6.2: Chemical Sedimentary Rocks
Whereas clastic sedimentary rocks are dominated by components that have been transported as solid clasts (clay, silt, sand, etc.), chemical sedimentary rocks are dominated by components that have been transported as ions in solution (Na + , Ca 2+ , HCO 3 − , etc.). There is some overlap between the two because almost all clastic sedimentary rocks contain cement formed from dissolved ions, and many chemical sedimentary rocks include some clasts. Since ions can stay in solution for tens of thousands of years (some much longer), and can travel for tens of thousands of kilometers, it is virtually impossible to relate chemical sediments back to their source rocks.
Chemical weathering and chemical sedimentary rocks
Many students confuse chemical weathering with chemical sedimentary rocks, or mistakenly assume that when and where chemical weathering is taking place, chemical sedimentary rocks will accumulate. Most ions in solution in rivers, lakes and the ocean are produced during chemical weathering, but those ions can remain in solution for millions of years, and during that time they can travel hundreds of thousands of km (yes, literally around the world, several times). They might eventually come out of solution as a result of a biological process or a change in the chemical conditions and will then become a mineral crystal that can settle to form a chemical sediment.
So the calcium ions that are part of a calcite mud on the sea floor near Australia’s Great Barrier Reef could literally have come from anywhere on Earth (and almost certainly came from many different places), and might have been in solution for as little as a few days or for as long as tens of millions of years.
The most common chemical sedimentary rock, by far, is limestone . Others include chert , banded iron formation , and evaporites . Biological processes are important in the formation of some chemical sedimentary rocks, especially limestone and chert. For example, limestone is made up almost entirely of fragments of marine [1] organisms that manufacture calcite for their shells and other hard parts, and most chert includes at least some of the silica tests (shells) of tiny marine organisms (such as diatoms and radiolarians).
Limestone
Almost all limestone forms in the oceans, and most of that forms on the shallow continental shelves, especially in tropical regions with coral reefs. Reefs are highly productive ecosystems populated by a wide range of organisms, many of which use calcium and bicarbonate ions in seawater to make carbonate minerals (especially calcite) for their shells and other structures. These include corals, of course, but also green and red algae, urchins, sponges, molluscs, and crustaceans. The hard parts of these organisms are eroded by waves and currents to produce carbonate fragments that accumulate in the surrounding region, as illustrated in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{2}\) shows a cross-section through a typical reef in a tropical environment (normally between 40° N and 40° S). Reefs tend to form in areas with clear water (e.g., not close to the mouths of large rivers), and near the edges of steep drop-offs because the reef organisms thrive on nutrient-rich upwelling currents. As the reef builds up, it is eroded by waves and currents to produce carbonate sediments that are transported into the steep offshore fore-reef area and the shallower inshore back-reef area. These sediments are dominated by reef-type carbonate fragments of all sizes, including mud. In many such areas, carbonate-rich sediments also accumulate in quiet lagoons, where mud and mollusc-shell fragments predominate (Figure \(\PageIndex{3}\)a) or in offshore areas with strong currents, where either foraminifera tests accumulate (Figure \(\PageIndex{3}\)b) or calcite crystallizes inorganically to form ooids —spheres of calcite that form in shallow tropical ocean water with strong currents (Figure \(\PageIndex{3}\)c).
Limestone also accumulates in deeper water, from the steady rain of the carbonate shells of tiny organisms that lived near the ocean surface. The lower limit for limestone accumulation is around 4,000 meters. Beneath that depth, calcite is soluble so limestone does not accumulate.
Calcite can also form on land in a number of environments. Tufa forms at springs (Figure \(\PageIndex{4}\)) and travertine forms at hot springs. Similar material precipitates within limestone caves to form stalactites , stalagmites , and a wide range of other speleothems . Tufa, travertine and speleothems make up only a tiny proportion of all limestone.
Dolomite (CaMg(CO 3 ) 2 ) is another carbonate mineral, but dolomite is also the name for a rock composed of the mineral dolomite (although some geologists use the term dolostone to avoid confusion). Dolomite rock is quite common (there’s a whole Italian mountain range named after it), which is surprising since marine organisms don’t make dolomite. All of the dolomite found in ancient rocks has been formed through magnesium replacing some of the calcium in the calcite in carbonate muds and sands. This process is known as dolomitization , and it is thought to take place where magnesium-rich water percolates through the sediments in carbonate tidal flat environments.
Chert
As we’ve seen, not all marine organisms make their hard parts out of calcite; some, like radiolarians and diatoms, use silica, and when they die their tiny shells (or tests) settle slowly to the bottom where they accumulate as chert. In some cases, chert is deposited along with limestone in the moderately deep ocean, but the two tend to remain separate, so chert beds within limestone are quite common (Figure \(\PageIndex{5}\)), as are nodules, like the flint nodules of the Cretaceous chalk of southeastern England. In other situations, and especially in very deep water, chert accumulates on its own, commonly in thin beds.
Banded iron formation
Banded iron formation (BIF) is a deep sea-floor deposit of iron oxide that is a common ore of iron (Figure \(\PageIndex{6}\)). BIF forms when iron dissolved in seawater is oxidized, becomes insoluble, and sinks to the bottom in the same way that silica tests do to form chert. BIF is prevalent in rocks dating from 2400 to 1800 Ma, a result off changes in the atmosphere and oceans that took place over that time period. Photosynthetic bacteria (i.e., cyanobacteria , a.k.a. blue-green algae) consume carbon dioxide from the atmosphere and use solar energy to convert it to oxygen. These bacteria first evolved around 3500 Ma, and for the next billion years, almost all of that free oxygen was used up by chemical and biological processes, but by 2400 Ma free oxygen levels started to increase in the atmosphere and the oceans. Over a period of 600 million years, that oxygen gradually converted soluble ferrous iron (Fe 2+ ) to insoluble ferric iron (Fe 3+ ), which combined with oxygen to form the mineral hematite (Fe 2 O 3 ), leading to the accumulation of BIFs on the sea floor. After 1800 Ma, little dissolved iron was left in the oceans and the formation of BIF essentially stopped.
Evaporites
In arid regions many lakes and inland seas have no stream outlet and the water that flows into them is removed only by evaporation. Under these conditions, the water becomes increasingly concentrated with dissolved salts, and eventually some of these salts reach saturation levels and start to crystallize (Figure \(\PageIndex{7}\)). Although all evaporite deposits are unique because of differences in the chemistry of the water, in most cases minor amounts of carbonates start to precipitate when the solution is reduced to about 50% of its original volume. Gypsum (CaSO 4 ·H 2 O) precipitates at about 20% of the original volume and halite (NaCl) precipitates at 10%. Other important evaporite minerals include sylvite (KCl) and borax (Na 2 B 4 O 7 ·10H 2 O). Sylvite is mined at numerous locations across Saskatchewan (Figure \(\PageIndex{8}\)) from evaporites that were deposited during the Devonian (~385 Ma) when an inland sea occupied much of the region.
This is an easy experiment that you can do at home. Pour about 50 mL (just less than 1/4 cup) of very hot water into a cup and add 2 teaspoons (10 mL) of salt. Stir until all or almost all of the salt has dissolved, then pour the salty water (leaving any undissolved salt behind) into a shallow wide dish or a small plate. Leave it to evaporate for a few days and observe the result. What is the size range and shape of the crystals you grew?
It may look a little like Figure \(\PageIndex{9}\). These crystals are up to about 3 millimeters across.
See Appendix 3 for Exercise 6.3 answers .
Media Attributions
- Figures 6.2.1, 6.2.2, 6.2.3ab 6.2.4, 6.2.5, 6.2.7, 6.2.9: © Steven Earle. CC BY.
- Figure \(\PageIndex{3}\)c: JoultersCayOoids by Wilson44691 . Public domain.
- Figure \(\PageIndex{6}\): © Andre Karwath. CC BY-SA.
- Figure \(\PageIndex{8}\): Photo courtesy of PotashCorp. All rights reserved. Used with permission.
- We use the word marine when referring to salt water (i.e., oceanic) environments, and the word aquatic when referring to freshwater environments. ↵


Each speech sound can be analyzed in terms of its phonetic features, the parts of the sound that can each be independently controlled by the articulators. We can represent the features of each sound using a feature matrix, or we can use a feature matrix to represent a class of sounds that have features in common.
A YouTube element has been excluded from this version of the text. You can view it online here: ecampusontario.pressbooks.pu...guistics/?p=88
Video Script
In our thinking about speech sounds so far, we’ve focused almost entirely on
segments
. Segments are the individual speech sounds, each of which gets transcribed with an individual symbol in the IPA. We’ve seen that any given segment can influence the segments that come before and after it, through coarticulation and other articulatory processes. And we’ve also seen that segments can be grouped together into
syllables
, which we look at in more detail in another unit. Within the grammar of any language, two different segments might contrast with each other or might not.
So we’ve been talking as if segments are the smallest unit in speech, but in fact, each speech segment is made up of smaller components called
features
. Each feature is an element of a sound that we can control independently. To see how features work, let’s look at a couple of examples. We can describe the segment [b], for example, as being made up of this set of features. First, [b] is a consonant (meaning it has some obstruction in the vocal tract), so it gets the feature
consonant
indicated with a plus sign to show that the consonant feature is present. Looking at the next feature, sonorant, notice that it’s indicated with a minus sign, meaning that [b] is not a sonorant. The feature
sonorant
, of course, has to do with sonority. We know that stops have very low sonority because the vocal tract is completely closed for stops, so stops are all coded as [-sonorant]. The next feature,
syllabic
, tells us whether a given segment is the nucleus of a syllable or not. Remember that the most common segments that serve as the nucleus of a syllable are vowels, but stops certainly cannot be the nucleus, so /b/ gets labelled as [-syllabic]. These first three features, consonant, sonorant, and syllabic allow us to group all speech segments into the major classes of consonants, vowels, and glides. We’ll see how in a couple of minutes.
