raw material in their textile mills, would ensure the South’s economic strength—and therefore victory in the war—by continuing to buy. Furthermore, the Confederate government hoped that Great Britain and France would make loans to their new nation in order to ensure the continued flow of raw materials. These hopes were never realized. Great Britain in particular did not wish to risk war with the United States, which would have meant the invasion of Canada. The United States was also a major source of grain for Britain and an important purchaser of British goods. Furthermore, the blockade made Southern trade with Europe difficult. Instead, Great Britain, the major consumer of American cotton, found alternate sources in India and Egypt, leaving the South without the income or alliance it had anticipated.
Dissent within the Confederacy also affected the South’s ability to fight the war. Confederate politicians disagreed over the amount of power that the central government should be allowed to exercise. Many states’ rights advocates, who favored a weak central government and supported the sovereignty of individual states, resented President Davis’s efforts to conscript troops, impose taxation to pay for the war, and requisition necessary resources. Governors in the Confederate states often proved reluctant to provide supplies or troops for the use of the Confederate government. Even Jefferson Davis’s vice president Alexander Stephens opposed conscription, the seizure of slave property to work for the Confederacy, and suspension of habeas corpus. Class divisions also divided Confederates. Poor whites resented the ability of wealthy slaveholders to excuse themselves from military service. Racial tensions plagued the South as well. On those occasions when free blacks volunteered to serve in the Confederate army, they were turned away, and enslaved African Americans were regarded with fear and suspicion, as whites whispered among themselves about the possibility of slave insurrections.
Union Mobilization
Mobilization for war proved to be easier in the North than it was in the South. During the war, the federal government in Washington, DC, like its Southern counterpart, undertook a wide range of efforts to ensure its victory over the Confederacy. To fund the war effort and finance the expansion of Union infrastructure, Republicans in Congress drastically expanded government activism, impacting citizens’ everyday lives through measures such as new types of taxation. The government also contracted with major suppliers of food, weapons, and other needed materials. Virtually every sector of the Northern economy became linked to the war effort.
In keeping with their longstanding objective of keeping slavery out of the newly settled western territories, the Republicans in Congress (the dominant party) passed several measures in 1862. First, the Homestead Act provided generous inducements for Northerners to relocate and farm in the West. Settlers could lay claim to 160 acres of federal land by residing on the property for five years and improving it. The act not only motivated free-labor farmers to move west, but it also aimed to increase agricultural output for the war effort. The federal government also turned its attention to creating a transcontinental railroad to facilitate the movement of people and goods across the country. Congress chartered two companies, the Union Pacific and the Central Pacific, and provided generous funds for these two businesses to connect the country by rail.
The Republican emphasis on free labor, rather than slave labor, also influenced the 1862 Land Grant College Act, commonly known as the Morrill Act after its author, Vermont Republican senator Justin Smith Morrill. The measure provided for the creation of agricultural colleges, funded through federal grants, to teach the latest agricultural techniques. Each state in the Union would be granted thirty thousand acres of federal land for the use of these institutions of higher education.
Congress paid for the war using several strategies. They levied a tax on the income of the wealthy, as well as a tax on all inheritances. They also put high tariffs in place. Finally, they passed two National Bank Acts, one in 1863 and one in 1864, calling on the U.S. Treasury to issue war bonds and on Union banks to buy the bonds. A Union campaign to convince individuals to buy the bonds helped increase sales. The Republicans also passed the Legal Tender Act of 1862, calling for paper money—known as greenbacks—to be printed Figure 15.3.2). Some $150 million worth of greenbacks became legal tender, and the Northern economy boomed, although high inflation also resulted.
Like the Confederacy, the Union turned to conscription to provide the troops needed for the war. In March 1863, Congress passed the Enrollment Act, requiring all unmarried men between the ages of twenty and twenty-five, and all married men between the ages of thirty-five and forty-five—including immigrants who had filed for citizenship—to register with the Union to fight in the Civil War. All who registered were subject to military service, and draftees were selected by a lottery system (Figure 15.3.3). As in the South, a loophole in the law allowed individuals to hire substitutes if they could afford it. Others could avoid enlistment by paying $300 to the federal government. In keeping with the Supreme Court decision in Dred Scott v. Sandford , African Americans were not citizens and were therefore exempt from the draft.
Like the Confederacy, the Union also took the step of suspending habeas corpus rights, so those suspected of pro-Confederate sympathies could be arrested and held without being given the reason. Lincoln had selectively suspended the writ of habeas corpus in the slave state of Maryland, home to many Confederate sympathizers, in 1861 and 1862, in an effort to ensure that the Union capital would be safe. In March 1863, he signed into law the Habeas Corpus Suspension Act, giving him the power to detain suspected Confederate operatives throughout the Union. The Lincoln administration also closed down three hundred newspapers as a national security measure during the war.
In both the North and the South, the Civil War dramatically increased the power of the belligerent governments. Breaking all past precedents in American history, both the Confederacy and the Union employed the power of their central governments to mobilize resources and citizens.
Women’s Mobilization
As men on both sides mobilized for the war, so did women. In both the North and the South, women were forced to take over farms and businesses abandoned by their husbands as they left for war. Women organized themselves into ladies’ aid societies to sew uniforms, knit socks, and raise money to purchase necessities for the troops. In the South, women took wounded soldiers into their homes to nurse. In the North, women volunteered for the United States Sanitary Commission, which formed in June 1861. They inspected military camps with the goal of improving cleanliness and reducing the number of soldiers who died from disease, the most common cause of death in the war. They also raised money to buy medical supplies and helped with the injured. Other women found jobs in the Union army as cooks and laundresses. Thousands volunteered to care for the sick and wounded in response to a call by reformer Dorothea Dix, who was placed in charge of the Union army’s nurses. According to rumor, Dix sought respectable women over the age of thirty who were “plain almost to repulsion in dress” and thus could be trusted not to form romantic liaisons with soldiers. Women on both sides also acted as spies and, disguised as men, engaged in combat.
