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12.8: Critical Thinking Questions
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How would your life be different if you had been ascribed a different gender category at birth? Would your parents, siblings, and other relatives have treated you differently? Would your experiences in school have been different? Would you be the same person?
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How did you learn to perform your assigned gender role? What lessons were explicitly taught, and what did you learn through observation and experience? Who taught you? Have you ever felt as though you were failing to successfully perform your gender role?
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What would life be like in a matriarchal society? How might family life be different? How would government and religion be different? What values might be emphasized? Why do you think so?
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Do all men benefit from patriarchy? How might men be limited or harmed by patriarchal beliefs and practices? Do some women benefit from patriarchy? How so?
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Do you live in a heteronormative culture? How can you tell? Are things changing? What same-sex practices and relationships might have existed in the past in your culture?
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What does the prevalence of transgender practices in many cultures tell us about human gender and sexuality?
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In your society, do men and women live in different subcultures? In what contexts are women and men segregated, either formally or informally? What forms of social interaction and bonding occur in those situations? What might be the function of the gendered subcultures found in many societies?
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Would it be possible to entirely eliminate the concept of gender in your society? What might be the advantages and disadvantages of a gender-free society? How would the ideologies and practices of sexuality be transformed by such a change?


9.5E: Finding New Antimicrobial Drugs
Antimicrobial resistance has created a public health crisis in the treatment of infectious diseases and necessitates the discovery of new drugs.
Learning Objectives
- Explain the reasons for low production of new antibiotics and discuss the proposed mechanisms to evade antimicrobial resistance
Key Points
- Finding new antimicrobial drugs requires researchers, pharmaceutical, and biotech companies to invest in new technology and discover new sources for antibiotic development.
- Proposed mechanisms to circumvent antimicrobial resistance range from exploring the list of resistance genes to antibody-based therapy and vaccines.
- Finding new candidates to target is essential but it needs to be accompanied by awareness on antibiotic misuse with the prospect to eliminate the root of the problem.
Key Terms
- mimetic : A substance with similar pharmacological effects to another substance.
Antimicrobial resistance: the problem
Antibiotics, more than any other medicines, have improved the life expectancy of mankind, however, multi-drug resistance has become common in pathogenic bacteria and multiple drugs are losing efficacy. Recent reports on the occurrence of panresistant gram-negative strains, i.e. strains resistant to every registered antibacterial drug, indicate that we are on the verge to lose the battle, taking us back to the pre-antibiotic era. There is world-wide consensus that the medical need for novel antiinfective drugs is enormous and that we are running out of time. Many achievements of modern medicine, not only treatment of infectious diseases, depend on the availability of efficacious antibiotics, still, the antibacterial development pipeline is slow and the number of new drugs reaching the market is alarmingly low. There are many reasons for this at all levels of the discovery and development process. Investments into antibiotic research and technologies is minimal; socioeconomic considerations together with regulatory hurdles have prompted pharmaceutical companies to exit the field and innovative biotech companies were confronted with problems beyond their control. Answers are needed as to where and how we can find new lead compounds with unprecedented activities?
Finding new antimicrobial drugs: the solution
Research on new antimicrobial compounds is geared towards innovative targets to circumvent resistance. Some of the proposed areas to investigate include: collecting and examining the list of antimicrobial resistance genes (e.g. exploring the resistome), targeting teichoic acid biosynthesis as a new method to compromise the bacterial wall integrity, producing ribosomal inhibitors to target protein synthesis, targeting outer-membrane transporters with protein epitope mimetics (e.g. mimetics of the cationic antimicrobial peptides that form part of the immune response to microbes), and developing antibody-based strategies and vaccines. Using bacteriophages for treatment of bacterial infection has also seen renewed interest and promise in recent years. In some cases, alternative treatments such as re-establishing normal flora can be extremely effective. For instance, fecal transplants have become a standard treatment for Clostridium difficile infections over the past decade or so.
The initiative to develop new antimicrobial agents is urgently needed but is a long process from invention, to development, to actual clinical application. It is also necessary to initiate a worldwide awareness on antibiotic misuse and overuse as a mean to address the root of the problem for antimicrobial resistance.


8.9: The Umayyad Caliphate
In 661, ‘Ali suffered the same fate as his predecessor when a Khariji stabbed him to death. And, just like with ‘Uthman, the murder of ‘Ali took place during prayers. ‘Ali’s death represented a deep loss for his followers, who saw him as an advocate of an egalitarian version of Islam and a believer in a just and righteous government. His martyrdom came to be regarded as a sacrifice in the service of God and prompted his supporters to pattern themselves after their champion, who, they insisted, had developed spiritual gifts that remained virtually unattainable for others.
The ‘Alids encouraged ‘Ali’s oldest son, Hasan, to succeed his father; however, Mu‘awiya threatened the Prophet’s grandson with continued warfare and convinced him to renounce his claim to the caliphate. Mu‘awiya promised Hasan that he would not appoint an heir so that election of future caliphs would return to the majlis . Handsomely compensated by Mu‘awiya, Hassan subsequently retired to Mecca and took up religion. He remained there until his death in 669. With this major obstacle removed, Mu‘awiya became the fifth caliph, ending the period of the four rightly guided caliphs, also known as the Rashidun Caliphate.
Mu‘awiya (661 – 680) founded the Umayyad Caliphate; the tribal ‘asabiyah of his Umayyad Clan contributed to their ascendance. And once ensconced in power, the Umayyad Caliphate ended the election of caliphs by consensus and established instead a hereditary principle of succession. Mu‘awiya established the caliphate to Damascus, where he previously served as ‘Uthman’s governor. In Syria, Mu‘awiya reformed the bureaucracy by eventually centralizing it. Unable to rely on the Arab tribal system or peninsula traditions to administer to an ever expanding empire, he depended on related Greek merchant families for administrators and adopted the existing administrative machinery of Byzantines, including their imperial customs and bureaucratic practices.
Mu‘awiya had received much recognition for his unfaltering determination to seek retribution for ‘Uthman’s death; however, he had squandered much of that good will in harassing ‘Ali. As anti-Umayyad sentiment increased, the rift that existed between the Sunnis and Shi‘a continued to expand, for recalcitrant ‘Alids continued to harbor resentment against the Umayyads. They remembered when the ruling aristocracy of Mecca had opposed Muhammad and the Muslim community. In fact, Mu‘awiya himself had fought against Muhammad until the Treaty of Hudaybiyyah, only to reverse course, convert to Islam, and become the Prophet’s secretary.
Unlike the two caliphs who preceded him, Mu‘awiya died peacefully in bed. Prior to his death, he designated his son Yazid (680 – 683) as his successor, thus violating his agreement with Hasan. Most notable for his well-deserved reputation as a fierce fighter, Yazid was also known for generally dissolute behavior that offended the religious sensibilities of many pious Muslims. Once ensconced as caliph, Yazid failed to secure an oath of allegiance from Husayn , brother of Hasan, one of the most important of Muslim leaders. Their rivalry escalated into a full-scale civil war.
A direct descendent of the Prophet Muhammad and the younger son of the Caliph ‘Ali, Husayn rejected the deal that his brother had negotiated, instead pursuing his own claim to the rightful leadership of the Islamic community. His ‘Alid supporters loathed the Umayyads and believed that the caliph must be closely related to the Prophet. Husayn’s refusal to recognize Yazid as the next caliph and their subsequent conflict culminated in 680 at the Battle of Karbala, located to the west of present day Baghdad. Yazid dispatched a military detachment to Iraq and overwhelmed Husayn’s small band of armed followers so that many of Husayn’s own men deserted him in his hour of need. The Shi‘a perceived this seminal event as a turning point in their history.
Much like the loss of ‘Ali, the death of Husayn shocked the incipient Shi‘a community, many of whom suffered from intense guilt for failing to assist his little band. Increasing numbers of Shi‘a became profoundly affected by his martyrdom, interpreting it as a sacrifice in the best interests of their community; over time, a passion narrative developed that commemorated his last hours. Through this commemoration of the Battle of Karbala on Ashura, the tenth day of the month of Muharram, they remember the terrible suffering and his untimely death and strive to experience an existential intimacy with their martyr.
Yazid had inherited an empire punctuated by civil war and rebellion. Another principle figure among those in revolt was ibn Zubayr , grandson of Caliph Abu Bakr. Following the death of Mu‘awiya, Ibn Zubayr had sworn allegiance to Husayn. He remained in Mecca, where he stood in opposition to the Umayyads. The general unpopularity of the Umayyads advanced his cause, and many Muslims considered him the rightful caliph. Indeed, much of his support came from Muslims who rejected the idea of hereditary succession and sought a return to the election of caliphs by consensus.
Yazid invaded the Hijaz in order to put an end to ibn Zubayr’s rebellion, but the caliph’s abrupt death in 683 halted the campaign. Marwan (684 – 685) followed his cousin Yazid but was not universally recognized as caliph, for many considered ibn Zubayr the legitimate successor. To garner support, Marwan exploited latent tribal animosities that existed between his Kalb Tribe, also known as the Yemen, and the Qays Tribe, who supported ibn Zubayr. At the Battle of Marj Rahit in 684, Marwan’s Kalb forces defeated the Qays, allowing him to consolidate Umayyad control over Syria and Egypt, thus shrinking ibn Zubayr’s rule down to Iraq and the Hijaz. Not until 691 did Abdul Malik (685 – 705), heir to Marwan, recover Iraq from ibn Zubayr. In the process, he also had to pacify Khariji and Shi‘a areas. Abdul Malik then dispatched General Hajjaj to the Hijaz. A brutal military leader, Hajjaj laid siege to the holy city of Mecca in 692 in order to secure the submission of ibn Zubayr’s men. He then beheaded ibn Zubayr and crucified his body. Abdul Malik rewarded the brutal general for his loyal service with the governorship of Iraq, where his ruthless reputation persisted.
Once he had assumed the throne, Abdul Malik promoted the Arabization of the caliphate. He rejected the use of Greek, Persian, Coptic, or Aramaic in government, decreeing that all bureaucracy had to be only in Arabic. Non-Arab administrators had to learn Arabic in order to keep their government jobs. Their integration did not lead to the complete Arabization of Umayyad society that Abdul Malik envisioned, however, and the spread of Arabic was not as great as the spread of Islam. Many Muslims continued to speak Berber, Turkish, Kurdish, and Persian. Although a separate process, Arabization only accompanied Islamization.
Abdul Malik also sought to Islamize the caliphate. First, he discontinued the earlier Byzantine coinage and created the first Islamic currency. Then he instituted a tax code based on the principles of Islam. Caliphs levied an additional tax on non-Muslims, known as the jizya , as was customary in Islam. Christians and Jews in conquered lands also paid a property tax called kharaj . By converting to Islam, one could avoid paying the jizya and kharaj altogether. Most important for ordinary citizens was the fact that Muslims bore lower tax rates than non-Muslims. As one could imagine, the thrust for conversion became primarily economic. Although the process of Islamization was relatively peaceful and gradual, Islam did become the dominant religion of the region. And the parallel processes of Arabization and Islamization helped to reestablish centralized rule after the second civil war.