This next set of features has to do with the manner of articulation. The feature
continuant
tells us how long a sound goes on. Stops are very short sounds; they last for only a brief moment, so [b] gets a minus sign for
continuant
. We also know that [b] is not made by passing air through the nasal cavity, so it also gets a minus sign for the feature
nasal
. And [b] is a voiced sound, made with vocal folds vibrating, so it is [+voice].
The last feature we list for [b] is [LABIAL]
because it’s made with the lips. (Stay tuned for an explanation of why some features are listed in lower-case and some in upper-case.)
This whole list of features is called a
feature matrix
; it’s the list of the individual features that describe the segment [b], in quite a lot of detail! Because features are at the phonetic level of representation, we use square brackets when we list them. You often see a feature matrix listed with a large pair of square brackets, like this, but we’ll just use individual square brackets on each feature.
Now I want you to notice something. If we take this whole feature matrix and change the value of just one feature, changing the feature
voice
from plus to minus, now we’re describing a different segment, [p]: [p] has every feature in common with [b] except for voicing. Likewise, if we take the feature matrix for [b] and change the value of the feature
continuant
from minus to plus, now we’re describing the segment [v], which has all the same features as [b] except that it can continue for a long time because it’s a fricative. Or if we take the feature matrix for [b] and change the feature
nasal
from minus to plus, this has the effect of also changing the
sonorant
feature to plus as well, because circulating air through the nasal cavity adds sonority. Now, this feature matrix describes the properties of the segment [m].
So each feature is something that we can control independently of the others with our articulators. And changing just one feature is enough to change the properties of a segment. That change might lead to a phonemic contrast within the mental grammar of a language, or it might just result in an allophone of the same phoneme.
It turns out that segments that have a lot of features in common tend to behave the same way within the mental grammar of a language. And we can use these features to group segments into natural classes that capture some of these similarities in their behaviour.
Let’s look again at the feature matrix for /b/. If we take away the feature that describes its place of articulation, we end up with a smaller list of features. This smaller list describes not just a single segment, but a class of segments: all the voiced stops. By not mentioning the place feature, we’ve allowed this matrix to include segments from any place of articulation, as long as they share all these other features. These three segments have all these features in common: they’re a natural class. If we remove another feature, the voicing feature, the natural class gets bigger: now we’ve got a feature matrix that describes all the stops in English, including those that are [+voice] and those that are [-voice]. So you can see that this system of features is very powerful for describing classes of segments that have things in common. We’ll learn more about natural classes in the next unit.


7.1: The Importance of Listening
selected template will load here
This action is not available.
Listening is the learned process of receiving, interpreting, recalling, evaluating, and responding to verbal and nonverbal messages. We begin to engage in the listening process from infancy, long before we engage in any recognizable verbal or nonverbal communication. In general, listening helps us achieve all the communication goals (physical, instrumental, relational, and identity) that we learned about in Chapter 1. Listening is also important in academic, professional, and personal contexts. In terms of academics, students with high scores for listening ability have greater academic achievement. Listening skills are highly sought after by potential employers, consistently ranking in the top ten in national surveys (National Association of Colleges and Employers, 2010). Listening also has implications for our personal lives and relationships. We shouldn’t underestimate the power of listening to make someone else feel better and to open our perceptual field to new sources of information. Listening can also help us expand our self and social awareness by learning from other people’s experiences and by helping us take on different perspectives.


4.2: Inequalities and modulus
The transition from school to university mathematics is in many ways marked by a shift from simple variables, equations and functions, to conditions and analysis involving inequalities and modulus.
Problem 100 What is | − x | equal to: x or − x ? (What if x is negative?)
4.2.1 Geometrical interpretation of modulus, of inequalities, and of modulus inequalities
Problem 101
(a) Mark on the coordinate line all those points x in the interval [0,1) which have the digit “1” immediately after the decimal point in their decimal expansion. What fraction of the interval [0,1) have you marked?
Note: “[0,1)” denotes the set of all points between 0 and 1, together with 0, but not including 1. [0,1] denotes the interval including both endpoints; and (0,1) denotes the interval excluding both endpoints.
(b) Mark on the interval [0,1) all those points x which have the digit “1” in at least one decimal place. What fraction of the interval [0,1) have you marked?
(c) Mark on the interval [0,1) all those points x which have a digit “1” in at least one position of their base 2 expansion. What fraction of the interval [0,1) have you marked?
(d) Mark on the interval [0,1) all those points x which have a digit “1” in at least one position of their base 3 expansion. What fraction of the interval [0,1) have you marked?
Problem 102 Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:
Problem 103 Mark on the coordinate line all those points x for which
Problem 104
(a) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:
(b) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:
Problem 105 Mark on the coordinate line all those points x for which
Problem 106 Find numbers a and b with the property that the set of solutions of the inequality
consists of the interval (−1, 2).
Problem 107
(a) Mark on the coordinate plane all points ( x , y ) satisfying the inequality
(b) Mark on the coordinate plane all points ( x , y ) satisfying the inequality
(c) Mark on the coordinate plane all points ( x , y ) satisfying the inequality
4.2.2 Inequalities
Problem 108 Suppose real numbers a , b, c, d satisfy .
(i) Prove that
(ii) If b, d > 0, prove that
Problem 109 (Farey series) When the fully cancelled fractions in [0,1] with denominator ≤ n are arranged in increasing order, the result is called the Farey series (or Farey sequence ) of order n.
Order 1:
Order 2:
Order 3:
Order 4:
(a) Write down the full Farey series (or sequence) of order 7 .
(b) (i) Imagine the points 0.1, 0.2,0.3,..., 0.9 dividing the interval [0,1] into ten subintervals of length . Now insert the eight points corresponding to
Into which of the ten subintervals do they fall?
(ii) Imagine the n points
dividing the interval [0,1] into n + 1 subintervals of length . Now insert the n − 1 points
Into which of the n + 1 subintervals do they fall?
(iii) In passing from the Farey series of order n to the Farey series of order n + 1, we insert fractions of the form between certain pairs of adjacent fractions in the Farey series of order n . If are adjacent fractions in the Farey series of order n , prove that, when adding fractions for the Farey series of order n + 1, at most one fraction is inserted between .
(c) Note: It is worth struggling to prove the two results in part (c). But do not be surprised if they prove to be elusive – in which case, be prepared to simply use the result in part (c)(ii) to solve part (d).
(i) In the Farey series of order n the first two fractions are , and the last two fractions are . Prove that every other adjacent pair of fractions in the Farey series of order n satisfies bd > n.
(ii) Let be adjacent fractions in the Farey series of order n . Prove (by induction on n ) that bc − ad = 1.
(d) Prove that if
are three successive terms in any Farey series, then
Problem 110 Solve the following inequalities.
(a)
(b)
(c)
Problem 111
(a) The sum of two positive numbers equals 5. Can their product be equal to 7?
(b) ( Arithmetic mean, Geometric mean, Harmonic mean, Quadratic mean ) Prove that, if a,b > 0, then
Problem 112 The two hundred numbers
are written on the board. Students take turns to replace two numbers a , b from the current list by their sum divided by . Eventually one number is left on the board. Prove that the final number must be less than 2000.


6.1: Course Materials
1.
“Age-Friendly Communities”
Debra Whitman, AARP Executive Vice President for Policy Strategy, and International Affairs, AARP International, The Journal,
2013.
Communities that adapt age-friendly features become more desirable places to live, to visit and to spend time.
2.
The Benefits of Intergenerational Programs
Generations United
,
2007
.
A fact sheet discussing Intergenerational Programs, which are social vehicles that offer younger and older generations the opportunities to interact and become engaged in issues concerning our society.
3.
“Culture Change in Aging Services (pdf)”
Michele Holleran, PhD; American Association of Homes and Services for the Aging, 2009
This article discusses the impact that culture change has had on aging services. It also reviews the many different culture change models that have been recently adapted into patient care.
4.
“Aging in Community: the Communitarian Alternative to Aging in Place”
Janice Blanchard. Generations: Journal of American Society on Aging, Winter 2013-2014.
In this article, the author talks about aging in community and the effects that this new approach has had on life expectancy, finances, and overall quality of life in the final stages.
5.
Indicators List: Essential Elements of an Elder Friendly Community
Center for Homecare Policy and Research
A short report discussing the benefits of community living, as well as reviewing all the components of this emerging lifestyle trend.


1.1: Overview
This week we will begin our semester long journey into the world of International Business with a look at the globalization of business. After reading/viewing the module content you will complete your first written assignment. Everything that you need to successfully complete this assignment is located in the Module Assessments folder, but if you have questions – please just ask!
LEARNING OUTCOMES
After you complete the required assignments you will be able to:
- Understand what is meant by “Global Business”
- Describe the general international business environment
- Identify potential impacts of globalization on business operations
CC licensed content, Original
- International Business. Authored by : Linda Williams. Provided by : Tidewater Community College. Located at : http://www.tcc.edu . Project : Z Degree Project. License : CC BY: Attribution


2: Language, Identity, and Culture- Exploring, Employing, Embracing
Figure \(2.1\) In a 19th-century lithograph, Cherokee leader Sequoyah, unable to read or write, is shown with a table depicting the writing system he created for his native Cherokee language. His invention of the Cherokee syllabary, a collection of symbols representing the syllables of the spoken language, would provide a divided Cherokee nation with a way to communicate and thus create a sense of identity and unity. (credit: “Sequoyah” by Lehman and Duval/ Wikimedia Commons, Public Domain)
Chapter Outline
Introduction
Based on his dealings with white people who could read and write, Sequoyah (c. 1775–1843), a Cherokee leader, believed that power resulted from the ability to share knowledge. A written language comprehensible and accessible to the Cherokee people could be far more effective in preserving their culture than relying on memory or oral tradition. Hoping to unite and strengthen the Cherokee politically after they were forced from their lands by the U.S. government, Sequoyah spent 10 years developing a method for his people to communicate in writing. His language system, based on written symbols representing syllables spoken in the Cherokee language, was treated skeptically at first but was later adopted and used in Cherokee newspapers, official documents, and descriptions of rituals and medicines.