EMANCIPATION
Early in the war, President Lincoln approached the issue of slavery cautiously. While he disapproved of slavery personally, he did not believe that he had the authority to abolish it. Furthermore, he feared that making the abolition of slavery an objective of the war would cause the border slave states to join the Confederacy. His one objective in 1861 and 1862 was to restore the Union.
President Lincoln wrote the following letter to newspaper editor Horace Greeley on August 22, 1862. In it, Lincoln states his position on slavery, which is notable for being a middle-of-the-road stance. Lincoln’s later public speeches on the issue take the more strident antislavery tone for which he is remembered.
I would save the Union. I would save it the shortest way under the Constitution. The sooner the national authority can be restored the nearer the Union will be “the Union as it was.” If there be those who would not save the Union unless they could at the same time save Slavery, I do not agree with them. If there be those who would not save the Union unless they could at the same time destroy Slavery, I do not agree with them. My paramount object in this struggle is to save the Union, and is not either to save or destroy Slavery. If I could save the Union without freeing any slave, I would do it, and if I could save it by freeing all the slaves, I would do it, and if I could save it by freeing some and leaving others alone, I would also do that. What I do about Slavery and the colored race, I do because I believe it helps to save this Union, and what I forbear, I forbear because I do not believe it would help to save the Union. I shall do less whenever I shall believe what I am doing hurts the cause, and I shall do more whenever I shall believe doing more will help the cause. I shall try to correct errors when shown to be errors; and I shall adopt new views so fast as they shall appear to be true views. I have here stated my purpose according to my view of official duty, and I intend no modification of my oft-expressed personal wish that all men, everywhere, could be free. Yours, A. LINCOLN.
How would you characterize Lincoln’s public position in August 1862? What was he prepared to do for slaves, and under what conditions?
Since the beginning of the war, thousands of slaves had fled to the safety of Union lines. In May 1861, Union general Benjamin Butler and others labeled these refugees from slavery contrabands. Butler reasoned that since Southern states had left the United States, he was not obliged to follow federal fugitive slave laws. Slaves who made it through the Union lines were shielded by the U.S. military and not returned to slavery. The intent was not only to assist slaves but also to deprive the South of a valuable source of manpower.
Congress began to define the status of these ex-slaves in 1861 and 1862. In August 1861, legislators approved the Confiscation Act of 1861, empowering the Union to seize property, including slaves, used by the Confederacy. The Republican-dominated Congress took additional steps, abolishing slavery in Washington, DC, in April 1862. Congress passed a second Confiscation Act in July 1862, which extended freedom to runaway slaves and those captured by Union armies. In that month, Congress also addressed the issue of slavery in the West, banning the practice in the territories. This federal law made the 1846 Wilmot Proviso and the dreams of the Free-Soil Party a reality. However, even as the Union government took steps to aid individual slaves and to limit the practice of slavery, it passed no measure to address the institution of slavery as a whole.
Lincoln moved slowly and cautiously on the issue of abolition. His primary concern was the cohesion of the Union and the bringing of the Southern states back into the fold. However, as the war dragged on and many thousands of contrabands made their way north, Republicans in Congress continued to call for the end of slavery. Throughout his political career, Lincoln’s plans for former slaves had been to send them to Liberia. As late as August 1862, he had hoped to interest African Americans in building a colony for former slaves in Central America, an idea that found favor neither with black leaders nor with abolitionists, and thus was abandoned by Lincoln. Responding to Congressional demands for an end to slavery, Lincoln presented an ultimatum to the Confederates on September 22, 1862, shortly after the Confederate retreat at Antietam. He gave the Confederate states until January 1, 1863, to rejoin the Union. If they did, slavery would continue in the slave states. If they refused to rejoin, however, the war would continue and all slaves would be freed at its conclusion. The Confederacy took no action. It had committed itself to maintaining its independence and had no interest in the president’s ultimatum.
On January 1, 1863, Lincoln made good on his promise and signed theEmancipation Proclamation. It stated “That on the first day of January, in the year of our Lord one thousand eight hundred and sixty-three, all persons held as slaves within any State or designated part of a State, the people whereof shall then be in rebellion against the United States, shall be then, thenceforward, and forever free.” The proclamation did not immediately free the slaves in the Confederate states. Although they were in rebellion against the United States, the lack of the Union army’s presence in such areas meant that the president’s directive could not be enforced. The proclamation also did not free slaves in the border states, because these states were not, by definition, in rebellion. Lincoln relied on his powers as commander-in-chief in issuing the Emancipation Proclamation. He knew the proclamation could be easily challenged in court, but by excluding the territories still outside his control, slaveholders and slave governments could not sue him. Moreover, slave states in the Union, such as Kentucky, could not sue because the proclamation did not apply to them. Slaveholders in Kentucky knew full well that if the institution were abolished throughout the South, it would not survive in a handful of border territories. Despite the limits of the proclamation, Lincoln dramatically shifted the objective of the war increasingly toward ending slavery. The Emancipation Proclamation became a monumental step forward on the road to changing the character of the United States.