Not all of Abdul Malik’s reforms adhered to the egalitarian principles set forth in Islam. Arab tribal elites did not want to recognize the mawali , non-Arab Muslims, as social equals, so did not afford them the same rights as Arab Muslims. However, the emerging power and influence of the mawali was apparent. They had become the intellectual elite of society and were the bureaucrats and commercial leaders of the umma . Nevertheless, they faced social discrimination. For example, Umayyad caliphs taxed the mawali as if they were non-Muslims. This inequitable practice became a social problem for the Umayyads, for it stood in stark relief against the values of justice and equality that had originally compelled them to convert.
An extremely devout and pious man, the Caliph ‘Umar II (r. 717 – 720) upended the Umayyad moral order. He considered it immoral to show prejudice against the mawali and to favor the Arabs, so he attempted to resolve the lingering hostilities of the mawali by advocating the equality of all Muslims. ‘Umar II declared an end to the practice of taxing the mawali like the Christians and Jews. His advisors warned him against this change because it precipitated numerous conversions of non-Muslims, so he decreased military expenditures to compensate for an expected drop in revenue. His reforms might have ended the official discrimination against the mawali , but they alienated the Umayyad privileged class, who paid a servant to poison ‘Umar II to death in 720.


7.3: Nouns, Verbs and Adjectives- Open Class Categories
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In Linguistics, we observe how parts of language behave. When we find a set of words that all behave similarly, we can group them into a category, specifically, into a syntactic category. You might have learned about some of these categories as “parts of speech”. This unit gives an overview of the behaviour of the biggest categories.
You’ve probably learned that
nouns
are words that describe a person, place or thing. But when we’re studying morphology and syntax, we categorize words according to their behaviour, not according to their meaning.There are two elements to a word’s behaviour:
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What inflectional morphemes does the word take?
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What is the word’s syntactic distribution? In other words, what position does it occupy in a sentence?
What behaviour can we observe that allows us to categorize words as nouns? Looking at the inflectional morphology, we observe that most nouns in English have a singular and a plural form:
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singular
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plural
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tree
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trees
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book
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books
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song
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songs
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idea
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ideas
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goal
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goals
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English uses a plural morpheme on a noun to indicate that there is more than one of something. But there is a
subcategory
of nouns that don’t have plural forms.
Mass nouns
like
rice
,
water
,
money
,
oxygen
refer to things that aren’t really countable, so the nouns don’t get pluralized. Nouns that refer to abstract things (such as
justice
,
beauty
,
happiness
) behave like mass nouns too. If they don’t have plural forms, why do we group them into the larger category of nouns? It’s because their syntactic distribution behaves like that of count nouns. Most English nouns, singular, plural, or mass, can appear in a phrase following the word
the
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the tree, the trees
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the book, the books
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the song, the songs
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the idea, the ideas
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the goal, the goals
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the rice
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the money
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the beauty (e.g., the beauty of the scenery)
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the happiness (e.g., the happiness of the children)
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3.8E: Inverses and Radical Functions (Exercises)
section 3.8 exercise
For each function, find a domain on which the function is one-to-one and non-decreasing, then find an inverse of the function on this domain.
1. \(f\left(x\right)=\left(x-4\right)^{2}\)
2. \(f\left(x\right)=\left(x+2\right)^{2}\)
3. \(f\left(x\right)=12-x^{2}\)
4. \(f\left(x\right)=9-x^{2}\)
5. \(f\left(x\right)=3x^{3} +1\)
6. \(f\left(x\right)=4-2x^{3}\)
Find the inverse of each function.
7. \(f\left(x\right)=9+\sqrt{4x-4}\)
8. \(f\left(x\right)=\sqrt{6x-8} +5\)
9. \(f\left(x\right)=9+2\sqrt[{3}]{x}\)
10. \(f\left(x\right)=3-\sqrt[{3}]{x}\)
11. \(f\left(x\right)=\dfrac{2}{x+8}\)
12. \(f\left(x\right)=\dfrac{3}{x-4}\)
13. \(f\left(x\right)=\dfrac{x+3}{x+7}\)
14. \(f\left(x\right)=\dfrac{x-2}{x+7}\)
15. \(f\left(x\right)=\dfrac{3x+4}{5-4x}\)
16. \(f\left(x\right)=\dfrac{5x+1}{2-5x}\)
Police use the formula \(v=\sqrt{20L}\) to estimate the speed of a car, \(v\), in miles per hour, based on the length, \(L\), in feet, of its skid marks when suddenly braking on a dry, asphalt road.
17. At the scene of an accident, a police officer measures a car’s skid marks to be 215 feet long. Approximately how fast was the car traveling?
18. At the scene of an accident, a police officer measures a car’s skid marks to be 135 feet long. Approximately how fast was the car traveling?
The formula \(v=\sqrt{2.7r}\) models the maximum safe speed, \(v\), in miles per hour, at which a car can travel on a curved road with radius of curvature \(r\), in feet.
19. A highway crew measures the radius of curvature at an exit ramp on a highway as 430 feet. What is the maximum safe speed?
20. A highway crew measures the radius of curvature at a tight corner on a highway as 900 feet. What is the maximum safe speed?
21. A drainage canal has a cross-section in the shape of a parabola. Suppose that the canal is 10 feet deep and 20 feet wide at the top. If the water depth in the ditch is 5 feet, how wide is the surface of the water in the ditch? [UW]
22. Brooke is located 5 miles out from the nearest point \(A\) along a straight shoreline in her sea kayak. Hunger strikes and she wants to make it to Kono’s for lunch; see picture. Br ooke can paddle 2 mph and walk 4 mph. [UW]
a. If she paddles along a straight line course to the shore, find an expression that computes the total time to reach lunch in terms of the location where Brooke beaches her kayak.
b. Determine the total time to reach Kono’s if she paddles directly to the point \(A\).
c. Determine the total time to reach Kono’s if she paddles directly to Kono’s.
d. Do you think your answer to b or c is the minimum time required for Brooke to reach lunch?
e. Determine the total time to reach Kono’s if she paddles directly to a point on the shore half way between point A and Kono’s. How does this time compare to the times in parts b or c? Do you need to modify your answer to part d?
23. Clovis is standing at the edge of a dropoff, which slopes 4 feet downward from him for every 1 horizontal foot. He launches a small model rocket from where he is standing. With the origin of the coordinate system located where he is standing, and the \(x\)-axis extending horizontally, the path of the rocket is described by the formula \(y=-2x^{2} +120x\). [UW]
a. Give a function \(h=f(x)\) relating the height \(h\) of the rocket above the sloping ground to its \(x\)-coordinate.
b. Find the maximum height of the rocket above the sloping ground. What is its \(x\)-coordinate when it is at its maximum height?
c. Clovis measures the height \(h\) of the rocket above the sloping ground while it is going up. Give a function \(x=g\left(h\right)\) relating the \(x\)-coordinate of the rocket to \(h\).
d. Does the function from (c) still work when the rocket is going down? Explain.
24 . A trough has a semicircular cross section with a radius of 5 feet. Water starts flowing into the trough in such a way that the depth of the water is increasing at a rate of 2 inches per hour. [UW]
a. Give a function \(w=f\left(t\right)\) relating the width w of the surface of the water to the time \(t\), in hours. Make sure to specify the domain and compute the range too.
b. After how many hours will the surface of the water have width of 6 feet?
c. Give a function \(t=f^{-1} \left(w\right)\) relating the time to the width of the surface of the water. Make sure to specify the domain and compute the range too.
- Answer
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1. Domain \((4, \infty)\) Inverse \(f^{-1} (x) = \sqrt{x} + 4\)
3. Domain \((-\infty, 0)\) Inverse \(f^{-1} (x) = -\sqrt{12 - x}\)
5. Domain \((-\infty, 0)\) Inverse \(f^{-1} (x) = \sqrt[3]{\dfrac{x - 1}{3}}\)
7. \(f^{-1} (x) = \dfrac{(x - 9)^2}{4} + 1\)
9. \(f^{-1} (x) = (\dfrac{x - 9}{2})^3\)
11. \(f^{-1} (x) = \dfrac{2 - 8x}{x}\)
13. \(f^{-1} (x) = \dfrac{3 - 7x}{x - 1}\)
15. \(f^{-1} (x) = \dfrac{5x - 4}{3 + 4x}\)
17. 65.574 mph
19. 34. 073 mph
21. 14.142 feet


18.3: Preparing Ethers
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Objectives
After completing this section, you should be able to
- write an equation to illustrate the industrial preparation of simple symmetrical ethers.
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write an equation to illustrate the Williamson synthesis of ethers.
- identify the ether obtained from the reaction of a given alkyl halide with a given alkoxide ion.
- identify the reagents needed to prepare a given ether through a Williamson synthesis.
- identify the limitations of the Williamson synthesis, and make the appropriate choices when deciding how best to synthesize a given ether.
- write an equation to describe the formation of an alkoxide from an alcohol.
- identify silver(I) oxide as a reagent which can be used in a Williamson synthesis.
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write an equation to show how an ether can be prepared by the alkoxymercuration-demercuration of an alkene.
- identify the product formed from the alkoxymercuration-demercuration of a given alkene.
- identify the alkene, the reagents, or both, needed to prepare a given ether by the alkoxymercuration-demercuration process.
- write the detailed mechanism of the reaction between an alkene, an alcohol and mercury(II) trifluoroacetate.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- alkoxymercuration
- oxymercuration
- Williamson ether synthesis
Study Notes
We studied oxymercuration as a method of converting an alkene to an alcohol in Section 8.5 . “Alkoxymercuration” is a very similar process, except that we are now converting an alkene into an ether. The two processes are compared below.
| Description | oxymercuration | alkoxymercuration |
|---|---|---|
| we react an | alkene | alkene |
| with | water | an alcohol |
| in the presence of | Hg(O 2 CCH 3 ) 2 | Hg(O 2 CCF 3 ) 2 |
| followed by treatment with | NaBH 4 | NaBH 4 |
| to produce an | alcohol | ether |
Review the mechanism of the oxymercuration reaction in Section 8.5 , paying particular attention to the regiochemistry and the stereochemistry of the reaction. The mechanism is identical to alkoxymercuration.
Ether Formation Though Dehydration
Acid-catalyzed dehydration of small 1º-alcohols constitutes a specialized industrial method of preparing symmetrical ethers. This reaction cannot be employed to prepare unsymmetrical ethers because a mixture of products is likely to be obtained. Also, 2 o and 3 o alcohols cannot be used for this reaction because they dehydrate to form alkenes by an E1 mechanism ( Section 17-6 ).
\[\ce{2 CH_3CH_2-OH + H_2SO_4 ->[130\;^oC] CH_3CH_2\bond{-}O\bond{-}CH_2CH_3 + H_2O} \tag{18.2.1} \]
Mechanism
In the first step of the reaction mechanism, one alcohol is protonated to become a good leaving group. In the second step, a second alcohol displaces water from the protonated alcohol during an S N 2 reaction yielding a protonated ether. In the final step, this intermediate is deprotonated to yield the symmetrical ether.