Although the use of the Cherokee language eventually gave way to English, Sequoyah remained true to all aspects of Cherokee culture. He spoke only Cherokee, wore the clothing of his people, and followed Cherokee religious customs, refusing to assimilate in any way. As evidenced by Sequoyah’s ideas about communication, language, writing, and culture are clearly linked. Although Sequoyah had political purposes in mind, a common language is an important element of shared culture and goes further in developing a sense of cultural identity than what Sequoyah may have envisioned at the time.
While all humans have the same basic needs—food, water, shelter, affection, and a sense of belonging—cultural identity is shaped by family, upbringing, language, and geographical location. You may have grown up in a small town, in a large city, or in the suburbs; you may have been raised with siblings and may have attended church; you may speak more than one language. You may be single, married, or dating; you may drive a car, SUV, or truck. All your lived experiences have shaped who you are today.  For example, if you were to respond to all of the scenarios presented in the sentences above, your answers would collectively make up at least some of your identity.
Having a variety of identities might result in both advantages and disadvantages associated with your lived experiences. Those advantages are called privileges, and the disadvantages discrimination . When you consider that people have multiple identities, also consider that these multiple identities lead to various points of intersection among lived experiences. The idea of intersectionality , then, helps you consider systems of privilege or discrimination projected onto people as a consequence of their identities.
Considering your own identity and position in the world, as well as the identities and positions of others, gives critical meaning to your personal and academic writing. Experiences make people who they are, teaching them how to move through the world and consider the effects their actions and words have on others.
In this chapter, you will explore the concept of cultural identity. What kind of lived experiences have you had? How have those experiences shaped and molded you? What experiences have others had, and how have those experiences shaped them? Whenever you consider lived experiences, think about the contexts in which they occur, and the systems of communication used in those contexts. The word culture includes the expressions, customs, practices, and experiences that connect a person to other people in their present, past, and future. Who you are is closely connected to the cultures you inhabit and the ways in which you communicate. Among the more noticeable markers of your identities are the languages you speak. In this chapter, language is defined as a system of words used to communicate. Just as you likely have multiple identities, you likely speak varieties of languages as part of each of those identities.
The goal of this chapter in particular, and this text in general, is to help you think critically about language, culture, and identity in ways that make room for everyone’s lived experiences. In this chapter, you will learn how authors use language to communicate their sense of cultural identity. You will read about Cathy Park Hong and W. E. B. Du Bois, activists who have used writing to explore identity and culture.


11.5: Applying the Nursing Process
Assessment
Nurses play an essential role in performing comprehensive pain assessment. Assessments include asking questions about the presence of pain, as well as observing for nonverbal indicators of pain, such as grimacing, moaning, and touching the painful area. It is especially important to observe for nonverbal indicators of pain in patients unable to self-report their pain, such as infants, children, patients who have a cognitive disorder, patients at end of life, non-English speaking patients, or patients who tend to be stoic due to cultural beliefs. See Figure 11.14 [1] for an image of a patient who is expressing pain nonverbally.
Recall that pain is defined as whatever the person experiencing it says it is. Subjective assessment includes asking questions regarding the severity rating, as well as obtaining comprehensive information by using the “PQRSTU” or “OLDCARTES” methods for assessing a chief complaint. For some patients who are unable to quantify the severity of their pain, a visual scale like the FACES scale is the best way to perform subjective assessment regarding the severity of pain.
Objective data includes observations of nonverbal indications of pain, such as restlessness, facial grimacing and wincing, moaning, and rubbing or guarding painful areas. For patients who cannot verbalize their pain, using a scale like the FLACC, COMFORT, or PAINAD is helpful to standardize observations across different staff members. Keep in mind that patients experiencing acute pain will also likely have vital signs changes, such as increased blood pressure, increased heart rate, and increased respiratory rate.
It is important to assess the impact of pain on a patient’s daily functioning. This can be accomplished by asking what effect the pain has on their ability to bathe, dress, prepare food, eat, walk, and complete other daily activities. Assessing the impact of pain on daily functioning is a new standard of care that assists the interdisciplinary team in tailoring treatment goals and interventions that are customized to the patient’s situation. For example, for some patients, chronic pain affects their ability to be employed, so effective pain management is vital so they can return to work. For other patients receiving palliative care, the ability to sit up and eat a meal with loved ones without pain is an important goal. [2]
When performing a patient assessment, any new complaints of pain or pain that is unresponsive to the current treatment plan should be reported to the health care provider. Instances of sudden, severe pain or chest pain require immediate notification or contact of emergency services.
Diagnoses
Commonly used NANDA-I nursing diagnoses for pain include Acute Pain (duration less than 3 months) and Chronic Pain . See Table 11.5 for more information regarding these diagnoses. [3] For more information about defining characteristics and related factors for other NANDA-I nursing diagnoses, refer to a current nursing diagnosis resource.
Table 11.5 Pain NANDA-I Nursing Diagnoses [4]
| NANDA-I Diagnosis | Definition | Defining Characteristics |
|---|---|---|
| Acute Pain | Unpleasant sensory and emotional experience associated with acute or potential tissue damage, or described in terms of such damage; sudden or slow onset of any intensity from mild to severe with an anticipated or predictable end, and with a duration of less than 3 months. |
|
| Chronic Pain | Unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage (International Association for the Study of Pain); sudden or slow onset of any intensity from mild to severe, constant or recurring without anticipated or predictable end, and with a duration of greater than 3 months. |
|
Outcome Identification
An overall goal when providing pain management is, “ The patient will report that the pain management treatment plan achieves their comfort-function goals. ” [5]
SMART outcomes are customized to the patient’s unique situation. An example of a SMART goal is, “ The patient will notify the nurse promptly for pain intensity level that is greater than their comfort-function goal throughout shift. ” [6]
Planning Interventions
Several pharmacological and nonpharmacological interventions have been described throughout this chapter. See the following box for a summarized list of interventions for acute pain management.
- Identify pain intensity during required recovery activities (e.g., coughing and deep breathing, ambulation, transfers to chair, etc.)
- Explore patient’s knowledge and beliefs about pain, including cultural influences
- Question patient regarding the level of pain that allows a state of comfort and desired function and attempt to keep pain at or lower than identified level
- Ensure that the patient receives prompt analgesic care before the pain becomes severe or before pain-inducing activities
- Administer analgesics around-the-clock as needed the first 24 to 48 hours after surgery, trauma, or injury except if sedation or respiratory status indicates otherwise
- Monitor sedation and respiratory status before administering opioids and at regular intervals when opioids are administered
- Follow agency protocols in selecting analgesia and dosage
- Use a combination of prescribed medications (e.g., opioids, nonopioids, and adjuvants), if pain level is severe
- Select and implement interventions tailored to the patient’s risks, benefits, and preferences (e.g., pharmacological and nonpharmacological) to facilitate pain relief
- Cautiously use analgesics that may have adverse effects in older adults
- Administer analgesics using the least invasive route available, avoiding the intramuscular route
- Advocate PCA, intrathecal, and epidural routes of administration when appropriate
- Modify pain control measures on the basis of the patient’s response to treatment
- Prevent and/or manage medication side effects
- Notify prescribing provider if pain control measures are unsuccessful
- Provide accurate information to family members or caregivers about the patient’s pain experience with the patient’s permission
See the following box for a summarized list of interventions for chronic pain management.
- Explore the patient’s knowledge and beliefs about pain, including cultural influences
- Determine the pain experience on quality of life (e.g., sleep, appetite, activity, cognition, mood, relationships, job performance, and role responsibilities)
- Evaluate the effectiveness of past pain control measures with the patient
- Question the patient regarding the level of pain that allows a state of comfort and appropriate functioning and attempt to keep pain at or lower than identified level
- Control environmental factors that may influence the patient’s pain experience
- Ensure that the patient receives prompt analgesic care before the pain becomes severe or before activities that are anticipated to be pain-inducing
- Select and implement intervention options tailored to the patient’s risks, benefits, and preferences (e.g., pharmacological, nonpharmacological, interpersonal) to facilitate pain relief, as appropriate
- Instruct the patient and family about principles of pain management
- Encourage the patient to monitor own pain and to use self-management approaches
- Encourage appropriate use of nonpharmacological techniques (e.g., biofeedback, TENS, hypnosis, relaxation, guided imagery, music therapy, distraction, play therapy, activity therapy, acupressure, heat and cold application, and massage) and pharmacological options as pain control measures
- Avoid use of analgesics that may have adverse effects on older adults
- Collaborate with the patient, family, and other health professionals to select and implement pain control measures
- Prevent or manage side effects
- Evaluate the effectiveness of pain control measures through ongoing monitoring of the pain experience
- Watch for signs of depression (e.g., sleeplessness, not eating, flat affect, statements of depression, or suicidal ideation)
- Watch for signs of anxiety or fear (e.g., irritability, tension, worry, or fear of movement)
- Modify pain control measures on the basis of the patient’s response to treatment
- Incorporate the family in the pain relief modality, when possible
- Utilize a multidisciplinary approach to pain management, when appropriate
- Consider referrals for the patient and family to support groups and other resources, as appropriate
- Evaluate patient satisfaction with pain management at specified intervals
- Evaluate barriers to adherence with past pain management care plans
Implementing Pharmacological Interventions
Patients should be involved and engaged in their plan of care to treat pain. By demonstrating empathy and collaborating with patients and the interdisciplinary team, it is more likely the treatment plan will be effective based on the patient’s goals.