Read through the full text of the Emancipation Proclamation at the National Archives website.
The proclamation generated quick and dramatic reactions. The news created euphoria among slaves, as it signaled the eventual end of their bondage. Predictably, Confederate leaders raged against the proclamation, reinforcing their commitment to fight to maintain slavery, the foundation of the Confederacy. In the North, opinions split widely on the issue. Abolitionists praised Lincoln’s actions, which they saw as the fulfillment of their long campaign to strike down an immoral institution. But other Northerners, especially Irish, working-class, urban dwellers loyal to the Democratic Party and others with racist beliefs, hated the new goal of emancipation and found the idea of freed slaves repugnant. At its core, much of this racism had an economic foundation: Many Northerners feared competing with emancipated slaves for scarce jobs.
In New York City, the Emancipation Proclamation, combined with unhappiness over the Union draft, which began in March 1863, fanned the flames of white racism. Many New Yorkers supported the Confederacy for business reasons, and, in 1861, the city’s mayor actually suggested that New York City leave the Union. On July 13, 1863, two days after the first draft lottery took place, this racial hatred erupted into violence. A volunteer fire company whose commander had been drafted initiated a riot, and the violence spread quickly across the city. The rioters chose targets associated either with the Union army or with African Americans. An armory was destroyed, as was a Brooks Brothers’ store, which supplied uniforms to the army.  On the fourth day of the riots, federal troops dispatched by Lincoln arrived in the city and ended the violence. Millions of dollars in property had been destroyed. More than one hundred people died, approximately one thousand were left injured, and about one-fifth of the city’s African American population fled New York in fear.
UNION ADVANCES
The war in the west continued in favor of the North in 1863. At the start of the year, Union forces controlled much of the Mississippi River. In the spring and summer of 1862, they had captured New Orleans—the most important port in the Confederacy, through which cotton harvested from all the Southern states was exported—and Memphis. Grant had then attempted to capture Vicksburg, Mississippi, a commercial center on the bluffs above the Mississippi River. Once Vicksburg fell, the Union would have won complete control over the river. A military bombardment that summer failed to force a Confederate surrender. An assault by land forces also failed in December 1862.
In April 1863, the Union began a final attempt to capture Vicksburg. On July 3, after more than a month of a Union siege, during which Vicksburg’s residents hid in caves to protect themselves from the bombardment and ate their pets to stay alive, Grant finally achieved his objective. The trapped Confederate forces surrendered. The Union had succeeded in capturing Vicksburg and splitting the Confederacy (Figure 15.3.5). This victory inflicted a serious blow to the Southern war effort.
As Grant and his forces pounded Vicksburg, Confederate strategists, at the urging of General Lee, who had defeated a larger Union army at Chancellorsville, Virginia, in May 1863, decided on a bold plan to invade the North. Leaders hoped this invasion would force the Union to send troops engaged in the Vicksburg campaign east, thus weakening their power over the Mississippi. Further, they hoped the aggressive action of pushing north would weaken the Union’s resolve to fight. Lee also hoped that a significant Confederate victory in the North would convince Great Britain and France to extend support to Jefferson Davis’s government and encourage the North to negotiate peace.
Beginning in June 1863, General Lee began to move the Army of Northern Virginia north through Maryland. The Union army—the Army of the Potomac—traveled east to end up alongside the Confederate forces. The two armies met at Gettysburg, Pennsylvania, where Confederate forces had gone to secure supplies. The resulting battle lasted three days, July 1–3 (Figure 15.3.6) and remains the biggest and costliest battle ever fought in North America. The climax of the Battle of Gettysburg occurred on the third day. In the morning, after a fight lasting several hours, Union forces fought back a Confederate attack on Culp’s Hill, one of the Union’s defensive positions. To regain a perceived advantage and secure victory, Lee ordered a frontal assault, known as Pickett’s Charge (for Confederate general George Pickett), against the center of the Union lines on Cemetery Ridge. Approximately fifteen thousand Confederate soldiers took part, and more than half lost their lives, as they advanced nearly a mile across an open field to attack the entrenched Union forces. In all, more than a third of the Army of Northern Virginia had been lost, and on the evening of July 4, Lee and his men slipped away in the rain. General George Meade did not pursue them. Both sides suffered staggering losses. Total casualties numbered around twenty-three thousand for the Union and some twenty-eight thousand among the Confederates. With its defeats at Gettysburg and Vicksburg, both on the same day, the Confederacy lost its momentum. The tide had turned in favor of the Union in both the east and the west.
Following the Battle of Gettysburg, the bodies of those who had fallen were hastily buried. Attorney David Wills, a resident of Gettysburg, campaigned for the creation of a national cemetery on the site of the battlefield, and the governor of Pennsylvania tasked him with creating it. President Lincoln was invited to attend the cemetery’s dedication. After the featured orator had delivered a two-hour speech, Lincoln addressed the crowd for several minutes. In his speech, known as the Gettysburg Address, which he had finished writing while a guest in David Wills’ home the day before the dedication, Lincoln invoked the Founding Fathers and the spirit of the American Revolution. The Union soldiers who had died at Gettysburg, he proclaimed, had died not only to preserve the Union, but also to guarantee freedom and equality for all.
Several months after the battle at Gettysburg, Lincoln traveled to Pennsylvania and, speaking to an audience at the dedication of the new Soldiers’ National Ceremony near the site of the battle, he delivered his now-famous Gettysburg Address to commemorate the turning point of the war and the soldiers whose sacrifices had made it possible. The two-minute speech was politely received at the time, although press reactions split along party lines. Upon receiving a letter of congratulations from Massachusetts politician and orator William Everett, whose speech at the ceremony had lasted for two hours, Lincoln said he was glad to know that his brief address, now virtually immortal, was not “a total failure.”