Williamson Ether Synthesis
One important procedure, known as the Williamson Ether Synthesis , proceeds by an S N 2 reaction of an alkoxide nucleophile with a primary alkyl halide or tosylate. As previously discussed in Section 17-2 , alkoxides are commonly created by deprotonating an alcohol with a strong base, such as sodium hydride (NaH). Simple alcohols can be used a solvent during a Williamson ether synthesis and with their alkoxide created through the addition of sodium metal (Na (s) ).
Planning a Williamson Ether Synthesis
The Williamson ether synthesis has the same limitations as other S N 2 reactions, as discussed in Section 11-3 . Since alkoxide anions are strong bases, utilizing 2 o or 3 o halogen leaving groups could possibly produce an E2 elimination product. When considering the synthesis of an unsymmetrical ether, there are two different combinations of reactants possible and each should be carefully considered. In general, the pathway which utilizes the least sterically hindered halogen will be preferred.
The key bond cleavage in the target molecule involves a C-O bond. Because unsymmetrical ethers have two unique C-O bonds, each can be broken to provide a unique set of reactants. After cleavage, the fragment with the oxygen will become an alkoxide. The other fragment will become a halogen or tosylate.
Worked Example \(\PageIndex{1}\)
How would you prepare the following molecule using a Williamson Ether Synthesis?
- Answer
-
Analysis: The ether is asymmetrical so each of the C-O bonds can be broken to create a different set of possible reactants. After cleavage of the C-O bond, pathway 1 shows a 3 o halogen as the starting material. This reaction will most likely not be effective due to alkoxides reacting with 3 o halogens to preferable form an alkenes by E2 elimination. Pathway 2 shows a 1 o halogen as a starting material which is favorable for S N 2 reactions.
Pathway 1
Solution 1
Pathway 2
Solution 2
Ether Synthesis Using Silver Oxide
A variation of the Williamson ether synthesis uses silver oxide (Ag 2 O) in the place of the strong base. The conditions of this variation are milder than the typical Willamson synthesis because a strong base and the formation of an alkoxide intermediate are not necessary. This reaction is particually useful when converting the -OH groups on a sugar into ethers.
Mechanism
During this reaction a partial positively charged silver in Ag 2 O gives draws electron density from the iodine in CH 3 I. This correspondingly removes electron density from the adjacent carbon increasing its partial positive charge which increases its electrophlicity. This allows the alcohol to act as a nucleophile in the subsequent S N 2 reaction.
Ether Synthesis Using Alkoxymercuration
Alkoxymercuration , is patterned after the oxymercuration reaction discussed in Section 8-4 . Reaction of an alkene with an alcohol in the presence of a trifluoroacetate mercury (II) salt [(CF 3 CO 2 ) 2 Hg] prodcues an alkoxymercuration product. Demercuration using sodium borohydride (NaBH 4 ) yields an ether product. Overall, this reaction allows for the Markovnikov addition of an alcohol to an alkene to create an ether. Note that the alcohol reactant is used as the solvent, and a trifluoroacetate mercury (II) salt is used in preference to the mercuric acetate (trifluoroacetate anion is a poorer nucleophile than acetate). Most 1 o , 2 o , 3 o alcohols can be successfully used for this reaction.
Mechanism
The mechanism of alkoxymercuration is similar to that of oxymercuration, with electrophillic addition of the mercuric species to the alkene. The alcohol nucleophile attacks the more substituted carbon of the three-membered ring via a S N 2 reaction. Finally, sodium borohydride (NaBH 4 ) provides a reductive demercuration to form the ether product.
Planning the Synthesis of an Ether using Alkoxymercuration
The key bond cleavage in the target molecule involves a C-O bond. Because unsymmetrical ethers have two unique C-O bonds, each can be broken to provide a unique set of reactants. After cleavage, the fragment with the oxygen will become an alcohol. The alkyl fragment will lose a hydrogen from a adjacent carbon to form an alkene. The main point to consider when choosing a possible synthesis pathways is the ability of the alkyl fragment to form an alkene.
Worked Example \(\PageIndex{2}\)
How would you prepare the following molecule using a alkoxymercuration?
- Answer
-
Analysis: The ether is symmetrical so each C-O bond of the ether can be cleaved to produce a set of starting materials for consideration. Pathway one shows a set of starting material which should work well for this reaction. The alcohol, methanol, can easily be used as a solvent. Although the alkene does not have a defined more and less substituted side, its symmetry will prevent a mixture of product from forming. The fragmentation for pathway 2 shows starting material which are not viable for this reaction. The alkyl fragment only has one carbon which cannot be used to form an alkene starting material. This means pathway 2 is not a viable method for the synthesis of the target molecule.
Pathway 1
Solution 1
Pathway 2
Exercise \(\PageIndex{1}\)
When preparing ethers using the Williamson ether synthesis, what factors are important when considering the nucleophile and the electrophile?
- Answer
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The nucleophile ideally should be very basic, yet not sterically hindered. This will minimize any elimination reactions from occurring. The electrophile should have the characteristics of a good S N 2 electrophile, preferably primary to minimize any elimination reactions from occurring.
Exercise \(\PageIndex{2}\)
How would you synthesize the following ethers? Keep in mind there are multiple ways. The Williamson ether synthesis, alkoxymercuration of alkenes, and also the acid catalyzed substitution.
(a) (b) (c) (d) (e)
- Answer
-
The Williamson ether syntheses require added catalytic base. Also, most of the halides can be interchanged, say for example for a -Br or a -Cl. Although, typically -I is the best leaving group.
(a)
(b)
(c)
(d)
Note, there is only one ether (also called a silyl ether, and often used as an alcohol protecting group.) The other group is an ester.
(e)
Exercise \(\PageIndex{3}\)
Draw the electron arrow pushing mechanism for the formation of diethyl ether in the previous problem.
- Answer
-
.
Exercise \(\PageIndex{4}\)
t-butoxycyclohexane can be prepared two different ways from an alkene and an alcohol, draw both possible reactions.
- Answer
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While both are possible, the top route is likely easier because both starting materials are a liquid.
Exercise \(\PageIndex{5}\)
Epoxides are often formed intramolecularly. Take for example this large ring, in a publication from 2016 [ J. Org. Chem. , 2016 , 81 (20) , pp 10029–10034]. If subjected to base, what epoxide would be formed? (Include stereochemistry)
- Answer
-
Exercise \(\PageIndex{6}\)
What reagents would you use to perform the following transformations?
(a)
(b)
(c)
- Answer
-
(a)
(b) Note the cis addition
(c)
An oxidation to an alcohol through hydroboration, and subsequent substitution with 2-bromopropane could also work, but this route provides the least likelihood of an elimination reaction occurring.
Exercise \(\PageIndex{7}\)
Predict the product of the following.
- Answer
-
The result is the production of dioxane, a common solvent.


8.6: Anyone Can Teach Writing
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- Cheryl E. Ball & Drew M. Loewe ed.
- West Virginia University via Digital Publishing Institute and West Virginia University Libraries
Author: Seth Kahn, English, West Chester University of Pennsylvania
Every time I hear somebody complain about the poor writing ability of today’s college graduates and students, I can’t help but wonder what people would think if they knew more about the circumstances of many college writing instructors, who go by the titles adjunct , contingent , term , or non-tenure-track faculty (I’ll use the word adjunct to stand in for all of those possibilities). In 2013, the Coalition on the Academic Workforce found that adjunct faculty members were teaching more than 70% of general education writing courses (first-year composition, or ENG 101, so to speak) in the United States, reinforcing results from the Association of Departments of English in 2007.
Who is Anybody?
Unlike the stereotype of a college professor—a giant office stuffed with books, an antique desk, expensive shabby chic clothes, you know the image—adjunct faculty often face difficult working conditions that, I believe, rest on the myth that anybody can teach writing. As of 2015, the current average salary for adjuncts in English is $2,700 per section; teaching 10 courses per year (which is a huge work load) would gross only $27,000 total. Yet, many campuses won’t offer full-time work (usually 8 to 10 courses a year) as a result of the Affordable Care Act’s requirement that employers provide health insurance to anybody working more than thirty hours per week. Because the pay is so low, it’s not unusual for adjuncts to become “freeway fliers” (teaching courses on multiple campuses) in order to cobble together enough money to live. Most adjuncts get no insurance or retirement benefits. Adjunct teaching loads can also change semester by semester (so that somebody might have two courses, then four, then one, then three, etc.). Often, adjuncts don’t know their schedules until a semester is about to begin, when it’s too late to find replacement work anywhere else, it’s too late to prepare for any new assignments, and it’s too late to update materials from prior semesters.
Adjuncts, almost by definition, have no job security or protection against being fired at will. On many campuses, adjuncts share incredibly cramped office space (I’ve heard of as many as 20 assigned to an office with four desks and one telephone), if they have offices at all; adjuncts often discover that the safest place to store books, laptops, phones and so on during classes is in their cars. Imagine the challenge of needing to have a confidential discussion with a student about a grade, or something sensitive somebody wrote in a paper, and not even having a semiprivate place to do it. Under those conditions, the truth of the matter is that nobody can teach writing, at least not well.
How we got to the point where so many faculty doing such important work can be treated so poorly is a long story. English Composition became an actual course in the late 1800s. According to historian Donna Strickland, for decades most teachers of writing were English professors trained primarily to teach literature, graduate students in literature (because there weren’t graduate programs in composition until the late 1970s), and faculty spouses or retirees who had at least taught high-school English. If you’ve never thought about specialized training for people who teach writing, that’s no surprise—the idea itself hasn’t been around for long. Because it’s low level (English 101 is about the lowest number a credit-bearing college course could have), and because of its content (traditionally grammar concerns, citation formats for research papers, and similar remediation that most people think students should have learned in high school), it’s not surprising that decision makers would conclude that anybody can do it.
Unfortunately, high-ranking administrators (deans, provosts, presidents, chancellors) on university campuses often use that conclusion to justify hiring and offering poor working conditions to adjunct faculty. If anybody can teach writing, the argument goes, then why pay experts well to do it? For whatever reason, even though it’s common to hear people complain about the poor writing skills of kids these days, it’s just as common to hear the assertion that teaching them to do better shouldn’t be hard. We hear these two arguments surprisingly often, and worse is that they reinforce each other. If the people who teach writing don’t need real professional training, then why treat them professionally? And if we’re not offering to treat them professionally, then why would anybody pursue the training necessary to do it well?
Poorly Trained Faculty Can’t Teach Writing Well
I wish it were obvious that people better trained to do something would do it better than people who aren’t trained as well. That feels like such a truism it’s hard to know what evidence to offer to support it. But here’s what we know: People without advanced training in writing pedagogy tend to rely on outdated ideas about writing, particularly that mastery of sentence-level skills like punctuation and word choice leads to mastery of more complex writing tasks. As early as the mid-1970s, researchers had established that this assumption was incorrect. (Several chapters on grammar instruction in this book show why.)