When administering analgesic medication, holistic nursing care is important. Begin by considering the patient’s goals for pain relief and ask if they have been met effectively by previously administered medications. If they have not been met, it may be necessary to advocate for additional or alternative medication with the health care provider. It is also important to consider if the patient is experiencing any side effects that may impact the patient’s desire to take additional pain medication.
When administering medications that have been ordered on an “as-needed” basis, it is vital for the nurse to verify the amount of medication the patient received in the past 24 hours and if any dosage limits have been met to ensure patient safety.
Prior to administration, consider the best route of administration for this patient at this particular time. For example, if the patient is nauseated and vomiting, then an oral route may not be effective. On the other hand, if a patient’s pain has improved when receiving intravenous medications during the recovery process, it may be possible for the patient to begin taking oral pain medications in preparation for discharge home. Keep the WHO ladder in mind when selecting medications to reach patient goals while also avoiding potential adverse effects when possible.
When preparing opioid medications, it is important to remember that these medications are controlled substances with special regulations regarding storage, count auditing, and disposal/wasting of medication. Follow agency policy regarding these issues. It is also important to assess the patient’s level of sedation and respiratory status before administering additional doses of opioids and withhold the medication if the patient is oversedated or their respiratory rate is less than 12/minute. However, when providing pain management during end-of-life care, these parameters no longer apply because the emphasis is on providing comfort according to the patient’s preferences. Read more about end-of-life care in the “ Grief and Loss ” chapter.
Evaluation
It is vital for the nurse to regularly evaluate if the established interventions are effectively meeting the pain management and function goals established collaboratively with the patient. Additionally, when administering analgesics, the patient should be reassessed in an hour (or other time frame based on the onset and peak of the medication) to determine if the medication was effective. If interventions are not effective, then follow-up interventions are required, which may include contacting the health care provider.
For patients living with chronic pain, it can be helpful for them or their caregiver to maintain a pain journal. In the journal they can document activities that precipitated pain, medications taken to manage the pain, and whether these medications were effective in helping them to meet their functional goals. This journal is shared with the health care provider during follow-up visits to enhance the treatment plan. [9]
The nurse must continually monitor for potential adverse effects of pain medications. For example, if a patient is receiving acetaminophen daily for chronic osteoarthritis pain, signs of liver dysfunction, such as jaundice and elevated liver function bloodwork, should be monitored. For older adults receiving NSAIDs, it is important to watch for early signs of gastrointestinal bleeding, such as melena. Patients receiving opioids should be continually monitored for oversedation, respiratory depression, constipation, nausea and vomiting, urinary retention, and pruritus. Side effects should be reported to the health care provider and orders received for treatment.
- “ 238074231_2485ed053b_o ” by Erik Ogan is licensed under CC BY-SA 2.0 ↵
- American Association of Colleges of Nursing. (n.d.). End-of-Life-Care (ELNEC) . https://www.aacnnursing.org/ELNEC ↵
- Herdman, T. H., & Kamitsuru, S. (Eds.). (2018). Nursing diagnoses: Definitions and classification, 2018-202 0. Thieme Publishers, New York, pp. 445-446. ↵
- Herdman, T. H., & Kamitsuru, S. (Eds.). (2018). Nursing diagnoses: Definitions and classification, 2018-202 0. Thieme Publishers, New York, pp. 445-446. ↵
- Ackley, B., Ladwig, G., Makic, M. B., Martinez-Kratz, M., & Zanotti, M. (2020). Nursing diagnosis handbook: An evidence-based guide to planning care (12th ed.). Elsevier, pp. 676-691. ↵
- Ackley, B., Ladwig, G., Makic, M. B., Martinez-Kratz, M., & Zanotti, M. (2020). Nursing diagnosis handbook: An evidence-based guide to planning care (12th ed.). Elsevier, pp. 676-691. ↵
- Butcher, H., Bulechek, G., Dochterman, J., & Wagner, C. (2018). Nursing Interventions Classification (NIC) . Elsevier, pp. 281-282. ↵
- Butcher, H., Bulechek, G., Dochterman, J., & Wagner, C. (2018). Nursing Interventions Classification (NIC) . Elsevier, pp. 281-282. ↵
- American Association of Colleges of Nursing. (n.d.). End-of-Life-Care (ELNEC) . https://www.aacnnursing.org/ELNEC ↵


11.1: Citation Generators
Unlike 10-15 years ago when college students had to try to memorize citation formats and create each citation by hand, there are now many online tools available that will automatically create citations for many different types of information sources. Citation generators are tools that allow users to type, paste, or import certain information (like a source’s type, title, author, and date), and then create a citation in whatever format is needed (e.g., APA, MLA, Chicago).
Library Databases
Almost all of your college library’s databases will have a convenient citation generator built right into them. In most cases, when you are reading an ebook, academic journal article, or another resource coming from a library database, there will be an icon somewhere on the page for a citation generator. These often offer a list of citation styles to choose from; they provide a citation for that particular source in whatever format is needed, so you can easily copy and paste the full citation into your assignment. The citation generator is not always located in the same spot, and does not always appear the same way in all library databases, but will often say “cite” or have quotation mark icons to indicate that it is a citation tool.
While these citation generators are incredibly helpful and convenient, they do sometimes make mistakes! Always be sure to double-check the citation generated for you—do not assume that all citations generated by a citation tool will be 100% correct.
OneSearch
OneSearch also has its own convenient tool located within the record of every item record. The tool is located in the “Send to” section of the record and looks like one set of quotation marks with the word “CITATION” underneath. The citation style can be selected on the left-hand side (but please note, some colleges may list these formats differently from what is shown in the image below). Clicking on “COPY CITATION TO CLIPBOARD” will copy the citation so you can easily paste it into your bibliography (in Microsoft Word, Google Docs, or elsewhere). Besides checking the citation for accuracy, it is also important to check the formatting of your pasted text. Some formatting, like italics, may be lost when copying and pasting. Citation generators are fantastic tools that save a lot of time, but are not always 100% accurate!
Word Processors
Two commonly used word processors—Microsoft Word and Google Docs—both have citation tools built into the software that make inserting in-text citations and compiling full citations quick and simple.
Microsoft Word’s citation generator is found under the toolbar section called “References,” and the “Insert Citation” tool allows users to plug in bibliographic information about each source used in their paper. The software will keep a running list of the sources, and allows for easy insertion of an in-text citation and a quick combination of citations into a bibliography in the appropriate citation style. An important thing to note is that Microsoft Word may not have the latest citation style format edition; for instance, it may only have the option to create citations in MLA 7th edition, but (as of the time this book was published) MLA 8th edition is the most up-to-date style.
The citation generator in Google Docs is located under the “Tools” menu item on the toolbar. Though this citation tool is much less robust than Microsoft Word’s, only having three citation styles to choose from, it still allows for quick insertion of both in-text citations and a bibliography at the end of the paper. As mentioned previously, the citation styles may not be the latest edition and it is important to always review any citation generated for you.


5.18: Poe, Edgar Allan. "The Fall of the House of Usher" (1839)
|
Son cœur est un luth suspendu;
Sitôt qu’on le touche il résonne. |
De Béranger.
D URING the whole of a dull, dark, and soundless day in the autumn of the year, when the clouds hung oppressively low in the heavens, I had been passing alone, on horseback, through a singularly dreary tract of country, and at length found myself, as the shades of the evening drew on, within view of the melancholy House of Usher. I know not how it was—but, with the first glimpse of the building, a sense of insufferable gloom pervaded my spirit. I say insufferable; for the feeling was unrelieved by any of that half-pleasurable, because poetic, sentiment, with which the mind usually receives even the sternest natural images of the desolate or terrible. I looked upon the scene before me—upon the mere house, and the simple landscape features of the domain—upon the bleak walls—upon the vacant eye-like windows—upon a few rank sedges—and upon a few white trunks of decayed trees—with an utter depression of soul which I can compare to no earthly sensation more properly than to the after-dream of the reveller upon opium—the bitter lapse into every-day life—the hideous dropping off of the veil. There was an iciness, a sinking, a sickening of the heart—an unredeemed dreariness of thought which no goading of the imagination could torture into aught of the sublime. What was it—I paused to think—what was it that so unnerved me in the contemplation of the House of Usher? It was a mystery all insoluble; nor could I grapple with the shadowy fancies that crowded upon me as I pondered. I was forced to fall back upon the unsatisfactory conclusion, that while, beyond doubt, there are combinations of very simple natural objects which have the power of thus affecting us, still the analysis of this power lies among considerations beyond our depth. It was possible, I reflected, that a mere different arrangement of the particulars of the scene, of the details of the picture, would be sufficient to modify, or perhaps to annihilate its capacity for sorrowful impression; and, acting upon this idea, I reined my horse to the precipitous brink of a black and lurid tarn that lay in unruffled lustre by the dwelling, and gazed down—but with a shudder even more thrilling than before—upon the remodelled and inverted images of the gray sedge, and the ghastly tree-stems, and the vacant and eye-like windows.