Four score and seven years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal.
Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battle-field of that war. We have come to dedicate a portion of that field, as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this.
It is for us the living . . . to be here dedicated to the great task remaining before us—that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion—that we here highly resolve that these dead shall not have died in vain—that this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth.
—Abraham Lincoln, Gettysburg Address, November 19, 1863
What did Lincoln mean by “a new birth of freedom”? What did he mean when he said “a government of the people, by the people, for the people, shall not perish from the earth”?
Acclaimed filmmaker Ken Burns has created a documentary about a small boys’ school in Vermont where students memorize the Gettysburg Address. It explores the value the address has in these boys’ lives, and why the words still matter.
Section Summary
The year 1863 proved decisive in the Civil War for two major reasons. First, the Union transformed the purpose of the struggle from restoring the Union to ending slavery. While Lincoln’s Emancipation Proclamation actually succeeded in freeing few slaves, it made freedom for African Americans a cause of the Union. Second, the tide increasingly turned against the Confederacy. The success of the Vicksburg Campaign had given the Union control of the Mississippi River, and Lee’s defeat at Gettysburg had ended the attempted Confederate invasion of the North.
Review Questions
Which of the following did the North not do to mobilize for war?
- institute a military draft
- form a military alliance with Great Britain
- print paper money
- pass the Homestead Act
B
Why is 1863 considered a turning point in the Civil War?
At the beginning of 1863, Abraham Lincoln issued the Emancipation Proclamation, which freed all slaves in areas under rebellion. This changed the war from one in which the North fought to preserve the Union to one in which it fought to free enslaved African Americans. On the battlefield, Union forces led by Grant captured Vicksburg, Mississippi, splitting the Confederacy in two and depriving it of a major avenue of transportation. In the east, General Meade stopped a Confederate invasion of the North at Gettysburg, Pennsylvania.
Glossary
- contrabands
- slaves who escaped to the Union army’s lines
- Emancipation Proclamation
- signed on January 1, 1863, the document with which President Lincoln transformed the Civil War into a struggle to end slavery
- Gettysburg Address
- a speech by Abraham Lincoln dedicating the military cemetery at Gettysburg on November 19, 1863
- greenbacks
- paper money the United States began to issue during the Civil War
- habeas corpus
- the right of those arrested to be brought before a judge or court to determine whether there is cause to hold the prisoner
- total war
- a state of war in which the government makes no distinction between military and civilian targets, and mobilizes all resources, extending its reach into all areas of citizens’ lives


11.6: Assignment- Global Organizations- The IMF and World Bank
Global Organizations: The IMF & World Bank
The assignment for this Module has you look at a project that has been or is currently being undertaken by either the World Bank or the IMF. ( LINKS to BOTH websites are located earlier in the module. )
Written Assignment:
-
Go to
either
the IMF or World Bank website and look at what they are doing. You will select one initiative from ONE of the two organizations as the subject of your two-page written paper this week.
- At the World Bank site: World Bank , click on the “Projects and Operations” tab from the Home Page. You may then select a project based on Sector, Theme or Country.
- At the IMF site: IMF , go to the Home Page and then select “Our Work” which will give you access to information about initiatives being undertaken by the IMF.
- Select ONE project or initiative of interest from either the IMF or World Bank.
-
Your written assignment should include the following information:
- A brief overview of the project or initiative. This should be an overview of the type of project, the partners involved in the project and the goals/outcomes of the project.
- How does this project/initiative reflect the role of the organization in the global business environment?
- What is the long term benefit of the project/initiative to the global business environment?
- Be certain to include a source (link/URL) for all of the sources that you used to prepare your assignment.
- Your assignment should follow the Written Assignment Guidelines posted in the course. It will be graded using the Written Assignment Grading Rubric that is also posted.
CC licensed content, Original
- Assignment: Global Organizations: The IMF & World Bank. Authored by : Linda Williams. Provided by : Tidewater Community College. Located at : http://www.tcc.edu . Project : Z Degree Project. License : CC BY: Attribution


1.1: Writing in an Age of Technology
In an age where technology, finance, and business reigns, you might ask about the relevance of writing and learning how to write essays. Knowing how to write is an essential skill that is as relevant today as ever! A recent survey of employers, conducted by the Association of American Colleges and Universities, found that 89 percent of employers believe that colleges and universities should place more emphasis on teaching students to "effectively communicate orally and in writing” (Seligo). Writing was the only 'most-favored' skill in this survey. In addition, several of the other valued skills mentioned by employers are grounded in written communication:
- “Critical thinking and analytical reasoning skills” (81 percent of employers)
- “The ability to analyze and solve complex problems” (75 percent of employers)
- “The ability to locate, organize, and evaluate information from multiple sources” (68 percent of employers).
If you want to be professional, you need to interact frequently with others. You have to be someone who can anticipate and solve complex problems and coordinate your work with others . Whether you're in business, an entrepreneur, or in a corporation, all of these skills depend on effective communication.