Mina Shaughnessy, who was a professor at the City University of New York during the period when the system became open admissions (so that anybody with a high-school diploma or equivalent could be admitted without question), published an influential book called Errors and Expectations in 1979. One of her key findings is that students struggle with sentence-level problems for any number of reasons that often have little to do with their mastery of mechanics; simply teaching them mechanical skills doesn’t solve those problems. Likewise, compositionist Patrick Hartwell reports in a well-regarded 1985 scholarly journal article titled “Grammar, Grammars, and the Teaching of Grammar” that students are much more likely to care about mechanical issues if they consider those issues in the context of their own writing purposes instead of worksheets and handouts.
Another outdated but still common practice among non-specialist writing teachers is teaching the modes of writing: narration, description, analysis, and argument. Teaching the modes suffers from the same basic problem as starting from the sentence level— the assumption that writers move neatly through these stages of complexity simply doesn’t hold. And Hartwell’s argument that students learn more when they work on writing they care about also applies here as well. A course built on a series of tasks students must do even if they have nothing to say that motivates or interests them is a course with, let’s just say, limited potential. But too many writing teachers are either required to teach such courses in programs designed by non-specialists, or they design courses this way because their models are the courses they took themselves.
I could keep listing practices that writing programs heavily dependent on adjunct faculty often use, but I hope the connection is clear—poorly treated instructors often work in programs without much regard for professional knowledge of the field, which both disempowers the instructors and reinforces the sense that what they do isn’t important. The system at too many colleges is stuck in a cycle of insisting that some work is lower value than other work, then using the fact that it abuses the people who do it as proof of its low value.
Exploring Alternatives
In its simplest form: Anybody who is trained and supported well and treated like a professional can teach writing. The key word is professional. The people teaching college writing courses have graduate degrees, often more than one— many have spent years on the job. Many conduct research into effective teaching or do other kinds of research that help them teach writing. Their training, experience, and expertise have earned them the support they need in order to do their work well. That support comes in many forms: resources they need like office space, computer access, photocopying and library privileges; engagement in their departments by being invited to participate in department meetings and curriculum development; job stability instead of constantly fluctuating schedules that may change suddenly and without explanation; and better compensation than most writing instructors currently receive.
All too often “Anybody can teach writing” translates to “It doesn’t matter who teaches writing,” and as a result, nobody needs to pay attention to writing instructors at all. In multiple surveys conducted over the last five years, adjunct faculty report that even more than pay and benefits, they want to be treated as professionals and colleagues. Also at the top of the list is job stability—knowing they have work from one semester, or better yet year, to the next helps them avoid freeway flying and alleviates the stress of uncertainty. It also helps them teach better by giving them time to reflect on and improve their teaching in a stable environment.
Finally, treating professionals as professionals means paying people more than many of them earn now. The Modern Language Association, which is one professional organization that represents faculty in English and Writing Studies, recommends a minimum salary of $10,700 per course. The National Council of Teachers of English, MLA, and many other organizations also recommend that faculty teaching at least half of a full-time load receive benefits (health insurance, retirement contributions) in proportion to their teaching load. The finances might be complex, but it’s clear that investing in faculty leads to better results; most recently, Amanda Griffith of Wake Forest University and Kevin Rask of Colorado College have found strong correlations between the instructional budgets of institutions and the earning power of graduates from those institutions. Investing wisely gets better results.
It’s fair to demand better for students, but not to demand magic from a system that’s currently built on a bad premise that anybody can teach writing. If you’re thinking about colleges for yourself or anybody you care about, ask how writing courses are staffed and how well supported the instructors are. The American Federation of Teachers offers a useful list of questions in their “Just Ask” brochure. If the answers sound like the college’s representatives think anybody can teach writing, think very hard about what they’re really saying.
Further Reading
Donna Strickland’s history The Managerial Unconscious in the History of Composition Studies helps illuminate the labor problems at the heart of commodified composition instruction. Laura McKenna’s “The Cost of an Adjunct,” Colleen Flaherty’s “The Case for Better Faculty Pay,” and the joint report by the Modern Language Association and Association of Departments of English, “Education in the Balance: A Report on the Academic Workforce in English,” illuminates the false economy behind under-paying and under-supporting writing teachers. Finally, the American Federation of Teachers webpage titled “Just Ask” gives students and parents information necessary to decide whether a particular college truly supports teaching and learning.
Keywords
academic labor, adjunct, contingent, non-tenure-track faculty, writing program administration/administrators, writing studies
Author Bio
Seth Kahn is an English professor at West Chester University of Pennsylvania. He researches and writes about academic labor, especially adjunct labor equity. He also serves on the board of the New Faculty Majority Foundation and has co-chaired labor/contingent faculty committees in several professional organizations.


1.2: Enumeration
Many basic problems in combinatorics involve counting the number of distributions of objects into cells—where we may or may not be able to distinguish between the objects and the same for the cells. Also, the cells may be arranged in patterns. Here are concrete examples.
Amanda has three children: Dawn, Keesha and Seth.
- Amanda has ten one dollar bills and decides to give the full amount to her children. How many ways can she do this? For example, one way she might distribute the funds is to give Dawn and Keesha four dollars each with Seth receiving the balance—two dollars. Another way is to give the entire amount to Keesha, an option that probably won't make Dawn and Seth very happy. Note that hidden within this question is the assumption that Amanda does not distinguish the individual dollar bills, say by carefully examining their serial numbers. Instead, we intend that she need only decide the amount each of the three children is to receive.
- The amounts of money distributed to the three children form a sequence which if written in non-increasing order has the form: \(a_1, a_2, a_3\) with \(a_1 \geq a_2 \geq a_3 \) and \(a_1 + a_2 + a_3 = 10\). How many such sequences are there?
- Suppose Amanda decides to give each child at least one dollar. How does this change the answers to the first two questions?
- Now suppose that Amanda has ten books, in fact the top 10 books from the New York Times best-seller list, and decides to give them to her children. How many ways can she do this? Again, we note that there is a hidden assumption—the ten books are all different.
- Suppose the ten books are labeled \(B_1, B_2, ... , B_{10} \). The sets of books given to the three children are pairwise disjoint and their union is {\(B_1, B_2, ... , B_{10} \)}. How many different sets of the form {\(S_1, S_2, S_3 \)} where \(S_1\), \(S_2\), and \(S_2\) are pairwise disjoint and \(S_1 \cup S_2 \cup S_3\) = {\(B_1, B_2, ... , B_{10} \)}?
- Suppose Amanda decides to give each child at least one book. How does this change the answers to the preceding two questions?
- How would we possibly answer these kinds of questions if ten was really ten thousand (OK, we're not talking about children any more!) and three was three thousand? Could you write the answer on a single page in a book?
A circular necklace with a total of six beads will be assembled using beads of three different colors. In Figure 1.1 , we show four such necklaces—however, note that the first three are actually the same necklace. Each has three red beads, two blues and one green. On the other hand, the fourth necklace has the same number of beads of each color but it is a different necklace.
- How many different necklaces of six beads can be formed using three reds, two blues and one green?
- How many different necklaces of six beads can be formed using red, blue and green beads (not all colors have to be used)?
- How many different necklaces of six beads can be formed using red, blue and green beads if all three colors have to be used?
- How would we possibly answer these questions for necklaces of six thousand beads made with beads from three thousand different colors? What special software would be required to find the exact answer and how long would the computation take?


4.5: Faucets, Fixtures, and Fixture Drains
Plumbing codes require that every house must have at least one toilet, lavatory sink, bathtub or shower, and a kitchen sink. These, and some other items referred to in this chapter, are known as fixtures. Clearances from finished walls and other fixtures are strictly regulated by code and most fixtures are provided with a manufacturer rough-in sheet and installation instructions. Codes regulate the materials that are used to manufacture plumbing fixtures, which must have smooth, impervious surfaces and be defect free. Fixtures that incorporate porcelain enameled surfaces must be capable of withstanding acid without damaging the fixture.
Pay attention to flow rates, volume, and water usage in selecting fixtures. One key aspect of sustainability in plumbing focuses on water conservation. Lavatory faucets and shower heads are two primary focal points of conserving water. The true water saving aspect of a shower is based on duration of each use. A handheld shower can be equipped with mechanisms to turn off the water flow for certain showering activities, such as lathering.
Faucets
Lavatory with 2 handles by Gwen Arkin is licensed under CC BY 4.0
Selecting which manufacturer’s product to install is based on cost, quality, and preferred faucet design which includes aesthetics, and functionality. Most master bathrooms and guest bathrooms have more expensive finishes than other bathrooms in a large house. Faucet finishes dictate the finishes used for drain assemblies and bathroom accessories which creates a color theme. Faucet accessories are available to create various themes. Water conservation faucets have aerators that reduce the flow of water to allow only 0.5 gallons per minute as opposed to a 1.0 gallons per minute. Single-handle faucets blend the hot and cold water with the use of one handle; two-handle designs require a user to operate both handles to achieve a desired temperature.
| Fixture | Single Handle | Two Handle | Three Handle |
|---|---|---|---|
| Bathtub | Yes | Yes | No |
| Bidet | No | Yes | No |
| Kitchen Sink | Yes | Yes | No |
| Large Capacity or Garden Tub | No | Yes | No |
| Laundry Sink | No | Yes | No |
| Lavatory Sink | Yes | Yes | No |
| Shower | Yes | Yes | No |
| Tub & Shower | Yes | Yes | *Yes |
Faucet Installation
A faucet installed on a sink through a countertop or through a tub platform, instead of through the fixture itself, is considered a deck mounted faucet. All faucets must be designed to prevent backflow of wastewater into the water distribution system and plumbing codes require that a faucet have an air gap or be protected with a vacuum breaker or approved check valve. Most faucets use a tightening nut and flat washer to secure the faucet to a fixture. A bathtub and shower faucet is usually installed during the rough-in phase of construction.
Faucet Water Supply: Connecting a water supply pipe to a faucet varies with the type of faucet and can be accomplished with many common connection methods. Some faucets connect to the water supply with male or female adapters to create a soldered connection, and others use a specially designed 3/8” OD supply tube that connects to the male threads of a faucet.
| Fixture | Male Thread | Female Thread | Solder | 3/8 Tubing |
|---|---|---|---|---|
| Bathtub | Yes | Yes | Yes | No |
| Bidet | Yes | No | No | Yes |
| Kitchen | Yes | No | No | Yes |
| Large Capacity or Whirlpool Tub | Yes | No | No | No |
| Laundry | Yes | No | No | No |
| Lavatory | Yes | No | No | Yes |
| Shower | Yes | Yes | Yes | No |
| Tub & Shower | Yes | Yes | Yes | No |
Drain Assemblies
Drain assemblies are purchased based on the specific fixture they serve. Codes dictate the minimum drain size serving a particular fixture, and all fixtures and drain assemblies are manufactured based on minimum code requirements. A trap adapter is installed to connect the stubout pipe to the p-trap outlet. Most codes dictate that the largest size foreign object that can enter a drainage system is 1/2” diameter.