Nevertheless, in this mansion of gloom I now proposed to myself a sojourn of some weeks. Its proprietor, Roderick Usher, had been one of my boon companions in boyhood; but many years had elapsed since our last meeting. A letter, however, had lately reached me in a distant part of the country—a letter from him—which, in its wildly importunate nature, had admitted of no other than a personal reply. The MS. gave evidence of nervous agitation. The writer spoke of acute bodily illness—of a mental disorder which oppressed him—and of an earnest desire to see me, as his best and indeed his only personal friend, with a view of attempting, by the cheerfulness of my society, some alleviation of his malady. It was the manner in which all this, and much more, was said—it was the apparent heart that went with his request—which allowed me no room for hesitation; and I accordingly obeyed forthwith what I still considered a very singular summons.
Although, as boys, we had been even intimate associates, yet I really knew little of my friend. His reserve had been always excessive and habitual. I was aware, however, that his very ancient family had been noted, time out of mind, for a peculiar sensibility of temperament, displaying itself, through long ages, in many works of exalted art, and manifested, of late, in repeated deeds of munificent yet unobtrusive charity, as well as in a passionate devotion to the intricacies, perhaps even more than to the orthodox and easily recognizable beauties, of musical science. I had learned, too, the very remarkable fact, that the stem of the Usher race, all time-honored as it was, had put forth, at no period, any enduring branch; in other words, that the entire family lay in the direct line of descent, and had always, with very trifling and very temporary variation, so lain. It was this deficiency, I considered, while running over in thought the perfect keeping of the character of the premises with the accredited character of the people, and while speculating upon the possible influence which the one, in the long lapse of centuries, might have exercised upon the other—it was this deficiency, perhaps, of collateral issue, and the consequent undeviating transmission, from sire to son, of the patrimony with the name, which had, at length, so identified the two as to merge the original title of the estate in the quaint and equivocal appellation of the “House of Usher”—an appellation which seemed to include, in the minds of the peasantry who used it, both the family and the family mansion.
I have said that the sole effect of my somewhat childish experiment—that of looking down within the tarn—had been to deepen the first singular impression. There can be no doubt that the consciousness of the rapid increase of my superstition—for why should I not so term it?—served mainly to accelerate the increase itself. Such, I have long known, is the paradoxical law of all sentiments having terror as a basis. And it might have been for this reason only, that, when I again uplifted my eyes to the house itself, from its image in the pool, there grew in my mind a strange fancy—a fancy so ridiculous, indeed, that I but mention it to show the vivid force of the sensations which oppressed me. I had so worked upon my imagination as really to believe that about the whole mansion and domain there hung an atmosphere peculiar to themselves and their immediate vicinity—an atmosphere which had no affinity with the air of heaven, but which had reeked up from the decayed trees, and the gray wall, and the silent tarn—a pestilent and mystic vapor, dull, sluggish, faintly discernible, and leaden-hued.
Shaking off from my spirit what must have been a dream, I scanned more narrowly the real aspect of the building. Its principal feature seemed to be that of an excessive antiquity. The discoloration of ages had been great. Minute fungi overspread the whole exterior, hanging in a fine tangled web-work from the eaves. Yet all this was apart from any extraordinary dilapidation. No portion of the masonry had fallen; and there appeared to be a wild inconsistency between its still perfect adaptation of parts, and the crumbling condition of the individual stones. In this there was much that reminded me of the specious totality of old wood-work which has rotted for long years in some neglected vault, with no disturbance from the breath of the external air. Beyond this indication of extensive decay, however, the fabric gave little token of instability. Perhaps the eye of a scrutinizing observer might have discovered a barely perceptible fissure, which, extending from the roof of the building in front, made its way down the wall in a zigzag direction, until it became lost in the sullen waters of the tarn.
Noticing these things, I rode over a short causeway to the house. A servant in waiting took my horse, and I entered the Gothic archway of the hall. A valet, of stealthy step, thence conducted me, in silence, through many dark and intricate passages in my progress to the studio of his master. Much that I encountered on the way contributed, I know not how, to heighten the vague sentiments of which I have already spoken. While the objects around me—while the carvings of the ceilings, the sombre tapestries of the walls, the ebony blackness of the floors, and the phantasmagoric armorial trophies which rattled as I strode, were but matters to which, or to such as which, I had been accustomed from my infancy—while I hesitated not to acknowledge how familiar was all this—I still wondered to find how unfamiliar were the fancies which ordinary images were stirring up. On one of the staircases, I met the physician of the family. His countenance, I thought, wore a mingled expression of low cunning and perplexity. He accosted me with trepidation and passed on. The valet now threw open a door and ushered me into the presence of his master.
The room in which I found myself was very large and lofty. The windows were long, narrow, and pointed, and at so vast a distance from the black oaken floor as to be altogether inaccessible from within. Feeble gleams of encrimsoned light made their way through the trellised panes, and served to render sufficiently distinct the more prominent objects around; the eye, however, struggled in vain to reach the remoter angles of the chamber, or the recesses of the vaulted and fretted ceiling. Dark draperies hung upon the walls. The general furniture was profuse, comfortless, antique, and tattered. Many books and musical instruments lay scattered about, but failed to give any vitality to the scene. I felt that I breathed an atmosphere of sorrow. An air of stern, deep, and irredeemable gloom hung over and pervaded all.
Upon my entrance, Usher rose from a sofa on which he had been lying at full length, and greeted me with a vivacious warmth which had much in it, I at first thought, of an overdone cordiality—of the constrained effort of the ennuyé man of the world. A glance, however, at his countenance convinced me of his perfect sincerity. We sat down; and for some moments, while he spoke not, I gazed upon him with a feeling half of pity, half of awe. Surely, man had never before so terribly altered, in so brief a period, as had Roderick Usher! It was with difficulty that I could bring myself to admit the identity of the man being before me with the companion of my early boyhood. Yet the character of his face had been at all times remarkable. A cadaverousness of complexion; an eye large, liquid, and luminous beyond comparison; lips somewhat thin and very pallid, but of a surpassingly beautiful curve; a nose of a delicate Hebrew model, but with a breadth of nostril unusual in similar formations; a finely moulded chin, speaking, in its want of prominence, of a want of moral energy; hair of a more than web-like softness and tenuity;—these features, with an inordinate expansion above the regions of the temple, made up altogether a countenance not easily to be forgotten. And now in the mere exaggeration of the prevailing character of these features, and of the expression they were wont to convey, lay so much of change that I doubted to whom I spoke. The now ghastly pallor of the skin, and the now miraculous lustre of the eye, above all things startled and even awed me. The silken hair, too, had been suffered to grow all unheeded, and as, in its wild gossamer texture, it floated rather than fell about the face, I could not, even with effort, connect its Arabesque expression with any idea of simple humanity.
In the manner of my friend I was at once struck with an incoherence—an inconsistency; and I soon found this to arise from a series of feeble and futile struggles to overcome an habitual trepidancy—an excessive nervous agitation. For something of this nature I had indeed been prepared, no less by his letter, than by reminiscences of certain boyish traits, and by conclusions deduced from his peculiar physical conformation and temperament. His action was alternately vivacious and sullen. His voice varied rapidly from a tremulous indecision (when the animal spirits seemed utterly in abeyance) to that species of energetic concision—that abrupt, weighty, unhurried, and hollow-sounding enunciation—that leaden, self-balanced and perfectly modulated guttural utterance, which may be observed in the lost drunkard, or the irreclaimable eater of opium, during the periods of his most intense excitement.
It was thus that he spoke of the object of my visit, of his earnest desire to see me, and of the solace he expected me to afford him. He entered, at some length, into what he conceived to be the nature of his malady. It was, he said, a constitutional and a family evil, and one for which he despaired to find a remedy—a mere nervous affection, he immediately added, which would undoubtedly soon pass off. It displayed itself in a host of unnatural sensations. Some of these, as he detailed them, interested and bewildered me; although, perhaps, the terms and the general manner of the narration had their weight. He suffered much from a morbid acuteness of the senses; the most insipid food was alone endurable; he could wear only garments of certain texture; the odors of all flowers were oppressive; his eyes were tortured by even a faint light; and there were but peculiar sounds, and these from stringed instruments, which did not inspire him with horror.
To an anomalous species of terror I found him a bounden slave. “I shall perish,” said he, “I must perish in this deplorable folly. Thus, thus, and not otherwise, shall I be lost. I dread the events of the future, not in themselves, but in their results. I shudder at the thought of any, even the most trivial, incident, which may operate upon this intolerable agitation of soul. I have, indeed, no abhorrence of danger, except in its absolute effect—in terror. In this unnerved, in this pitiable, condition I feel that the period will sooner or later arrive when I must abandon life and reason together, in some struggle with the grim phantasm, F EAR.”
I learned, moreover, at intervals, and through broken and equivocal hints, another singular feature of his mental condition. He was enchained by certain superstitious impressions in regard to the dwelling which he tenanted, and whence, for many years, he had never ventured forth—in regard to an influence whose supposititious force was conveyed in terms too shadowy here to be re-stated—an influence which some peculiarities in the mere form and substance of his family mansion had, by dint of long sufferance, he said, obtained over his spirit—an effect which the physique of the gray walls and turrets, and of the dim tarn into which they all looked down, had, at length, brought about upon the morale of his existence.
He admitted, however, although with hesitation, that much of the peculiar gloom which thus afflicted him could be traced to a more natural and far more palpable origin—to the severe and long-continued illness—indeed to the evidently approaching dissolution—of a tenderly beloved sister, his sole companion for long years, his last and only relative on earth. “Her decease,” he said, with a bitterness which I can never forget, “would leave him (him the hopeless and the frail) the last of the ancient race of the Ushers.” While he spoke, the lady Madeline (for so was she called) passed slowly through a remote portion of the apartment, and, without having noticed my presence, disappeared. I regarded her with an utter astonishment not unmingled with dread; and yet I found it impossible to account for such feelings. A sensation of stupor oppressed me as my eyes followed her retreating steps. When a door, at length, closed upon her, my glance sought instinctively and eagerly the countenance of the brother; but he had buried his face in his hands, and I could only perceive that a far more than ordinary wanness had overspread the emaciated fingers through which trickled many passionate tears.