Taking these ideas a step further, CEO of Amazon, Jeff Bezos, has done away with PowerPoint presentations altogether and has reverted back to writing longer essay-style memos. In a famous email sent out to his employees on June 9, 2004, he explains why: “The reason writing a 4-page memo is harder than “writing” a 20-page PowerPoint is because the narrative structure of a good memo forces better thought and better understanding of what’s more important than what, and how things are related.” At a typical Amazon meeting, employees present a long narrative style memo detailing their ideas. During the first part of the meeting everyone is silent, reading the memo and taking notes, before discussing the memo. The act of writing and annotating, whether it is in business or technology, forces the writer to think through ideas fully.
The process that Amazon employees use to write memos and sharpen their ideas is not much different than that used by students in first year composition classes. In college writing classes, students study specific topics in depth, research ideas, and then, in a process that involves planning, reading, writing, researching, and revising, present their ideas in a well-organized format. This process is used in writing an essay. Most of the writing that we will be doing in a first year Composition class is expository, detailing and explaining a theory. Students will learn how to make a statement and convince their readers about its validity using logic and persuasive writing.
All these skills are what companies are looking for in their future employees, whether these employees are working in technology, finance, the arts, or business. Jeffrey Seligo, author of College Unbound , writes about the dearth of college graduates with these skills. He writes, "Hiring managers complain that they often find today's college graduates lacking in interpersonal skills, problem solving, effective written and oral communication skills, the ability to work together in teams, and critical and analytical thinking" and that employees who have the ability to "come up with novel solutions to problems and [are] better able to sort through information to filter out the most critical pieces are the ones who will be of great value." Writing expository essays in college will help build critical thinking and inter-personal skills as students work in groups to debate their ideas and fine-tune their essays based on helpful feedback from their instructor and, often, peers.
By taking first year college composition, you are on the way to becoming a better thinker and writer -- skills that can make you highly desirable, both as an employee but also for your own self-development as the benefits of learning how to write go beyond getting a job. Writing effectively helps to expand your world by making you think critically about issues in the news. And it enables you to make more informed decisions as a member of a family or a community.
Now that we have established the need for learning how to write effectively, a process that goes hand-in-hand with reading and critical thinking, let's get started with making sure we have all we need to succeed in the course.
Contributors and Attributions
- Revision and adaptation of the page " Why Write " in College Writing . Authored by : Susan Oaks. Provided by : Empire State College, SUNY OER Services. License : CC BY-NC-SA .
NON CC LICENSED CONTENT:
- Quotes taken from Jeffrey J.Seligo's College Unbound , Amazon Publishing, Las Vegas, 2013.
This page most recently updated on May 31, 2020.


3.6: Exercises
EXERCISE 3.1. Prove de Morgan’s laws, (3.3) and (3.4). (Hint: There are four possible assignments of truth values 0 and 1 to the two statements \(P\) and \(Q\) . For each such assignment, evaluate the truth values of the left-hand and right-hand sides of (3.3) and show they are always the same.)
EXERCISE 3.2. Prove that compound statements \(P\) and \(Q\) are propositionally equivalent iff \(P \Longleftrightarrow Q\) .
EXERCISE 3.3. Give an example of a true conditional statement in which the consequence is false. ExERCISE 3.4. If
\(P, Q\)
and
\(R\)
are statements, prove that the following are true:
a)
\(P \wedge \neg P \Rightarrow Q\)
b)
\([(P \Rightarrow Q) \wedge(Q \Rightarrow R)] \Rightarrow(P \Rightarrow R)\)
c)
\([P \Rightarrow(Q \wedge \neg Q)] \Rightarrow \neg P\)
d)
\([P \wedge(P \Rightarrow Q)] \Rightarrow Q\)
e)
\(P \Rightarrow(Q \vee \neg Q)\)
.
EXERCISE 3.5. Let
\(P\)
and
\(Q\)
be statements. Prove that there are statements using only
\(P, Q, \neg\)
and
\(\wedge\)
that are propositionally equivalent to
a)
\(P \wedge Q\)
b)
\(P \vee Q\)
c)
\(P \Rightarrow Q\)
.
Prove that there are statements using only \(P, Q, \neg\) and \(\vee\) that are equivalent to the above.
EXERCISE 3.6. Prove the distributive laws for propositional logic: If
\(P, Q\)
and
\(R\)
are statements, then
a)
\(P \vee(Q \wedge R) \equiv(P \vee Q) \wedge(P \vee R)\)
b)
\(P \wedge(Q \vee R) \equiv(P \wedge Q) \vee(P \wedge R)\)
.
EXERCISE 3.7. Prove the distributive law for sets: If
\(X, Y\)
and
\(Z\)
are sets, then
a)
\(X \cup(Y \cap Z)=(X \cup Y) \cap(X \cup Z)\)
b)
\(X \cap(Y \cup Z)=(X \cap Y) \cup(X \cap Z)\)
.
EXERCISE 3.8. Let sets \(X, Y\) and \(Z\) be characteristic sets of formulas \(P(x), Q(x)\) and \(R(x)\) respectively. For each possible region of the Venn diagram of \(X, Y\) and \(Z\) give a compound formula (with atomic formulas \(P, Q\) and \(R\) ) that has that region as its characteristic set.
EXERCISE 3.9. Write a formula in one variable that defines the even integers.
EXERCISE 3.10. Write a formula that defines perfect squares. EXERCISE 3.11. Write a formula in two variables that defines the points in \(\mathbb{R}^{2}\) that have distance 1 from the point \((\pi, e)\) .
EXERCISE 3.12. Can you write a formula in one variable using only addition, multiplication, exponentiation, integers and equality, to define the set of all roots of a given polynomial with integer coefficients? How about the set of roots of all polynomials with integer coefficients?