Fixture Types and Assemblies
Toilet
Toilets are also known as water closets. Residential toilets must be self-cleaning during their flushing cycle and have a toilet seat installed. Water conservation fixtures ate rated at 1.6 gallons per flush (gpf) and 1.28 gpf. The most common residential toilet bowl design uses a siphon-jet flushing action. A tank handle activates a flushing cycle, and the water flows from the tank and enters the rim of the bowl. Small holes in the rim are angled to allow the water to create the vortex. The vortex (swirl) begins a siphoning action to evacuate waste from the bowl. The jet stream exits the rim and thrusts into the passageway of the toilet providing the initial thrust in the flushing process.
The water supply is located on the left side of all toilets. The stop and escutcheon are installed the same for a toilet as for other fixtures. The water is connected to a toilet using a a tank supply. A tank supply is a chrome-plated soft copper tubing having a flat end that connects the stop to the fill valve. More modern supply lines are compression fit, plastic tube, with some having a braided stainless steel sleeve that protects the line from swelling and bursting. The toilet is installed onto the closet (toilet) flange and sealed with a wax ring. Some wax seals have a plastic accessory called a horn molded into the wax.
Toilet ADA Requirements: Handicap fixtures must comply with the Americans with Disabilities Act (ADA). The tank handle of an ADA-compliant toilet must be located on the side of the tank that has the greatest distance from a sidewall. A handle located on the top of a tank typically meets ADA handle location regulations. The height of a toilet bowl from the floor, which includes the seat, is regulated by code. ADA codes dictate that the minimum height from a floor to a toilet seat is 16-1/2”; the maximum is 19-1/2”.
Toilet Piping Locations: The outlet distance of a standard toilet is 12” from the finished wall located behind the toilet (back wall). Lesser used 10” and 14” rough toilets are available in some toilet designs. Typically, one-piece toilets have different water rough-in location requirements than two-piece toilets. When selecting a one-piece toilet, always request the manufacturer’s data sheet, known as a roughin sheet, to confirm the water and drain pipes’ installation locations.
Toilet Bowl Shapes & Seats:
Residential toilets have round style bowls. Elongated bowls installed in many homes are usually considered a fixture upgrade. Codes require that an elongated (oval) bowl design be installed in commercial applications. When selecting an elongated bowl in place of a round bowl, be sure to check your local codes pertaining to the
minimum clearance in front of a toilet. Codes require that commercial seats be an open front type and not have a lid.
How To Replace and Install a Toilet
Query \(\PageIndex{1}\)
Fixing a Noisy Toilet & Other Problems with Fluidmaster
Query \(\PageIndex{2}\)
Lavatory sink
Also known as a lavatory, lav, or basin. Many types, shapes, and colors are available. Many homebuilders install cultured marble solid surface countertops with pre-molded sink basins, so the plumber does not install a separate sink. Other models of lavatory sinks include drop-in, under-mount, and vessel (similar to bowl setting on the countertop fed by a separate counter mounted faucet), The stub-out piping serving a lavatory is either 1 – 1/4” or 1 -1/2”. Most residential lavatories use a pop-up drain assembly. The overflow drain on a residential lavatory is an integral feature provided by the sink manufacturer.
- Lavatory ADA Requirements: Many lavatories are sold specifically for ADA compliance, but they are used more for commercial applications. ADA requirements require specific codes relating to the countertop height from the floor, the knee space under the sink, and the distances from the side and back walls.
-
Lavatory Styles:
Lavatory sinks are ordered based on shape, size, color, and mounting requirements, as well as the number of faucet holes and the distance between them.
- One-Piece: Cultured marble or other approved material, incorporates the countertop and sink being formed as one unit.
- Drop-in: A typical residential home utilizes a drop-in style lavatory that is either round or oval. A drop-in type lavatory sink requires a specific size hole cut into the countertop for the particular sink to be installed into the hole.
- Under Counter Mount: Considered an upgrade, under counter mounted sinks attach to the underside of finish grade holes based on templates of the sink in granite, composite, and other hard countertops. While some of these sinks are secured with retaining clip, many rely on modern adhesive caulks to adhere them to the countertop and support their full weight during use.
- Pedestal: A pedestal sink is a wall-hung sink with a decorative vertical leg known as a pedestal. The bowl is supported with brackets that are anchored to pieces of wood installed in the wall framing. The pedestal is not designed to be the sole support of the basin (bowl), but instead conceals the drain piping below the sink while providing a decorative styling.
- Lavatory Faucets: A lavatory faucet installs onto a plumbing fixture with various methods depending on the faucet type and manufacturer’s design. They can be installed in either fixture mount style sinks or deck mount (through the countertop) applications. The standard, and most common, lavatory sink has a 4” hole spread between the hot and cold handles. The term spread refers to the distance between the hot and cold faucet inlets. The two most common handle designs are a single-handle and a two-handle faucet. The middle hole of a three-hole lavatory sink is located in the center in the sink to receive the drainage operating assembly (pop-up) and/or a faucet spout connection.
- Lavatory Drains: The drain assembly for a lavatory faucet is known as a pop-up. The pop-up assembly consists of several different operating pieces that function as one unit. The pop-up rod is inserted through the faucet within the spout and connects to a linkage assembly below the sink that operates the pop-up plunger. All lavatory sink pop-up assemblies are 1 -1/4” tubular size. Most lavatories have an overflow port to eliminate water from rising over the rim of the fixture.
How To Install a Bathroom Vanity
Query \(\PageIndex{3}\)
Bathtub
Bathtubs are also known simply as tubs. A standard tub in a home is 5-0” in length and averages 30” wide. The depth of water a tub can hold varies with each specific tub design. Some tubs are sold separately, while others are sold with wall kits as a one-piece tub and shower unit or with various whirlpool features. A one-piece tub and shower unit is a fiberglass tub molded with the walls as a single unit. A tub is typically installed during the rough-in phase of a project. However, some drop-in style tubs, whirlpool tubs, and large-capacity garden tubs, are installed on top of tile or other solid surface are installed after the finished surface is complete.
The drain location is typically on one end of the tub that is known as the head wall. This is usually the same wall where the faucet is located and tubs are selected based on a left-hand or right-hand head wall design. The bottom surface of a tub slopes toward the drain. Every tub has an overflow hole (port) where a waste and overflow is installed.
- Bathtub Faucets: A tub faucet is intended to fill a tub. A tub faucet can either be deck-mounted or installed in a wall. Deck-mounted faucets are common for large-capacity tubs and whirlpool tubs. A faucet serving a tub or a shower is commonly referred to as a tub valve. Tub/shower valves can be used for tubs without a shower by installing a plug or cap in the shower riser port.
- Tub & Shower Faucets: A tub and shower combination faucet is capable of providing water for bathing or showering with the use of a diverter. Many different designs are available to divert the water flowing through a tub spout to flow through a showerhead. Diversion methods of a single-handle tub and shower faucet use either a diverter style tub spout or a push button diverter usually located directly below the faucet handle. A three-handle Tub and shower faucet design uses the middle handle as the diverter.
- Bathtub Drains: A bathtub drain assembly is called a bath waste and overflow (BW&O). A bathtub has an overflow port hole and a drain port that are always connected and installed as a pair (usually aligned with each other). Large capacity whirlpool tubs are more likely to have the holes in varying locations than a standard bathtub.
Installing a 1-Handle Posi-Temp Shower Valve: Copper to Copper
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Installation – Sterling Ensemble Medley
Query \(\PageIndex{5}\)
Shower
Not part of a tub and shower combination.
- Shower ADA Requirements: ADA-compliant shower bases have a lower or no threshold and a larger square foot area than a typical non-handicap shower base. Many shower designs use a seat within the shower, and most handicap shower designs must have a seat. If a seat is constructed in a tiled shower on a wood floor, the plumber must provide waterproofing (ex: PVC liner) to the seat as well as the shower base. One-piece shower units that have a seat are typically premolded into the design of the shower.
- Shower Faucets: A shower faucet is intended to serve a shower head. Shower faucets and tub valves are interchangeable, however a combination tub/shower valve that does not incorporate an integrated diverter may be used with the tub port plugged or capped.
- Shower Drains: A shower base constructed on a wooden, and sometimes concrete, floor prepared to be covered with ceramic tile requires a safety pan. Most safety pans use a polyvinyl chloride (PVC) liner. A three-piece shower drain is required to ensure that water does not seep around the drain. The threaded top portion is adjustable to allow various tile thicknesses. The middle portion receives the top threaded portion and is bolted to the bottom portion. The bottom portion rests flush with the wood floor and connects the piping to the p-trap.
Oatey Shower Pan Liner Installation
Query \(\PageIndex{6}\)
Kitchen sink
The most common residential kitchen sinks have either a single bowl or a double bowl. Most kitchen sinks are surface mounted and are installed into a countertop during the trim-out phase of construction. Surface-mounted sinks are also known as self-rimming sinks and typically have holes for installing the faucet directly onto the sink. Solid surface countertops can utilize a sink mounted from under the countertop and incorporate deck mounted faucets. The common types of kitchen sinks used in residential construction are stainless steel and cast iron. The weight of a cast iron sink provides the necessary stability to maintain its permanent position on the countertop and does not require retainer clips when installed with an appropriate adhesive caulk. A plumber applies caulking to the edge of the cutout area of the countertop and places the cast iron sink into the hole. A stainless steel sink requires the sink to be fastened to the countertop using fastening clips provided. A kitchen sink can also serve a garbage disposal and dishwasher
- Kitchen Faucets: A kitchen faucet is usually installed before installing a kitchen sink. A kitchen faucet has a swivel spout that allows the water flow to be used in each bowl of a kitchen sink. The most common type of kitchen faucet requires a sink to have three faucet holes that are 4” apart or 8” from the hot and cold water supply connections to the faucet. Most kitchen sinks are offered as a three-hole design, but many popular faucet designs have a pull-out spout that is also the spray unit. If a separate handheld sprayer is used, a four-hole model sink must be selected. Most pull-out spout faucets are manufactured with an integral check valve. The center hole of a three-hole sink is normally aligned with the center of the sink.
- Kitchen Sink Drains: The drainage system serving a kitchen sink is connected to the fixture with a basket strainer. Regardless of the type or style of kitchen sink, the connection of the drainage system is the same, and all have 1 -1/2” drain connections. A rubber gasket is placed over the basket strainer from under the sink. A fiber (cardboard)-type gasket is placed between the tightening nut and rubber gasket.
Delta Faucets-How to Install a Single Handle Kitchen Faucet
Query \(\PageIndex{7}\)
Installing a Pfister 2-Handle Kitchen Faucet with a Sidespray – Harbor Collection
Query \(\PageIndex{8}\)
How-to Install a Stainless Steel Drop-In Sink | Moen Installation Video
Query \(\PageIndex{9}\)
How-to Install a Stainless Steel Undermount Kitchen Sink | Moen Installation
Query \(\PageIndex{10}\)
Laundry Sink
Also known as a laundry tray or utility sink. They are typically installed in the same room as a washing machine, or in a garage or workshop. Two most common types of laundry sink
designs are wall mounted, which requires a plumber to install wood backing in the
wall during the rough-in and a hanger bracket installed during trim-out; or with the four legs secured to the floor which requires a drill to install anchors into the floor.