The disease of the lady Madeline had long baffled the skill of her physicians. A settled apathy, a gradual wasting away of the person, and frequent although transient affections of a partially cataleptical character were the unusual diagnosis. Hitherto she had steadily borne up against the pressure of her malady, and had not betaken herself finally to bed; but on the closing in of the evening of my arrival at the house, she succumbed (as her brother told me at night with inexpressible agitation) to the prostrating power of the destroyer; and I learned that the glimpse I had obtained of her person would thus probably be the last I should obtain—that the lady, at least while living, would be seen by me no more.
For several days ensuing, her name was unmentioned by either Usher or myself; and during this period I was busied in earnest endeavors to alleviate the melancholy of my friend. We painted and read together, or I listened, as if in a dream, to the wild improvisations of his speaking guitar. And thus, as a closer and still closer intimacy admitted me more unreservedly into the recesses of his spirit, the more bitterly did I perceive the futility of all attempt at cheering a mind from which darkness, as if an inherent positive quality, poured forth upon all objects of the moral and physical universe in one unceasing radiation of gloom.
I shall ever bear about me a memory of the many solemn hours I thus spent alone with the master of the House of Usher. Yet I should fail in any attempt to convey an idea of the exact character of the studies, or of the occupations, in which he involved me, or led me the way. An excited and highly distempered ideality threw a sulphureous lustre over all. His long improvised dirges will ring forever in my ears. Among other things, I hold painfully in mind a certain singular perversion and amplification of the wild air of the last waltz of Von Weber. From the paintings over which his elaborate fancy brooded, and which grew, touch by touch, into vagueness at which I shuddered the more thrillingly, because I shuddered knowing not why—from these paintings (vivid as their images now are before me) I would in vain endeavor to educe more than a small portion which should lie within the compass of merely written words. By the utter simplicity, by the nakedness of his designs, he arrested and overawed attention. If ever mortal painted an idea, that mortal was Roderick Usher. For me at least, in the circumstances then surrounding me, there arose out of the pure abstractions which the hypochondriac contrived to throw upon his canvas, an intensity of intolerable awe, no shadow of which felt I ever yet in the contemplation of the certainly glowing yet too concrete reveries of Fuseli.
One of the phantasmagoric conceptions of my friend, partaking not so rigidly of the spirit of abstraction, may be shadowed forth, although feebly, in words. A small picture presented the interior of an immensely long and rectangular vault or tunnel, with low walls, smooth, white, and without interruption or device. Certain accessory points of the design served well to convey the idea that this excavation lay at an exceeding depth below the surface of the earth. No outlet was observed in any portion of its vast extent, and no torch or other artificial source of light was discernible; yet a flood of intense rays rolled throughout, and bathed the whole in a ghastly and inappropriate splendor.
I have just spoken of that morbid condition of the auditory nerve which rendered all music intolerable to the sufferer, with the exception of certain effects of stringed instruments. It was, perhaps, the narrow limits to which he thus confined himself upon the guitar which gave birth, in great measure, to the fantastic character of the performances. But the fervid facility of his impromptus could not be so accounted for. They must have been, and were, in the notes, as well as in the words of his wild fantasias (for he not unfrequently accompanied himself with rhymed verbal improvisations), the result of that intense mental collectedness and concentration to which I have previously alluded as observable only in particular moments of the highest artificial excitement. The words of one of these rhapsodies I have easily remembered. I was, perhaps, the more forcibly impressed with it as he gave it, because, in the under or mystic current of its meaning, I fancied that I perceived, and for the first time, a full consciousness on the part of Usher of the tottering of his lofty reason upon her throne. The verses, which were entitled “The Haunted Palace,” ran very nearly, if not accurately, thus:—
|
I. |
|
|
In the greenest of our valleys,
By good angels tenanted, Once a fair and stately palace— Radiant palace—reared its head. In the monarch Thought’s dominion— It stood there! Never seraph spread a pinion Over fabric half so fair. |
|
|
II. |
|
|
Banners yellow, glorious, golden,
On its roof did float and flow; (This—all this—was in the olden Time long ago); And every gentle air that dallied, In that sweet day, Along the ramparts plumed and pallid, A winged odor went away. |
|
|
III. |
|
|
Wanderers in that happy valley
Through two luminous windows saw Spirits moving musically To a lute’s well-tunèd law; Round about a throne, where sitting (Porphyrogene!) In state his glory well befitting, The ruler of the realm was seen. |
|
|
IV. |
|
|
And all with pearl and ruby glowing
Was the fair palace door, Through which came flowing, flowing, flowing And sparkling evermore, A troop of Echoes whose sweet duty Was but to sing, In voices of surpassing beauty, The wit and wisdom of their king. |
|
|
V. |
|
|
But evil things, in robes of sorrow,
Assailed the monarch’s high estate; (Ah, let us mourn, for never morrow Shall dawn upon him, desolate!) And, round about his home, the glory That blushed and bloomed Is but a dim-remembered story Of the old time entombed. |
|
|
VI. |
|
|
And travellers now within that valley,
Through the red-litten windows see Vast forms that move fantastically To a discordant melody; While, like a rapid ghastly river, Through the pale door, A hideous throng rush out forever, And laugh—but smile no more. |
I well remember that suggestions arising from this ballad, led us into a train of thought wherein there became manifest an opinion of Usher’s which I mention not so much on account of its novelty (for other men * have thought thus), as on account of the pertinacity with which he maintained it. This opinion, in its general form, was that of the sentience of all vegetable things. But, in his disordered fancy, the idea had assumed a more daring character, and trespassed, under certain conditions, upon the kingdom of inorganization. I lack words to express the full extent, or the earnest abandon of his persuasion. The belief, however, was connected (as I have previously hinted) with the gray stones of the home of his forefathers. The conditions of the sentience had been here, he imagined, fulfilled in the method of collocation of these stones—in the order of their arrangement, as well as in that of the many fungi which overspread them, and of the decayed trees which stood around—above all, in the long undisturbed endurance of this arrangement, and in its reduplication in the still waters of the tarn. Its evidence—the evidence of the sentience—was to be seen, he said, (and I here started as he spoke), in the gradual yet certain condensation of an atmosphere of their own about the waters and the walls. The result was discoverable, he added, in that silent yet importunate and terrible influence which for centuries had moulded the destinies of his family, and which made him what I now saw him—what he was. Such opinions need no comment, and I will make none.
Our books—the books which, for years, had formed no small portion of the mental existence of the invalid—were, as might be supposed, in strict keeping with this character of phantasm. We pored together over such works as the “Ververt et Chartreuse” of Gresset; the “Belphegor” of Machiavelli; the “Heaven and Hell” of Swedenborg; the “Subterranean Voyage of Nicholas Klimm” by Holberg; the “Chiromancy” of Robert Flud, of Jean D’Indaginé, and of De la Chambre; the “Journey into the Blue Distance” of Tieck; and the “City of the Sun” of Campanella. One favorite volume was a small octavo edition of the “Directorium Inquisitorium,” by the Dominican Eymeric de Gironne; and there were passages in Pomponius Mela, about the old African Satyrs and Œgipans, over which Usher would sit dreaming for hours. His chief delight, however, was found in the perusal of an exceedingly rare and curious book in quarto Gothic—the manual of a forgotten church—the Vigiliæ Mortuorum Secundum Chorum Ecclesiæ Maguntinæ .
I could not help thinking of the wild ritual of this work, and of its probable influence upon the hypochondriac, when, one evening, having informed me abruptly that the lady Madeline was no more, he stated his intention of preserving her corpse for a fortnight (previously to its final interment), in one of the numerous vaults within the main walls of the building. The worldly reason, however, assigned for this singular proceeding, was one which I did not feel at liberty to dispute. The brother had been led to his resolution (so he told me) by consideration of the unusual character of the malady of the deceased, of certain obtrusive and eager inquiries on the part of her medical men, and of the remote and exposed situation of the burial-ground of the family. I will not deny that when I called to mind the sinister countenance of the person whom I met upon the staircase, on the day of my arrival at the house, I had no desire to oppose what I regarded as at best but a harmless, and by no means an unnatural, precaution.
At the request of Usher, I personally aided him in the arrangements for the temporary entombment. The body having been encoffined, we two alone bore it to its rest. The vault in which we placed it (and which had been so long unopened that our torches, half smothered in its oppressive atmosphere, gave us little opportunity for investigation) was small, damp, and entirely without means of admission for light; lying, at great depth, immediately beneath that portion of the building in which was my own sleeping apartment. It had been used, apparently, in remote feudal times, for the worst purposes of a donjon-keep, and, in later days, as a place of deposit for powder, or some other highly combustible substance, as a portion of its floor, and the whole interior of a long archway through which we reached it, were carefully sheathed with copper. The door, of massive iron, had been, also, similarly protected. Its immense weight caused an unusually sharp, grating sound, as it moved upon its hinges.