EXERCISE 3.13. Which of the following statements are true?
a)
\((\forall x \in \mathbb{R}) x+1>x\)
b)
\((\forall x \in \mathbb{Z}) x^{2}>x\)
c)
\((\exists x \in \mathbb{Z})(\forall y \in \mathbb{Z}) x \leq y\)
d)
\((\forall y \in \mathbb{Z})(\exists x \in \mathbb{Z}) x \leq y\)
e)
\((\forall \varepsilon>0)(\exists \delta>0)(\forall x \in \mathbb{R})[0<|x-1|<\delta] \Rightarrow\left[\left|x^{2}-1\right|<\varepsilon\right]\)
.
EXERCISE 3.14. What is the negation of each statement in Exercise 3.13? Which of the negations are true?
EXERCISE 3.15. Let \(a, L \in \mathbb{R}\) and \(f\) be a real function. Prove that the statements \[(\forall \varepsilon>0)(\exists \delta>0)(\forall x \in \operatorname{Dom}(f))[0<|x-a|<\delta] \Rightarrow[|f(x)-L|<\varepsilon]\] and \[(\exists \delta>0)(\forall \varepsilon>0)(\forall x \in \operatorname{Dom}(f))[0<|x-a|<\delta] \Rightarrow[|f(x)-L|<\varepsilon]\] are not equivalent. Which statement is a consequence of the other?
EXERCISE 3.16. Let \(P(x, y)\) be a formula in two variables. Show that in general \((\forall x)(\exists y) P(x, y)\) need not be equivalent to \((\exists y)(\forall x) P(x, y)\) . Show that \((\forall x)(\forall y) P(x, y)\) is equivalent to \((\forall y)(\forall x) P(x, y)\) . What about \((\exists x)(\exists y) P(x, y)\) and \((\exists y)(\exists x) P(x, y)\) ?
EXERCISE 3.17. Consider the following statements. Write down the contrapositive and the converse to each one.
(i) All men are mortal.
(ii) I mean what I say. (iii) Every continuous function on the interval \([0,1]\) attains its maximum
(iv) The sum of the angles of a triangle is \(180^{\circ}\) .
EXERCISE 3.18. Prove that a number is divisible by 4 if and only if its last two digits are.
EXERCISE 3.19. Prove that a number is divisible by 8 iff its last three digits are.
EXERCISE 3.20. Prove that a number is divisible by \(2^{n}\) iff its last \(n\) digits are.
EXERCISE 3.21. Suppose \(m\) is a number with the property that any natural number is divisible by \(m\) iff its last three digits are. What does this say about \(m\) ? Prove your assertion.
ExERCISE 3.22. Prove that an integer is divisible by 11 iff the sum of the oddly placed digits minus the sum of the evenly placed digits is divisible by 11. (So \(11 \mid 823493\) iff 11 divides \((2+4+3)-(8+3+9)\) .)
EXERCISE 3.23. Show that every interval contains rational and irrational numbers.
EXERCISE 3.24. Prove that \(\sqrt{3}\) is irrational.
EXERCISE 3.25. Prove that \(\sqrt{10}\) is irrational.
EXERCISE 3.26. Prove that the square root of any natural number is either an integer or irrational.
EXERCISE 3.27. Prove that there exist irrational numbers \(x\) and \(y\) so that \(x^{y}\) is rational. (Hint: consider \(\sqrt{2}^{\sqrt{2}}\) and \(\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}\) .)
EXERCISE 3.28. Prove or disprove the following assertion: Any 4 points in the plane, no three of which are collinear, lie on a circle.
EXERCISE 3.29. Prove that there are an infinite number of primes. ExERCISE 3.30. For \(k=0,1,2\) , let \(P_{k}\) be the set of prime numbers that are congruent to \(k \bmod 3\) . By Exercise 3.29, \(P_{0} \cup P_{1} \cup P_{2}\) is infinite. Can you say which of the sets \(P_{0}, P_{1}\) and \(P_{2}\) are infinite?
(Remark: For two of the three sets, this problem is not too difficult. For the third one, it is extremely difficult, and is a special case of a celebrated theorem of Dirichlet. See \(e . g\) . [8] for a treatment of Dirichlet’s theorem.)
EXERCISE 3.31. Let the points in \(\mathbb{R}^{2}\) be colored red, green and blue. Prove that either there are two points of the same color a distance 1 apart, or there is an equilateral triangle of side length \(\sqrt{3}\) all of whose vertices are the same color.
EXERCISE 3.32. Prove that \[e=\sum_{n=0}^{\infty} \frac{1}{n !}\] is irrational. (Hint: Argue by contradiction. Assume \(e=\frac{p}{q}\) , and multiply both sides by \(q !\) Rearrange the equation to get an integer equal to an infinite sum of rational numbers that converges to a number in \((0,1)\) . )


2.3: Histograms, Frequency Polygons, and Time Series Graphs
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For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.
A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.
The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.(Remember, frequency is defined as the number of times an answer occurs.) If:
- \(f\) is frequency
- \(n\) is total number of data values (or the sum of the individual frequencies), and
- \(RF\) is relative frequency,
then:
\[RF=\dfrac{f}{n} \label{2.3.1}\]
For example, if three students in Mr. Ahab's English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = fn = 340 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.
To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is \(6.05 (6.1 – 0.05 = 6.05)\). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is \(1.495 (1.5 – 0.005 = 1.495)\). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is \(0.9995 (1.0 – 0.0005 = 0.9995)\). If all the data happen to be integers and the smallest value is two, then a convenient starting point is \(1.5 (2 - 0.5 = 1.5)\). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.
Example 2.3.1
The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured.
60; 60.5; 61; 61; 61.5
63.5; 63.5; 63.5
64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5
66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5
68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5
70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71
72; 72; 72; 72.5; 72.5; 73; 73.5
74
The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.
60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.
The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.
Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.
\[\dfrac{74.05−59.95}{8}=1.76\]We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.
The boundaries are:
- 59.95
- 59.95 + 2 = 61.95
- 61.95 + 2 = 63.95
- 63.95 + 2 = 65.95
- 65.95 + 2 = 67.95
- 67.95 + 2 = 69.95
- 69.95 + 2 = 71.95
- 71.95 + 2 = 73.95
- 73.95 + 2 = 75.95
The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.
The following histogram displays the heights on the x -axis and relative frequency on the y -axis.
Figure 2.3.1: Histogram of somethingExercise 2.3.1
continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.
9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5
11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5
12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14
Answer
Smallest value: 9
Largest value: 14
Convenient starting value: 9 – 0.05 = 8.95
Convenient ending value: 14 + 0.05 = 14.05
\(\frac{14.05-8.95}{6}\) = 0.85
The calculations suggests using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one.
Example 2.3.2
The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3
4; 4; 4; 4; 4; 4
5; 5; 5; 5; 5
6; 6
Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.
Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.
Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .
Answer
Calculate the number of bars as follows:
\(\frac{6.5 - 0.5}{\text{number of bars}}\) = 1
where 1 is the width of a bar. Therefore, bars = 6.
The following histogram displays the number of books on the x -axis and the frequency on the y -axis.
Figure 2.3.2.Exercise 2.3.2
The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3
20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.
Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____.
Answer
1.5
1.5 to 2.5
2.5 to 3.5
Example 2.3.3
Using this data set, construct a histogram.
| 9.95 | 10 | 2.25 | 16.75 | 0 |
| 19.5 | 22.5 | 7.5 | 15 | 12.75 |
| 5.5 | 11 | 10 | 20.75 | 17.5 |
| 23 | 21.9 | 24 | 23.75 | 18 |
| 20 | 15 | 22.9 | 18.8 | 20.5 |
Answer
Figure 2.3.3.Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.
Query \(\PageIndex{1}\)
Exercise 2.3.3
The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.
22; 35; 15; 26; 40; 28; 18; 20; 25; 34; 39; 42; 24; 22; 19; 27; 22; 34; 40; 20; 38 and 28
Use 10–19 as the first interval.
Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.
Frequency Polygons
Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.
Example 2.3.4
A frequency polygon was constructed from the frequency table below.
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 5 | 5 |
| 59.5 | 69.5 | 10 | 15 |
| 69.5 | 79.5 | 30 | 45 |
| 79.5 | 89.5 | 40 | 85 |
| 89.5 | 99.5 | 15 | 100 |
The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.
Exercise 2.3.4
Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in Table .
| Age at Inauguration | Frequency |
|---|---|
| 41.5–46.5 | 4 |
| 46.5–51.5 | 11 |
| 51.5–56.5 | 14 |
| 56.5–61.5 | 9 |
| 61.5–66.5 | 4 |
| 66.5–71.5 | 2 |
Answer
The first label on the x -axis is 39. This represents an interval extending from 36.5 to 41.5. Since there are no ages less than 41.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 44 represents the next interval, or the first “real” interval from the table, and contains four scores. This reasoning is followed for each of the remaining intervals with the point 74 representing the interval from 71.5 to 76.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.
.
Figure 2.3.5.
Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.
Example 2.3.5
We will construct an overlay frequency polygon comparing the scores from Example with the students’ final numeric grade.
| Frequency Distribution for Calculus Final Test Scores | |||
|---|---|---|---|
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
| 49.5 | 59.5 | 5 | 5 |
| 59.5 | 69.5 | 10 | 15 |
| 69.5 | 79.5 | 30 | 45 |
| 79.5 | 89.5 | 40 | 85 |
| 89.5 | 99.5 | 15 | 100 |
| Frequency Distribution for Calculus Final Grades | |||
|---|---|---|---|
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
| 49.5 | 59.5 | 10 | 10 |
| 59.5 | 69.5 | 10 | 20 |
| 69.5 | 79.5 | 30 | 50 |
| 79.5 | 89.5 | 45 | 95 |
| 89.5 | 99.5 | 5 | 100 |
Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.
One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.
Constructing a Time Series Graph
To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.
Example 2.3.6
The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.