- Laundry Sink Faucets: Laundry tub faucets usually have a 4” spread design and most laundry sink faucets have a swivel spout. Some laundry sink faucets have a hose thread on the outlet portion of a spout to allow a garden hose to be connected. For a hose-end spout to be legally installed, it must have a vacuum breaker to prevent backflow into the potable water supply.
- Laundry Sink Drains: The minimum size drain allowed by code to serve a laundry sink is 1 -1/2”. Most quality types of laundry sink basket strainers have a removable strainer. Because a laundry sink can receive discharge from a washing machine, a removable strainer should be installed.
How to Install the UTILATUB® Laundry/Utility Tub
Query \(\PageIndex{11}\)
Bidet
A bidet is a personal hygiene fixture that is usually matched in style and color with, and placed adjacent to, a toilet. A bidet’s faucet and drain assembly are sold based on the fixture design and typically sold as a pair with the toilet. The water supply must be protected against backflow with a vacuum breaker assembly. The water supply serving a bidet is typically a 3/8” supply tube similar to a lavatory. The base of a bidet, like a toilet, typically has two mounting holes to anchor the bidet to the floor.
- Bidet Faucets: A bidet faucet must be compatible with the fixture based on the faucet hole design for installing a particular faucet and a vacuum breaker if required. A vacuum breaker is required by code if the hygiene sprayer is located in the bowl area of a bidet because it is below the flood level rim of the bidet.
- Bidet Drains: A bidet’s drain assembly is very similar to a lavatory’s pop-up assembly. Most bidets that use vacuum breakers have a dedicated hole in the fixture while others are served with the backflow device installed in the piping system.


8.4: Comparing Two Independent Population Proportions
Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance in the sampling. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the two population proportions.
Like the case of differences in sample means, we construct a sampling distribution for differences in sample proportions: \(\left(P_{A}^{\prime}-P_{B}^{\prime}\right)\) where \(P_{A}^{\prime}={\frac{x_A}{n_{A}}}\) and \(P_{B}^{\prime}={\frac{x_B}{n_{B}}}\) are the sample proportions for the two sets of data in question. \(X_A\) and \(X_B\) are the number of observations in each sample group of interest, respectively, and \(n_A\) and \(n_B\) are the respective sample sizes from the two groups. Again we go the Central Limit theorem to find the distribution of this sampling distribution for the differences in sample proportions. And again we find that this sampling distribution, like the ones past, are normally distributed as proved by the Central Limit Theorem.
Generally, the null hypothesis allows for the test of a difference of a particular value, \(\delta_{0}\), just as we did for the case of differences in means.
\[H_{0} : P_{A}-P_{B}=\delta_{0}\nonumber\]
\[H_{1} : P_{A}-P_{B} \neq \delta_{0}\nonumber\]
\[\textbf{The confidence interval formula is:}\nonumber\]
\[\left(P_{2}^{\prime}-P_{1}^{\prime}\right) \pm z_\frac{\alpha}{2} * \sqrt{\frac{P_{1}^{\prime} *\left(1-P_{1}^{\prime}\right)}{n_{1}}+\frac{P_{2}^{\prime} *\left(1-P_{2}^{\prime}\right)}{n_{2}}}\nonumber\]
Example \(\PageIndex{1}\)
A bank has recently acquired a new branch and thus has customers in this new territory. They are interested in the default rate in their new territory. They wish to test the hypothesis that the default rate is different from their current customer base. They sample 200 files in area A, their current customers, and find that 20 have defaulted. In area B, the new customers, another sample of 200 files shows 12 have defaulted on their loans. At a 1% level of significance can we say that the default rates are the same or different?
- Answer
-
This is a test of proportions. We know this because the underlying random variable is binary, default or not default. Further, we know it is a test of differences in proportions because we have two sample groups, the current customer base and the newly acquired customer base. Let A and B be the subscripts for the two customer groups. Then p A and p B are the two population proportions we wish to test.
Random Variable:
\(P_{A}^{\prime}-P_{B}^{\prime}\) = difference in the proportions of customers who defaulted in the two groups.
\(H_{0} : P_{A}=P_{B}\)
\(H_{a} : P_{A} \neq P_{B}\)
The words "is different" tell you the test is two-tailed.
Distribution for the test:
Estimated proportion for group A: \(P^{\prime}_{A}=\frac{x_{A}}{n_{A}}=\frac{20}{200}=0.1\)
Estimated proportion for group B: \(P^{\prime}_{B}=\frac{x_{B}}{n_{B}}=\frac{12}{200}=0.06\)
The estimated confidence interval for the difference between the two groups is: \[\left(P_{2}^{\prime}-P_{1}^{\prime}\right) \pm z_\frac{\alpha}{2} * \sqrt{\frac{P_{1}^{\prime} *\left(1-P_{1}^{\prime}\right)}{n_{1}}+\frac{P_{2}^{\prime} *\left(1-P_{2}^{\prime}\right)}{n_{2}}} = [-.03, .11]\nonumber\]
Make a decision: Since the calculated confidence interval contains 0, we cannot reject \(H_0\).
Conclusion: At a 1% level of significance, from the sample data, there is not sufficient evidence to conclude that there is a difference between the proportions of customers who defaulted in the two groups.
Exercise \(\PageIndex{1}\)
Two types of valves are being tested to determine if there is a difference in pressure tolerances. Fifteen out of a random sample of 100 of Valve A cracked under 4,500 psi. Six out of a random sample of 100 of Valve B cracked under 4,500 psi. Test at a 5% level of significance.


3.2: Inversion
Inversion offers a way to reflect points across a circle. This transformation plays a central role in visualizing the transformations of non-Euclidean geometry, and this section is the foundation of much of what follows.
Suppose \(C\) is a circle with radius \(r\) and center \(z_0\text{.}\) Inversion in the circle \(C\) sends a point \(z \neq z_0\) to the point \(z^*\) defined as follows: First, construct the ray from \(z_0\) through \(z\text{.}\) Then, let \(z^*\) be the unique point on this ray that satisfies the equation
\[ |z-z_0|\cdot|z^*-z_0| = r^2\text{.} \]
The point \(z^*\) is called the symmetric point to \(z\) with respect to \(C\text{.}\)
Inversion in a circle centered at \(z_0\) is a transformation on the set \(\mathbb{C}-\{z_0\}\) consisting of all complex numbers except \(z_0\text{.}\) We usually denote inversion in the circle \(C\) by by \(i_C(z) = z^*\text{.}\) In the next section we will discuss how to extend this transformation in a way to include the center \(z_0\text{.}\)
You will work through several features of circle inversions in the exercises, including how to construct symmetry points with compass and ruler (see Figure \(\PageIndex{5}\)). We note here that \(i_C\) fixes all the points on the circle \(C\text{,}\) and points inside the circle get mapped to points outside the circle and vice versa. The closer \(z\) gets to the center of the circle, the further \(i_C(z)\) gets from the circle.
The unit circle in \(\mathbb{C}\text{,}\) denoted \(\mathbb{S}^1\text{,}\) is the circle with center \(z_0 = 0\) and radius \(r = 1\text{.}\) The equation for the point \(z^*\) symmetric to a point \(z \neq 0\) with repsect to \(\mathbb{S}^1\) thus reduces from \(|z-z_0|\cdot|z^*-z_0| = r^2\) to
\[ |z|\cdot|z^*| = 1\text{.} \]
Moreover, \(z^*\) is just a scaled version of \(z\) since they are on the same ray through the origin. That is, \(z^* = kz\) for some positive real number \(k\text{.}\) Plug this description of \(z^*\) into the symmetry point equation to see \(|z| \cdot |kz| = 1\text{,}\) which implies \(k = 1/|z|^2\text{.}\) Thus, \(z^*=(1/|z|^2)z\text{.}\) Moreover, \(|z|^2 = z \cdot \overline{z}\text{,}\) so inversion in the unit circle \(\mathbb{S}^1\) may be written as
\[ i_{\mathbb{S}^1}(z) = 1/\overline{z}\text{.} \]
The following formula for inversion about an arbitrary circle can be obtained by composition of inversion in the unit circle with some general linear transformations. The details are left to Exercise \(3.2.1\).
Inversion in the circle \(C\) centered at \(z_0\) with radius \(r\) is given by
\begin{align*} i_C(z) & = \frac{r^2}{(\overline{z-z_0})} + z_0\text{.} \end{align*}
Below we have inverted a circle, the letter ‘M,’ and a small grid across the circle \(C\) centered at \(z_0\text{.}\) It looks as if the image of the circle is another circle, which we will soon prove is the case. We will also prove that lines not intersecting the center of \(C\) get inverted into circles. It follows that the line segments in the ‘M’ get mapped to arcs of circles.
As Example \(\PageIndex{2}\) suggests, the distinction between lines and circles gets muddied a bit by inversion. A line can get mapped to a circle and vice versa. In what follows, it will be helpful to view reflection in a line and inversion in a circle as special cases of the same general map. To arrive at this view we first make lines and circles special cases of the same general type of figure.
A cline is a Euclidean circle or line. Any cline can be described algebraically by an equation of the form
\[ cz\overline{z} + \alpha z + \overline{\alpha}\overline{z} + d = 0 \]
where \(z = x + yi\) is a complex variable, \(\alpha\) is a complex constant, and \(c, d\) are real numbers. If \(c = 0\) the equation describes a line, and if \(c \neq 0\) and \(|\alpha|^2 > cd\) the equation describes a circle.
The word “cline” (pronounced ‘Klein’) might seem a bit forced, but it represents the shift in thinking we aim to achieve. We need to start thinking of lines and circles as different manifestations of the same general class of objects. What class? The class of clines.
Letting \(\alpha = a + bi\) and \(z = x + yi\text{,}\) the cline equation \(cz\overline{z} + \alpha z + \overline{\alpha} \overline{z} + d = 0\) can be written as
\[ c(x^2+y^2)+[ax - by + (ay+bx)i]+[ax-by - (ay+bx)i] + d = 0 \]
which simplifies to
\[ c(x^2+y^2) + 2(ax - by) + d = 0\text{.} \]
If \(c = 0\) then we have the equation of a line, and if \(c \neq 0\) we have the equation of a circle, so long as \(a^2 + b^2 > cd\text{.}\) In this case, the equation can be put into standard form by completing the square. Let's run through this.
If \(c \neq 0\text{,}\)
and we have the equation of a circle so long as the right-hand side (the radius term) is positive. In other words, we have the equation of a circle so long as \(a^2 + b^2>cd\text{.}\) We summarize this information below.