Having deposited our mournful burden upon tressels within this region of horror, we partially turned aside the yet unscrewed lid of the coffin, and looked upon the face of the tenant. A striking similitude between the brother and sister now first arrested my attention; and Usher, divining, perhaps, my thoughts, murmured out some few words from which I learned that the deceased and himself had been twins, and that sympathies of a scarcely intelligible nature had always existed between them. Our glances, however, rested not long upon the dead—for we could not regard her unawed. The disease which had thus entombed the lady in the maturity of youth, had left, as usual in all maladies of a strictly cataleptical character, the mockery of a faint blush upon the bosom and the face, and that suspiciously lingering smile upon the lip which is so terrible in death. We replaced and screwed down the lid, and, having secured the door of iron, made our way, with toil, into the scarcely less gloomy apartments of the upper portion of the house.
And now, some days of bitter grief having elapsed, an observable change came over the features of the mental disorder of my friend. His ordinary manner had vanished. His ordinary occupations were neglected or forgotten. He roamed from chamber to chamber with hurried, unequal, and objectless step. The pallor of his countenance had assumed, if possible, a more ghastly hue—but the luminousness of his eye had utterly gone out. The once occasional huskiness of his tone was heard no more; and a tremulous quaver, as if of extreme terror, habitually characterized his utterance. There were times, indeed, when I thought his unceasingly agitated mind was laboring with some oppressive secret, to divulge which he struggled for the necessary courage. At times, again, I was obliged to resolve all into the mere inexplicable vagaries of madness, for I beheld him gazing upon vacancy for long hours, in an attitude of the profoundest attention, as if listening to some imaginary sound. It was no wonder that his condition terrified—that it infected me. I felt creeping upon me, by slow yet certain degrees, the wild influences of his own fantastic yet impressive superstitions.
It was, especially, upon retiring to bed late in the night of the seventh or eighth day after the placing of the lady Madeline within the donjon, that I experienced the full power of such feelings. Sleep came not near my couch—while the hours waned and waned away. I struggled to reason off the nervousness which had dominion over me. I endeavored to believe that much, if not all of what I felt, was due to the bewildering influence of the gloomy furniture of the room—of the dark and tattered draperies, which, tortured into motion by the breath of a rising tempest, swayed fitfully to and fro upon the walls, and rustled uneasily about the decorations of the bed. But my efforts were fruitless. An irrepressible tremor gradually pervaded my frame; and, at length, there sat upon my very heart an incubus of utterly causeless alarm. Shaking this off with a gasp and a struggle, I uplifted myself upon the pillows, and, peering earnestly within the intense darkness of the chamber, hearkened—I know not why, except that an instinctive spirit prompted me—to certain low and indefinite sounds which came, through the pauses of the storm, at long intervals, I knew not whence. Overpowered by an intense sentiment of horror, unaccountable yet unendurable, I threw on my clothes with haste (for I felt that I should sleep no more during the night), and endeavored to arouse myself from the pitiable condition into which I had fallen, by pacing rapidly to and fro through the apartment.
I had taken but few turns in this manner, when a light step on an adjoining staircase arrested my attention. I presently recognized it as that of Usher. In an instant afterward he rapped, with a gentle touch, at my door, and entered, bearing a lamp. His countenance was, as usual, cadaverously wan—but, moreover, there was a species of mad hilarity in his eyes—an evidently restrained hysteria in his whole demeanor. His air appalled me—but anything was preferable to the solitude which I had so long endured, and I even welcomed his presence as a relief.
“And you have not seen it?” he said abruptly, after having stared about him for some moments in silence—“you have not then seen it?—but, stay! you shall.” Thus speaking, and having carefully shaded his lamp, he hurried to one of the casements, and threw it freely open to the storm.
The impetuous fury of the entering gust nearly lifted us from our feet. It was, indeed, a tempestuous yet sternly beautiful night, and one wildly singular in its terror and its beauty. A whirlwind had apparently collected its force in our vicinity; for there were frequent and violent alterations in the direction of the wind; and the exceeding density of the clouds (which hung so low as to press upon the turrets of the house) did not prevent our perceiving the life-like velocity with which they flew careering from all points against each other, without passing away into the distance. I say that even their exceeding density did not prevent our perceiving this—yet we had no glimpse of the moon or stars, nor was there any flashing forth of the lightning. But the under surfaces of the huge masses of agitated vapor, as well as all terrestrial objects immediately around us, were glowing in the unnatural light of a faintly luminous and distinctly visible gaseous exhalation which hung about and enshrouded the mansion.
“You must not—you shall not behold this!” said I, shuddering, to Usher, as I led him, with a gentle violence, from the window to a seat. “These appearances, which bewilder you, are merely electrical phenomena not uncommon—or it may be that they have their ghastly origin in the rank miasma of the tarn. Let us close this casement;—the air is chilling and dangerous to your frame. Here is one of your favorite romances. I will read, and you shall listen:—and so we will pass away this terrible night together.”
The antique volume which I had taken up was the “Mad Trist” of Sir Launcelot Canning; but I had called it a favorite of Usher’s more in sad jest than in earnest; for, in truth, there is little in its uncouth and unimaginative prolixity which could have had interest for the lofty and spiritual ideality of my friend. It was, however, the only book immediately at hand; and I indulged a vague hope that the excitement which now agitated the hypochondriac, might find relief (for the history of mental disorder is full of similar anomalies) even in the extremeness of the folly which I should read. Could I have judged, indeed, by the wild overstrained air of vivacity with which he hearkened, or apparently hearkened, to the words of the tale, I might well have congratulated myself upon the success of my design.
I had arrived at that well-known portion of the story where Ethelred, the hero of the Trist, having sought in vain for peaceable admission into the dwelling of the hermit, proceeds to make good an entrance by force. Here, it will be remembered, the words of the narrative run thus:
“And Ethelred, who was by nature of a doughty heart, and who was now mighty withal, on account of the powerfulness of the wine which he had drunken, waited no longer to hold parley with the hermit, who, in sooth, was of an obstinate and maliceful turn, but, feeling the rain upon his shoulders, and fearing the rising of the tempest, uplifted his mace outright, and, with blows, made quickly room in the plankings of the door for his gauntleted hand; and now pulling therewith sturdily, he so cracked, and ripped, and tore all asunder, that the noise of the dry and hollow-sounding wood alarumed and reverberated throughout the forest.”
At the termination of this sentence I started and, for a moment, paused; for it appeared to me (although I at once concluded that my excited fancy had deceived me)—it appeared to me that, from some very remote portion of the mansion, there came, indistinctly to my ears, what might have been, in its exact similarity of character, the echo (but a stifled and dull one certainly) of the very cracking and ripping sound which Sir Launcelot had so particularly described. It was, beyond doubt, the coincidence alone which had arrested my attention; for, amid the rattling of the sashes of the casements, and the ordinary commingled noises of the still increasing storm, the sound, in itself, had nothing, surely, which should have interested or disturbed me. I continued the story:
“But the good champion Ethelred, now entering within the door, was sore enraged and amazed to perceive no signal of the maliceful hermit; but, in the stead thereof, a dragon of a scaly and prodigious demeanor, and of a fiery tongue, which sat in guard before a palace of gold, with a floor of silver; and upon the wall there hung a shield of shining brass with this legend enwritten—
|
Who entereth herein, a conqueror hath bin;
Who slayeth the dragon, the shield he shall win. |
And Ethelred uplifted his mace, and struck upon the head of the dragon, which fell before him, and gave up his pesty breath, with a shriek so horrid and harsh, and withal so piercing, that Ethelred had fain to close his ears with his hands against the dreadful noise of it, the like whereof was never before heard.”
Here again I paused abruptly, and now with a feeling of wild amazement—for there could be no doubt whatever that, in this instance, I did actually hear (although from what direction it proceeded I found it impossible to say) a low and apparently distant, but harsh, protracted, and most unusual screaming or grating sound—the exact counterpart of what my fancy had already conjured up for the dragon’s unnatural shriek as described by the romancer.
Oppressed, as I certainly was, upon the occurrence of this second and most extraordinary coincidence, by a thousand conflicting sensations, in which wonder and extreme terror were predominant, I still retained sufficient presence of mind to avoid exciting, by any observation, the sensitive nervousness of my companion. I was by no means certain that he had noticed the sounds in question; although, assuredly, a strange alteration had, during the last few minutes, taken place in his demeanor. From a position fronting my own, he had gradually brought round his chair, so as to sit with his face to the door of the chamber; and thus I could but partially perceive his features, although I saw that his lips trembled as if he were murmuring inaudibly. His head had dropped upon his breast—yet I knew that he was not asleep, from the wide and rigid opening of the eye as I caught a glance of it in profile. The motion of his body, too, was at variance with this idea—for he rocked from side to side with a gentle yet constant and uniform sway. Having rapidly taken notice of all this, I resumed the narrative of Sir Launcelot, which thus proceeded:
“And now, the champion, having escaped from the terrible fury of the dragon, bethinking himself of the brazen shield, and of the breaking up of the enchantment which was upon it, removed the carcass from out of the way before him, and approached valorously over the silver pavement of the castle to where the shield was upon the wall; which in sooth tarried not for his full coming, but fell down at his feet upon the silver floor, with a mighty great and terrible ringing sound.”
No sooner had these syllables passed my lips, than—as if a shield of brass had indeed, at the moment, fallen heavily upon a floor of silver—I became aware of a distinct, hollow, metallic, and clangorous, yet apparently muffled, reverberation. Completely unnerved, I leaped to my feet; but the measured rocking movement of Usher was undisturbed. I rushed to the chair in which he sat. His eyes were bent fixedly before him, and throughout his whole countenance there reigned a stony rigidity. But, as I placed my hand upon his shoulder, there came a strong shudder over his whole person; a sickly smile quivered about his lips; and I saw that he spoke in a low, hurried, and gibbering murmur, as if unconscious of my presence. Bending closely over him, I at length drank in the hideous import of his words.