| Year | Jan | Feb | Mar | Apr | May | Jun | Jul |
|---|---|---|---|---|---|---|---|
| 2003 | 181.7 | 183.1 | 184.2 | 183.8 | 183.5 | 183.7 | 183.9 |
| 2004 | 185.2 | 186.2 | 187.4 | 188.0 | 189.1 | 189.7 | 189.4 |
| 2005 | 190.7 | 191.8 | 193.3 | 194.6 | 194.4 | 194.5 | 195.4 |
| 2006 | 198.3 | 198.7 | 199.8 | 201.5 | 202.5 | 202.9 | 203.5 |
| 2007 | 202.416 | 203.499 | 205.352 | 206.686 | 207.949 | 208.352 | 208.299 |
| 2008 | 211.080 | 211.693 | 213.528 | 214.823 | 216.632 | 218.815 | 219.964 |
| 2009 | 211.143 | 212.193 | 212.709 | 213.240 | 213.856 | 215.693 | 215.351 |
| 2010 | 216.687 | 216.741 | 217.631 | 218.009 | 218.178 | 217.965 | 218.011 |
| 2011 | 220.223 | 221.309 | 223.467 | 224.906 | 225.964 | 225.722 | 225.922 |
| 2012 | 226.665 | 227.663 | 229.392 | 230.085 | 229.815 | 229.478 | 229.104 |
| Year | Aug | Sep | Oct | Nov | Dec | Annual |
|---|---|---|---|---|---|---|
| 2003 | 184.6 | 185.2 | 185.0 | 184.5 | 184.3 | 184.0 |
| 2004 | 189.5 | 189.9 | 190.9 | 191.0 | 190.3 | 188.9 |
| 2005 | 196.4 | 198.8 | 199.2 | 197.6 | 196.8 | 195.3 |
| 2006 | 203.9 | 202.9 | 201.8 | 201.5 | 201.8 | 201.6 |
| 2007 | 207.917 | 208.490 | 208.936 | 210.177 | 210.036 | 207.342 |
| 2008 | 219.086 | 218.783 | 216.573 | 212.425 | 210.228 | 215.303 |
| 2009 | 215.834 | 215.969 | 216.177 | 216.330 | 215.949 | 214.537 |
| 2010 | 218.312 | 218.439 | 218.711 | 218.803 | 219.179 | 218.056 |
| 2011 | 226.545 | 226.889 | 226.421 | 226.230 | 225.672 | 224.939 |
| 2012 | 230.379 | 231.407 | 231.317 | 230.221 | 229.601 | 229.594 |
Answer
Figure 2.3.7.Exercise 2.3.5
The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.
| CO2 Emissions | |||
|---|---|---|---|
| Ukraine | United Kingdom | United States | |
| 2003 | 352,259 | 540,640 | 5,681,664 |
| 2004 | 343,121 | 540,409 | 5,790,761 |
| 2005 | 339,029 | 541,990 | 5,826,394 |
| 2006 | 327,797 | 542,045 | 5,737,615 |
| 2007 | 328,357 | 528,631 | 5,828,697 |
| 2008 | 323,657 | 522,247 | 5,656,839 |
| 2009 | 272,176 | 474,579 | 5,299,563 |
Uses of a Time Series Graph
Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.
Chapter Review
A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.Glossary
References
- Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker
- “Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/a...-us-presidents (accessed April 3, 2013).
- “Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).
- “Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).
- “Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).
- “CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).
- “Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/stati...me-series.html (accessed April 3, 2013).
- “Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).
- Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.
- “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).
- Frequency
- the number of times a value of the data occurs
- Histogram
- a graphical representation in \(x-y\) form of the distribution of data in a data set; \(x\) represents the data and \(y\) represents the frequency, or relative frequency. The graph consists of contiguous rectangles.
- Relative Frequency
- the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes
Contributors and Attributions
-
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


7.4: Ampere’s Circuital Law (Magnetostatics) - Integral Form
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Ampere’s circuital law (ACL) relates current to the magnetic field associated with the current. In the magnetostatic regime, the law is (see also Figure \(\PageIndex{1}\)):
\[\boxed{ \oint_{\mathcal C}{ {\bf H} \cdot d{\bf l} } = I_{encl} } \label{m0019_eACL} \]
That is, the integral of the magnetic field intensity \({\bf H}\) over a closed path \({\mathcal C}\) is equal to the current enclosed by that path, \(I_{encl}\). Before proceeding to interpret this law, it is useful to see that it is dimensionally correct. That is, \({\bf H}\) (units of A/m) integrated over a distance (units of m) yields a quantity with units of current (i.e., A).
Figure \(\PageIndex{1}\): Reference directions for Ampere’s Circuital Law (Equation \ref{m0019_eACL}). ( CC BY SA 4.0 ; K. Kikkeri)In general, \(I_{encl}\) may be either positive or negative. The direction corresponding to positive current flow must be correctly associated with \({\mathcal C}\). The relationship follows the right-hand rule of Stokes’ Theorem summarized as follows. The direction of positive \(I_{encl}\) is the direction in which the fingers of the right hand intersect any surface \({\mathcal S}\) bordered by \({\mathcal C}\) when the thumb of the right hand points in the direction of integration. The connection to Stokes’ Theorem is not a coincidence. See Section 7.9 for more about this.
Note that \({\mathcal S}\) can be any surface that is bounded by \({\mathcal C}\) – not just the taut surface implied in Figure \(\PageIndex{1}\).
The integral form of Ampere’s Circuital Law for magnetostatics (Equation \ref{m0019_eACL}) relates the magnetic field along a closed path to the total current flowing through any surface bounded by that path.
ACL plays a role in magnetostatics that is very similar to the role played by the integral form of Gauss’ Law in electrostatics (Section 5.5). That is, ACL provides a means calculate the magnetic field given the source current. ACL also has a similar limitation. Generally, symmetry is required to simplify the problem sufficiently so that the integral equation may be solved. Fortunately, a number of important problems fall in this category. Examples include the problems addressed in Sections 7.5 (magnetic field of a line current), 7.6 (magnetic field inside a straight coil), 7.7 (magnetic field of a toroidal coil), 7.8 (magnetic field of a current sheet), and 7.11(boundary conditions on the magnetic field intensity). For problems in which the necessary symmetry is not available, the differential form of ACL may be required (Section 7.9).
Finally, be aware that the form of ACL addressed here applies to magnetostatics only. In the presence of a time-varying electric field, the right side of ACL includes an additional term known as the displacement current (Section 8.9).