Given \(c, d \in \mathbb{R}\text{,}\) \(\alpha \in \mathbb{C}\text{,}\) if \(c \neq 0\text{,}\) the cline equation
\[ cz\overline{z} + \alpha z + \overline{\alpha} \overline{z} + d = 0 \]
gives a circle with center \(z_0\) and radius \(r\text{,}\) where
\[ z_0=\bigg(-\frac{\text{Re}(\alpha)}{c}, \frac{\text{Im}(\alpha)}{c}\bigg)~~~\text{and}~~~ r = \sqrt{\frac{|\alpha|^2-cd}{c^2}}\text{,} \]
so long as \(|\alpha|^2 > cd\text{.}\) If \(c = 0\text{,}\) the cline equation gives a line.
From now on, if you read the phrase “inversion in a cline,” know that this means inversion in a circle or reflection about a line, and if someone hands you a cline \(C\text{,}\) you might say, “Thanks! By the way, is this a line or a circle?”
We note here the construction of a cline through three points in \(\mathbb{C}\text{.}\) This construction is used often in later chapters to generate figures in non-Euclidean geometry.
There exists a unique cline through any three distinct points in \(\mathbb{C}\text{.}\)
- Proof
-
Suppose \(u\text{,}\) \(v\text{,}\) and \(w\) are distinct complex numbers. If \(v\) is on the line through \(u\) and \(w\) then this line is the unique cline through the three points. Otherwise, the three points do not lie on a single line, and we may build a circle through these three points as demonstrated in Figure \(\PageIndex{1}\). Construct the perpendicular bisector to segment \(uv\text{,}\) and the perpendicular bisector to segment \(vw\text{.}\) These bisectors will intersect because the three points are not collinear. If we call the point of intersection \(z_0\text{,}\) then the circle centered at \(z_0\) through \(w\) is the unique cline through the three points.
Inversion in a circle maps clines to clines. In particular, if a cline goes through the center of the circle of inversion, its image will be a line; otherwise the image of a cline will be a circle.
- Proof
-
We prove the result in the case of inversion in the unit circle. The general proof will then follow, since any inversion is the composition of this particular inversion together with translations and dilations, which also preserve clines by Theorem \(3.1.2\).
Suppose the cline \(C\) is described by the cline equation
\[ cz\overline{z}+\alpha z + \overline{\alpha} \overline{z} + d = 0\text{,} \]
where \(c, d \in \mathbb{R}\text{,}\) \(\alpha \in \mathbb{C}\text{.}\)
We want to show that the image of this cline under inversion in the unit circle, \(i_{\mathbb{S}^1}(C)\text{,}\) is also a cline. Well, \(i_{\mathbb{S}^1}(C)\) consists of all points \(w = 1/\overline{z}\text{,}\) where \(z\) satisfies the cline equation for \(C\text{.}\) We show that all such \(w\) live on a cline.
If \(z \neq 0\) then we may multiply each side of the cline equation by \(1/(z\cdot \overline{z})\) to obtain
\[ c +\alpha \frac{1}{\overline{z}} + \overline{\alpha} \frac{1}{z} +d\frac{1}{z}\frac{1}{\overline{z}}=0\text{.} \]
But since \(w = 1/\overline{z}\) and \(\overline{w} = 1/z\text{,}\) this equation reduces to
\[ c +\alpha\cdot w + \overline{\alpha} \cdot \overline{w} + dw\overline{w}=0\text{,} \]
or
\[ dw\overline{w} + \alpha \cdot w + \overline{\alpha} \cdot \overline{w} + c = 0\text{.} \]
Thus, the image points \(w\) form a cline equation. If \(d = 0\) then the original cline \(C\) passed through the origin, and the image cline is a line. If \(d \neq 0\) then \(C\) did not pass through the origin, and the image cline is a circle. (In fact, we must also check that \(|\alpha |^2 > dc\text{.}\) This is the case because the original cline equation ensures \(|\alpha|^2 > cd\text{.}\))
We will call two clines orthogonal if they intersect at right angles. For instance, a line is orthogonal to a circle if and only if it goes through the center of the circle. One very important feature of inversion in \(C\) is that clines orthogonal to \(C\) get inverted to themselves. To prove this fact, we first prove the following result, which can be found in Euclid's Elements (Book III, Proposition \(36\)).
Suppose \(C\) is the circle with radius \(r\) centered at \(o\text{,}\) and \(p\) is a point outside \(C\text{.}\) Let \(s = |p-o|\text{.}\) If a line through \(p\) intersects \(C\) at points \(m\) and \(n\text{,}\) then
\[ |p-m|\cdot|p-n|=s^2-r^2\text{.} \]
- Proof
-
Suppose the line through \(p\) does not pass through the center of \(C\text{,}\) as in the diagram below. Let \(q\) be the midpoint of segment \(mn\text{,}\) and let \(d = |q - o|\) as in the diagram. Note also that the line through \(q\) and \(o\) is the perpendicular bisector of segment \(mn\text{.}\) In particular, \(|m-q|=|q-n|\text{.}\)
The Pythagorean theorem applied to \(\Delta pqo\) gives
\[|p-q|^2 + d^2 = s^2, \label{3.2.1}\]
and the Pythagorean theorem applied to \(\Delta nqo\) gives
\[ |q-n|^2 + d^2 = r^2. \label{3.2.2}\]
By subtracting equation (\(\ref{3.2.2}\)) from (\(\ref{3.2.1}\)), we have
\[ |p-q|^2-|q-n|^2 = s^2-r^2\text{,} \]
which factors as
\[ (|p-q|-|q-n|)(|p-q|+|q-n|) = s^2-r^2\text{.} \]
Since \(|p-q|-|q-n| = |p-m|\) and \(|p-q| + |q-n| = |p-n|\text{,}\) the result follows.
The case that the line through \(p\) goes through the center of \(C\) is left as an exercise.
We note that the quantity \(s^2-r^2\) in the previous lemma is often called the power of the point \(p\) with respect to the circle \(C\text{.}\) That is, if circle \(C\) has radius \(r\) and a point \(p\) is a distance \(s\) from the center of \(C\) then the quantity \(s^2-r^2\) is called the power of the point \(p\text{.}\)
Suppose \(C\) is a circle in \(\mathbb{C}\) centered at \(z_0\text{,}\) and \(z\neq z_0\) is not on \(C\text{.}\) A cline through \(z\) is orthogonal to \(C\) if and only if it goes through \(z^*\text{,}\) the point symmetric to \(z\) with respect to \(C\text{.}\)
- Proof
-
Assume \(C\) is the circle of radius \(r\) centered at \(z_0\text{,}\) and \(D\) is a cline through a point \(z \neq z_0\) not on \(C\text{.}\) Let \(z^*\) denote the point symmetric to \(z\) with respect to \(C\text{.}\)
First, suppose \(D\) is a line through \(z\text{.}\) A line through \(z\) passes through \(z^*\) if and only if it passes through the center of \(C\text{,}\) which is true if and only if the line is orthogonal to \(C\text{.}\) Thus, the line \(D\) through \(z\) contains \(z^*\) if and only if it is orthogonal to \(C\text{,}\) and the theorem is proved in this case.
Now assume \(D\) is a circle through \(z\text{.}\) Let \(o\) and \(k\) denote the center and radius of \(D\text{,}\) respectively. Set \(s = |z_o-o|\text{,}\) and let \(t\) denote a point of intersection of \(C\) and \(D\) as pictured below.
We must argue that \(C\) and \(D\) are orthogonal if and only if \(z^*\) is on \(D\text{.}\) Now, \(C\) and \(D\) are orthogonal if and only if \(\angle otz_0\) is right, which is the case if and only if \(r^2 = s^2-k^2\) by the Pythagorean theorem. Applying Lemma \(3.2.1\) to the point \(z_0\) (which is outside \(D\)) and the line through \(z_0\) and \(z\text{,}\) we see that
\[ |z_0-z|\cdot|z_0-w| = s^2-k^2, \label{3.2.3}\]
where \(w\) is the second point of intersection of the line with circle \(D\text{.}\)
Note also that as symmetric points, \(z\) and \(z^*\) satisfy the equation
\[ |z_0-z|\cdot|z_0-z^*| = r^2. \label{3.2.4}\]
Thus, if we assume \(z^*\) is on \(D\text{,}\) then it must be equal to the point \(w\text{,}\) in which case equations (\(\ref{3.2.3}\)) and (\(\ref{3.2.4}\)) above tell us \(s^2-k^2 = r^2\text{.}\) It follows that \(D\) is orthogonal to \(C\text{.}\) Conversely, if \(D\) is orthogonal to \(C\text{,}\) then \(s^2-k^2=r^2\text{,}\) so \(|z_0-w|=|z_0-z^*|\text{.}\) Since \(z^*\) and \(w\) are both on the ray \(\overrightarrow{z_0z}\) it must be that \(z^* = w\text{.}\) In other words, \(z^*\) is on \(D\text{.}\)
Inversion in \(C\) takes clines orthogonal to \(C\) to themselves.
Inversion in a cline preserves angle magnitudes.
- Proof
-
The result was stated for lines in Theorem \(3.1.5\). Here we assume \(C\) is a circle of inversion. Consider two curves \(\boldsymbol{r_1}\) and \(\boldsymbol{r_2}\) that intersect at a point \(z\) that is not on \(C\) or at the center of \(C\text{.}\) Recall, \(\angle(\boldsymbol{r_1},\boldsymbol{r_2}) = \angle(L_1,L_2)\) where \(L_i\) is the line tangent to curve \(\boldsymbol{r_i}\) at \(z\text{,}\) for \(i = 1,2\text{.}\) We may describe this angle with two circles \(C_1\) and \(C_2\) tangent to the tangent lines \(L_1\) and \(L_2\text{,}\) respectively, with the additional feature that the circles meet the circle of inversion \(C\) at right angles, as in Figure \(3.2.2\). Indeed, \(C_1\) is the circle through \(z\) and \(z^*\) whose center is at the intersection of lines \(m_1\) and \(k\text{,}\) where \(m_1\) is the line through \(z\) that is perpendicular to \(L_1\text{,}\) and \(k\) is the perpendicular bisector of segment \(zz^*\text{.}\) Circle \(C_2\) also goes through \(z\) and \(z^*\text{,}\) and its center is on the intersection of \(k\) and the line \(m_2\) through \(z\) that is perpendicular to \(L_2\text{.}\)
The advantage to describing \(\angle(L_1,L_2)\) with these circles is that the image of the angle, \(\angle(i_C(L_1),i_C(L_2))\text{,}\) is also described by these two circles, at their other intersection point \(z^*\text{.}\) Notice that these angles will have opposite signs. For instance, in Figure \(3.2.2\), our initial angle is negative, described by sweeping arc \(C_1\) clockwise onto \(C_2\text{,}\) but in the image, we sweep \(i_C(C_1)\) counterclockwise onto \(i_C(C_2)\text{.}\) We leave it as an exercise for the reader to check that the angle of intersection of \(C_1\) and \(C_2\) at \(z^*\) is the same magnitude as the angle between \(C_1\) and \(C_2\) at \(z\text{.}\)
Now we show that inversion preserves angle magnitudes for angles that occur on the circle \(C\) (i.e., \(z\) is on \(C\)). Let \(C^\prime\) be a concentric circle to \(C\text{.}\) Then \(i_C(z) = S\circ i_{C^\prime}\) where \(S\) is a dilation of \(\mathbb{C}\) whose fixed point is the common center of circles \(C\) and \(C^\prime\) (see Exercise \(3.2.12\)). Since our angle is not on circle \(C^\prime\text{,}\) \(i_{C^\prime}\) preserves the magnitude of the angle by reason of the preceding argument. The dilation \(S\) preserves angles according to Theorem \(3.1.5\). Thus \(i_C\) preserves angle magnitudes as well. We leave the case of the angle occurring at the origin to the next section. Bearing that exception in mind, this completes the proof.
Another important feature of inversion in a cline is that it preserves symmetry points.
Let \(i_C\) denote inversion in a cline \(C\text{.}\) If \(p\) and \(q\) are symmetric with respect to a cline \(D\text{,}\) then \(i_C(p)\) and \(i_C(q)\) are symmetric with respect to the cline \(i_C(D)\text{.}\)
- Proof
-
Assume \(C\) is the cline of inversion, and assume \(p\) and \(q\) are symmetric with respect to a cline \(D\) as in Figure \(3.2.3\) (where \(C\) and \(D\) are represented as circles).
We may construct two clines \(E\) and \(F\) that go through \(p\) and \(q\text{.}\) In the figure, cline \(E\) is a circle and cline \(F\) is a line. These clines intersect \(D\) at right angles (Theorem \(3.2.3\)). Since inversion preserves clines and angle magnitudes, we know that \(E^*=i_C(E)\) and \(F^*=i_C(F)\) are clines intersecting the cline \(D^*=i_C(D)\) at right angles. Both \(E^*\) and \(F^*\) contain \(p^*=i_C(p)\text{,}\) so they both contain the point symmetric to \(p^*\) with respect to \(D^*\) (Theorem \(3.2.3\)), but the only other point common to both \(E^*\) and \(F^*\) is \(q^* = i_C(q)\text{.}\) Thus, \(p^*\) and \(q^*\) are symmetric with respect to \(D^*\text{.}\)
We close the section with two applications of inversion.
Let \(p,q\) be distinct points in \(\mathbb{C}\text{,}\) and \(k > 0\) a positive real number. Let \(D\) consist of all points \(z\) in \(\mathbb{C}\) such that \(|z-p| = k|z - q|.\) Then \(D\) is a cline.
- Proof
-
If \(k = 1\text{,}\) the set \(D\) is a Euclidean line, according to Theorem \(2.4.1\), so we assume \(k \neq 1\text{.}\) Let \(C\) be the circle centered at \(p\) with radius \(1\). Suppose \(z\) is an arbitrary point in the set \(D\text{.}\) Inverting about \(C\text{,}\) let \(z^* = i_C(z)\) and \(q^* = i_C(q)\) as in the following diagram.
Observe first that \(\Delta pz^*q^*\) and \(\Delta pqz\) are similar.
Indeed, \(|p-z|\cdot|p-z^*|=1=|p-q|\cdot|p-q^*|\) by the definition of the inversion transformation, so we have equal side-length ratios
\[ \frac{|p-z^*|}{|p-q|}=\frac{|p-q^*|}{|p-z|}\text{,} \]
and the included angles are equal, \(\angle q^*pz^* = \angle qpz\text{.}\)
It follows that
\[ \frac{|z-q|}{|p-q|}=\frac{|z^*-q^*|}{|z^*-p|}\text{,} \]
from which we derive
\begin{align*} |z^*-q^*|& =|z^*-p|\cdot\frac{|z-q|}{|p-q|}\\ & =[|z^*-p|\cdot|z-p|]\cdot\frac{|z-q|}{|z-p|}\cdot\frac{1}{|p-q|}\\ & = 1 \cdot \frac{1}{k} \cdot \frac{1}{|p-q|}\text{.} \end{align*}
Thus, the set \(D\) of all points \(z\) satisfying \(|z-p|=k|z-q|\) has image \(i_C(D)\) under this inversion consisting of all points \(z^*\) on a circle centered at \(q^*\) with radius \((k|p-q|)^{-1}\text{.}\) Since inversion preserves clines and \(p\) is not on \(i_C(D)\text{,}\) it follows that \(D\) itself is a circle.
As we let \(k\) run through all positive real numbers, we obtain a family of clines, called the circles of Apollonius of the points \(\boldsymbol{p}\) and \(\boldsymbol{q}\) . We note that \(p\) and \(q\) are symmetric with respect to each cline in this family (see Exercise \(3.3.2\)).
Suppose we have two clines that do not intersect, and at least one of them is a circle. Then there exist two points, \(p\) and \(q\text{,}\) that are symmetric with respect to both clines.
- Proof
-
First, assume one cline is a line \(L\text{,}\) and the other is a circle \(C\) centered at the point \(z_0\) as pictured in Figure \(3.2.4\). Let \(L_1\) be the line through \(z_0\) that is perpendicular to \(L\text{,}\) and let \(z_1\) be the point of intersection of \(L\) and \(L_1\text{.}\) Next, construct the circle \(C_1\) having the diameter \(z_0z_1\text{.}\) Circle \(C_1\) intersects circle \(C\) at some point, which we call \(t\text{.}\) Notice that \(\angle z_0 t z_1\) is right, and so the circle \(C_2\) centered at \(z_1\) through \(t\) is orthogonal to \(C\text{.}\) Furthermore, the center of \(C_2\text{,}\) \(z_1\text{,}\) lies on line \(L\text{,}\) so \(C_2\) is orthogonal to \(L\text{.}\) Let \(p\) and \(q\) be the two points at which \(C_2\) intersects \(L_1\text{.}\) By construction, and by using Theorem \(3.2.3\), \(p\) and \(q\) are symmetric to both \(C\) and \(L\text{.}\)
Now, suppose \(C_1\) and \(C_2\) are circles that do not intersect. We may first perform an inversion in a circle \(C\) that maps \(C_1\) to a line \(C_1^*\text{,}\) and \(C_2\) to another circle, \(C_2^*\text{,}\) as suggested in Figure \(3.2.5\) (any circle \(C\) centered on a point of \(C_1\) works). Then by reason of the preceeding argument, there exist two points \(p\) and \(q\) that are symmetric with respect to \(C_1^*\) and \(C_2^*\text{.}\) Since inversion preserves symmetry points, \(i_C(p)\) and \(i_C(q)\) are symmetric with respect to both \(i_C(C_1^*)\) and \(i_C(C_2^*)\text{.}\) But \(i_C(C_1^*) = C_1\) and \(i_C(C_2^*) = C_2\) so we're found two points symmetric to both \(C_1\) and \(C_2\text{.}\) (In fact, we have one exception. If \(C_1\) and \(C_2\) are concentric circles, this strategy will produce points \(i_C(p)\) and \(i_C(q)\text{,}\) one of which is the center of \(C\text{,}\) and we have not yet extended the notion of inversion to include the center. We do so in the next section in such a way that the theorem applies to this exceptional case as well.)
Exercises
Prove the general formula for inversion in a circle \(C\) centered at \(z_0\) with radius \(r\text{.}\) In particular, show in this case that
\[ i_C(z) = \frac{r^2}{(\overline{z-z_0})} + z_0. \]
Constructing the symmetric point to \(z\) when \(z\) is inside the circle of inversion.
Prove that for a point \(z\) inside the circle \(C\) with center \(z_0\) (Figure \(3.2.6\)(a)), the following construction finds the symmetry point of \(z\text{.}\)
- Draw the ray from \(z_0\) through \(z\text{.}\)
- Construct the perpendicular to this ray at \(z\text{.}\) Let \(t\) be a point of intersection of this perpendicular and \(C\text{.}\)
- Construct the radius \(z_0t\text{.}\)
- Construct the perpendicular to this radius at \(t\text{.}\) The symmetric point \(z^*\) is the point of intersection of this perpendicular and ray \(\overrightarrow{z_oz}\text{.}\)
Constructing the symmetric point to \(z\) when \(z\) is outside the circle of inversion.
Prove that for a point \(z\) outside the circle \(C\) with center \(z_0\) (Figure \(3.2.6\)(b)), the following construction finds the symmetry point of \(z\text{.}\)
- Construct the circle having diameter \(z_0z\text{.}\) Let \(t\) be a point of intersection of the two circles.
- Construct the perpendicular to \(z_0z\) through \(t\text{.}\) Let \(z^*\) be the intersection of this perpendicular with segment \(z_0z\text{.}\)
Suppose \(T_1\) is inversion in the circle \(|z| = r_1\text{,}\) and \(T_2\) is inversion in the circle \(|z| = r_2\text{,}\) where \(r_1, r_2 > 0\text{.}\) Prove that \(T_2 \circ T_1\) is a dilation. Conversely, show any dilation is the composition of two inversions.
Determine the image of the line \(y = mx + b\) (when \(b \neq 0)\) under inversion in the unit circle. In particular, show that the image is a circle with center \((-\dfrac{m}{2b}, \dfrac{1}{2b})\) and radius \(\sqrt{\dfrac{(m^2 + 1)}{4b^2}}\text{.}\)
- Hint
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Refer to Exercise \(2.4.1\).
Determine the image of the line \(L\) given by \(y = 3x + 4\) under inversion in the unit circle. Give a careful plot of the unit circle, the line \(L\text{,}\) and the image of \(L\) under the inversion.
Prove that inversion in the unit circle maps the circle \((x-a)^2 + (y-b)^2 = r^2\) to the circle
\[ \bigg(x-\frac{a}{d}\bigg)^2 + \bigg(y-\frac{b}{d}\bigg)^2 = \bigg(\frac{r}{d}\bigg)^2 \]
where \(d = a^2+b^2-r^2\text{,}\) provided that \(d \neq 0\text{.}\)
Determine in standard form the image of the circle \(C\) given by \((x-1)^2 + y^2 = 4\) under inversion in the unit circle. Give a careful plot of the unit circle, the circle \(C\text{,}\) and the image of \(C\) under the inversion.
True or False? If a circle \(C\) gets mapped to another circle under inversion in the unit circle, then the center of \(C\) gets mapped to the center of the image circle, \(i_{\mathbb{S}^1}(C)\text{.}\) If the statement is true, prove it; if it is false, provide a counterexample.
Suppose \(C\) and \(D\) are orthogonal circles. Corollary \(3.2.1\) tells us that inversion in \(C\) maps \(D\) to itself. Prove that this inversion also takes the interior of \(D\) to itself.
Finish the proof of Theorem \(3.2.4\) by showing that the angle of intersection at \(z^*\) equals the angle of intersection at \(z\) in Figure \(3.2.2\).
Suppose \(C\) is the circle \(|z - z_0| = r\) and \(C^\prime\) is the circle \(|z - z_0| = r^\prime\text{.}\) Find the stretch factor \(k\) in the dilation \(S(z) = k(z-z_0) + z_0\) so that \(i_C = S \circ i_{C^\prime}\text{.}\)
Complete the proof of Lemma \(3.2.1\) by proving the case in which the line through \(p\) passes through the center of \(C\text{.}\)