“Not hear it?—yes, I hear it, and have heard it. Long—long—long—many minutes, many hours, many days, have I heard it—yet I dared not—oh, pity me, miserable wretch that I am!—I dared not—I dared not speak! We have put her living in the tomb! Said I not that my senses were acute? I now tell you that I heard her first feeble movements in the hollow coffin. I heard them—many, many days ago—yet I dared not— I dared not speak! And now—to-night—Ethelred—ha! ha!—the breaking of the hermit’s door, and the death-cry of the dragon, and the clangor of the shield!—say, rather, the rending of her coffin, and the grating of the iron hinges of her prison, and her struggles within the coppered archway of the vault! Oh! whither shall I fly? Will she not be here anon? Is she not hurrying to upbraid me for my haste? Have I not heard her footstep on the stair? Do I not distinguish that heavy and horrible beating of her heart? Madman!”—here he sprang furiously to his feet, and shrieked out his syllables, as if in the effort he were giving up his soul— “Madman! I tell you that she now stands without the door!”
As if in the superhuman energy of his utterance there had been found the potency of a spell, the huge antique panels to which the speaker pointed threw slowly back, upon the instant, their ponderous and ebony jaws. It was the work of the rushing gust—but then without those doors there did stand the lofty and enshrouded figure of the lady Madeline of Usher.  For a moment she remained trembling and reeling to and fro upon the threshold—then, with a low moaning cry, fell heavily inward upon the person of her brother, and in her violent and now final death-agonies, bore him to the floor a corpse, and a victim to the terrors he had anticipated.
From that chamber, and from that mansion, I fled aghast. The storm was still abroad in all its wrath as I found myself crossing the old causeway. Suddenly there shot along the path a wild light, and I turned to see whence a gleam so unusual could have issued; for the vast house and its shadows were alone behind me. The radiance was that of the full, setting, and blood-red moon which now shone vividly through that once barely-discernible fissure of which I have before spoken as extending from the roof of the building, in a zigzag direction, to the base. While I gazed, this fissure rapidly widened—there came a fierce breath of the whirlwind—the entire orb of the satellite burst at once upon my sight—my brain reeled as I saw the mighty walls rushing asunder—there was a long tumultuous shouting sound like the voice of a thousand waters—and the deep and dank tarn at my feet closed sullenly and silently over the fragments of the “ House of Usher .”


17.1: Chapter Introduction
-
- Last updated
- Save as PDF
While psychodynamic theory was developing in Europe, American psychology was largely under the influence of behaviorism. The American psychologist John B. Watson (of “Little Albert” fame) is considered to be the father of behaviorism. Although he is not known for addressing issues of personality development, he did feel it was important for behaviorists to do so. His approach involved reducing personality to smaller and smaller units of behavior referred to as habit systems, suggesting that personality was very consistent. Nonetheless, through further conditioning the personality of an individual could change, leading Watson to make the bold statement that if he was given a dozen healthy infants he could take any one at random and train him or her for any career, including “doctor, lawyer, artist, merchant-chief and yes, even beggar-man and thief” (cited in Stagner, 1988; see also Lundin, 1979).
As the scientific study of behavior continued, it became common to try determining behavior on the basis of mathematical models. This work led to an era of grand learning theories, which culminated in the highly complex models of behavior proposed by Clark Hull (see Bower & Hilgard, 1981). This research took the behavioral study of personality in a very different direction than psychodynamic theory. The direction in which B. F. Skinner took personality theory, however, was so different that it became known as “ radical behaviorism .” Skinner rejected anything he could not directly observe, so concepts such as consciousness, thought, reasoning, and the “mind,” were all considered irrelevant to the study of individuals. Only the specific behaviors performed by the individual were open to being examined in Skinner’s form of behaviorism.
In contrast, John Dollard and Neal Miller tried to find some common ground between psychodynamic theory and learning theory. Dollard was a true generalist, with interests in anthropology and sociology in addition to psychology. Miller studied with two renowned learning theorists, Edwin Guthrie and Clark Hull, and was psychoanalyzed by Heinz Hartman in Vienna, while studying in Europe in the 1930s. Together, Dollard and Miller tried to develop a theory that would encompass psychodynamic theory, learning theory, and the influence of sociocultural factors. Their effort to develop what might be called a unified theory of personality stands in stark contrast to the constraints of radical behaviorism. Most importantly, they set the stage for the social learning theorists who followed.


3.4: Disproofs
The idea of a “disproof” is really just semantics – in order to disprove a statement we need to prove its negation.
So far we’ve been discussing proofs quite a bit, but have paid very little attention to a really huge issue. If the statements we are attempting to prove are false, no proof is ever going to be possible. Really, a prerequisite to developing a facility with proofs is developing a good “lie detector.” We need to be able to guess, or quickly ascertain, whether a statement is true or false.  If we happen across a value that satisfies the statement’s hypotheses but doesn’t satisfy the conclusion, we’ve found what is known as a counterexample .
Consider the following statement about integers and divisibility:
\[∀a, b, c ∈ \mathbb{Z}, a| bc \implies a| b ∨ a| c.\]
This is phrased as a UCS, so the hypothesis is clear, we’re looking for three integers so that the first divides the product of the other two. In the following table, we have collected several values for \(a\), \(b\) and \(c\) such that \(a| bc\).
| \(a\) | \(b\) | \(c\) | \(a| b ∨ a| c ?\) |
|---|---|---|---|
| \(2\) | \(7\) | \(6\) | \(\text{yes}\) |
| \(2\) | \(4\) | \(5\) | \(\text{yes}\) |
| \(3\) | \(12\) | \(11\) | \(\text{yes}\) |
| \(3\) | \(5\) | \(15\) | \(\text{yes}\) |
| \(5\) | \(4\) | \(15\) | \(\text{yes}\) |
| \(5\) | \(10\) | \(3\) | \(\text{yes}\) |
| \(7\) | \(2\) | \(14\) | \(\text{yes}\) |
As noted in Section 1.2 the statement above is related to whether or not a is prime. Note that in the table, only prime values of \(a\) appear. This is a rather broad hint. Find a counterexample to Conjecture \(3.4.1\).
There can be times when the search for a counterexample starts to feel really futile. Would you think it likely that a statement about natural numbers could be true for (more than) the first \(50\) numbers a yet still be false?
\(∀n ∈ \mathbb{Z} + n^2 − 79n + 1601\) is prime.
Find a counterexample to Conjecture \(3.4.2\)
Hidden within Euclid’s proof of the infinitude of the primes is a sequence. Recall that in the proof we deduced a contradiction by considering the number \(N\) defined by
\[N = 1 + \prod_{k=1}^{n} p_k. \]
Define a sequence by
\[N_n = 1 + \prod_{k=1}^{n} p_k, \]
where \(\{p_1, p_2, . . . , p_n\}\) are the actual first \(n\) primes. The first several values of this sequence are:
| \(n\) | \(N_n\) |
|---|---|
| \(1\) | \(1 + (2) = 3\) |
| \(2\) | \(1 + (2 · 3) = 7\) |
| \(3\) | \(1 + (2 · 3 · 5) = 31\) |
| \(4\) | \(1 + (2 · 3 · 5 · 7) = 211\) |
| \(5\) | \(1 + (2 · 3 · 5 · 7 · 11) = 2311\) |
| \(⋮\) | \(⋮\) |
Now, in the proof, we deduced a contradiction by noting that \(N_n\) is much larger than \(p_n\), so if \(p_n\) is the largest prime it follows that \(N_n\) can’t be prime – but what really appears to be the case (just look at that table!) is that \(N_n\) actually is prime for all \(n\).
Find a counterexample to the conjecture that \(1 + \prod^{n}_{k=1} p_k\) is itself always a prime.
Exercises:
Find a polynomial that assumes only prime values for a reasonably large range of inputs.
Find a counterexample to Conjecture \(3.4.1\) using only powers of \(2\).
The alternating sum of factorials provides an interesting example of a sequence of integers.
\(1! = 1 \\ 2! − 1! = 1 \\ 3! − 2! + 1! = 5 \\ 4! − 3! + 2! − 1! = 19 \\ \text{et cetera}\)
Are they all prime? (After the first two \(1\)’s.)
It has been conjectured that whenever \(p\) is prime, \(2^p − 1\) is also prime. Find a minimal counterexample.
True or false: The sum of any two irrational numbers is irrational. Prove your answer.
True or false: There are two irrational numbers whose sum is rational. Prove your answer.
True or false: The product of any two irrational numbers is irrational. Prove your answer.
True or false: There are two irrational numbers whose product is rational. Prove your answer.
True or false: Whenever an integer n is a divisor of the square of an integer, \(m^2\), it follows that \(n\) is a divisor of \(m\) as well. (In symbols, \(∀n ∈ \mathbb{Z}, ∀m ∈ \mathbb{Z}, n | m^2 \implies n | m.)\) Prove your answer.
In an exercise in Section 3.2 , we proved that the quadratic equation \(ax^2 + bx + c = 0\) has two solutions if \(ac < 0\). Find a counterexample which shows that this implication cannot be replaced with a biconditional.


8: Sequences, Probability, and Counting Theory
In this chapter, we will explore the mathematics behind situations involving probabilities and counting. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning.
-
- 8.1: Prelude to Sequences, Probability and Counting Theory
- A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a villa in Saint Barthélemy? A luxury convertible? A cruise around the world? The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over the next 30 years or so.
-
- 8.2: Sequences and Their Notations
- One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.
-
- 8.3: Arithmetic Sequences
- In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation. For example, companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
-
- 8.5: Series and Their Notations
- The sum of the terms of a sequence is called a series. Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, ∑, to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. In this section, we will learn how to use series to address annuity problems.
Contributors
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .