2.3: Narrative of the Third Voyage, 1498-1500 (Excerpt)
I have always read that the world comprising the land and the water was spherical, and the recorded experiences of Ptolemy and all others have proved this by the eclipses of the moon and other observations made from East to West, as well as the elevation of the Pole from North to South. But as I have already described, I have now seen so much irregularity, that I have come to another conclusion respecting the Earth, namely, that it is not round as they describe, but of the form of a pear, which is very round except where the stalk grows, at which part it is most prominent; or like a round ball upon part of which is a prominence like a woman’s nipple, this protrusion being the highest and nearest the sky, situated under the equinoctial line, and at the eastern extremity of this sea. . . .
Ptolemy and the other philosophers who have written upon the globe thought that it was spherical, believing that this [western] hemisphere was round as well as that in which they themselves dwelt, the centre of which was in the island of Arin, which is under the equinoctial line between the Arabian Gulf and the Gulf of Persia; and the circle passes over Cape St. Vincent in Portugal westward, and eastward by Cangara and the Seras;—in which hemisphere I make no difficulty as to its being a perfect sphere as they describe; but this western half of the world I maintain is like half of a very round pear, having a raised projection for the stalk, as I have already described, or like a woman’s nipple on a round ball. Ptolemy and the others who have written on the globe had no information respecting this part of the world, which was then unexplored; they only established their own hemisphere, which, as I have already said, is half of a perfect sphere. And now that your Highnesses have commissioned me to make this voyage of discovery, the truths which I have stated are evidently proved, because in this voyage, when I was off the island of Hargin and its vicinity, which is twenty degrees to the North of the equinoctial line, I found the people black and the land very much burnt; and when after that I went to the Cape Verde Islands I found the people there very much darker still, and the more southward we went, the more they approach the extreme of blackness; so that when I reached the parallel of Sierra Leone, where, as night came on, the North star rose five degrees, the people there were excessively black, and as I sailed westward the heat became extreme. But after I had passed the meridian or line which I have already described, I found the climate became gradually more temperate; so that when I reached the island of Trinidad, where the North star rose five degrees as night came on, there, and in the land of Gracia, I found the temperature exceedingly mild; the fields and the foliage likewise were remarkably fresh and green, and as beautiful as the gardens of Valencia in April. The people there are very graceful in form, less dark than those whom I had before seen in the Indies, and wear their hair long and smooth; they are also more shrewd, intelligent, and courageous. The sun was then in the sign of Virgo over our heads and theirs; therefore all this must proceed from the extreme blandness of the temperature, which arises, as I have said, from this country being the most elevated in the world and the nearest to the sky. On these grounds, therefore, I affirm that the globe is not spherical, but that there is the difference in its form which I have described; the which is to be found in this hemisphere at the point where the Indies meet the ocean, the extremity of the hemisphere being below the equinoctial line. And a great confirmation of this is, that when our Lord made the sun, the first light appeared in the first point of the East, where the most elevated point of the globe is.


9.1: Introduction
- What is the strategic management process?
- What is the difference between a firm’s vision and its mission?
- Why is strategic analysis important to strategy formulation?
- What are strategic objectives, levels of strategy, and a grand strategy? How are they related?
- How and why do managers plan? Why are goals important in the planning process?
- How and why do managers evaluate the effectiveness of strategic plans?
Chieh Huang, Boxed
As a successful entrepreneur, Chieh Huang knows how spot a business opportunity. His current business venture, Boxed, ships warehouse-club type products directly to customers’ homes. The company has grown from $8 million in revenue to more than $100 million in just three years. How has Huang achieved this success? He started “basically trying to solve a problem that I myself have. I grew up in the burbs, and every other weekend would go to the Price Club, and then I went to the city and didn’t have a car anymore. Am I just supposed to get ripped off?” 1 Huang thought that other millennials might be in the same situation and developed a company to offer bulk items like paper towels and energy bars at warehouse-club prices to millennials who want app-based shopping convenience.
Exhibit 9.2 Chieh Huang Chieh Huang, founder and CEO of Boxed. (Credit: Boxed.com / Attribution 2.0 Generic (CC BY 2.0))
Huang explained his entrepreneurial approach this way: “With repeat entrepreneurs, you not only solve a problem, you look for changes taking place in the world that become tailwinds to help the business exponentially grow.” 2 Environmental analysis might reveal an opportunity, but strategic planning is what makes it grow. Huang has managed growth by obtaining the resources necessary to serve more customers. As CEO of Boxed, Huang has raised money to build distribution centers, hired employees, developed private-label products to offer customers low prices, and expanded supplier relationships.
What is the strategy at Boxed? Huang discussed the company’s position in an interview on CNBC. He acknowledged that today’s selling environment is focused on “value, convenience, and brand,” 3 and said that when companies sell at similar low prices and offer similar delivery services, the only real differentiator left is brand. Huang has worked hard to develop the Boxed brand, promoting it on CNN, MSNBC, and the Today Show. The Boxed brand is also enhanced by reports of the benefits Huang offers employees. The CEO pays college tuition for employees’ children and even pays for employee weddings. For millennials, a company’s values have become part of its value, and if the price and convenience offered by Boxed match other sellers, Boxed’s values may be their best asset in attracting customers. This YouTube video is a CNBC story about Boxed with an interview with Huang.
References
1. Feldman, Amy (2016). “Costco For Millennials: How Chieh Huang Built Boxed, A Mobile Juggernaut With $100M+ In Revenue.” Forbes . www.forbes.com/sites/forbest.../#7870fb552adb Accessed September 4, 2017.
2. Mccausland, Christianna (2016). “Boxed founder Chieh Huang shares lessons in entrepreneurship.” Johns Hopkins Magazine . https://hub.jhu.edu/magazine/2016/su...ang-interview/ . Accessed September 4, 2017.
3. Huang, Chieh (2016). Interview on Mad Money, MSNBC. Available on Youtube: https://www.youtube.com/ watch?v=3ANAe1vLAIw


4.1: Lab 4- Brain and Cranial Nerves
Measurable Outcomes
- Complete the dissection of the sheep brain and identify (with a pin) all of the structures of the brain using the corresponding vocabulary list.
- Locate structures of the brain and cranial nerves on the various models in the lab.
- Identify the 12 cranial nerves as well as their target organs.
- Determine the composition of the of the brain from dissections.
- Determine/trace the path of cerebrospinal fluid through the brain.
- Differentiate histology from different regions of the brain.
- Demonstrate an adequate understand of the material in this section.
Background
The central nervous system entails all neurons of the brain and spinal cord. The brain is the central processing organ of the body and contains 100 billion neurons and a remarkable 1 trillion glial cells. It is estimated that cortical neurons alone consume around 5 billion ATP molecules per second. Whats more, some neurons can have axons that extend several feet. Unlike the spinal cord, the gray and white matter in the brain are arranged in three segments. From deep to superficial, the innermost region is made of gray matter which is surrounded by the myelinated axons of the white matter. The thin layer of the cerebral cortex responsible for higher order cognition is the outermost layer of gray matter. The brain is divided into four major regions, the brainstem, diencephalon, cerebellum, and cerebrum. The brainstem contains the medulla oblongata, pons, and midbrain (which houses the pineal gland). Caudal to the forebrain is the diencephalon, a region which contains the epithalamus, hypothalamus, thalamus and third ventricle.
There are four cavities in the brain called ventricles; here cerebrospinal fluid (CSF) is produced and circulated by ependymal cells and the choroid plexuses. The two largest ventricles lie within each cerebral hemispheres and are known as the lateral ventricles. Cerebrospinal fluid drains from the lateral ventricles, through the interventricular foramen and into the third ventricle. The third ventricle lies between the halves of the thalamus. From here, it flows through the cerebral aqueduct (aqueduct of sylvius) and into the fourth ventricle, which lies between the cerebellum and the pons. Cerebrospinal fluid drains from the fourth ventricle, into the lateral and median apertures and down through the central canal of the spinal cord. Cerebrospinal fluid leaks out through foramina into the subarachnoid space where it is reabsorbed by veins on the surface of the brain and spinal cord.
Like the spinal cord, the brain is protected by three meninx, the dura, arachnoid and pia mater. Unlike the spinal meninges, the cranial dura mater is subdivided into two distinct layers; the periosteal layer, which is the superficial mot layer, and the inner meningeal dura mater. The two dural layers form the superior sagittal sinus which collectively channels venous blood from the brain. The falx cerebri divides the cerebrum into left and right hemispheres, the falx cerebelli divides the cerebellum into left and right hemispheres, and the tentorium cerebelli forms a physical barrier between the cerebrum and the cerebellum.
Vocabulary for the Brain and Cranial Nerves on page(s)
162-163
.


7.2: Career Development as a Lifelong Process
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Entire books have been written, and graduate-level courses are taught about theories of career development. Most of the recently-published theories indicate that career development is a life-long, nonlinear process. Many students think the process starts with at least a little bit of career research, progresses to a singular career decision and then continues with education, employment and perhaps professional development in that single career field. The traditional idea is that we pick a career and work in that career for the rest of our professional lives. Unfortunately, that view is incorrect, and in fact, we need to be open and flexible plus plan to continue career development and decision-making throughout our lives.
Most of the widely-accepted career development theories indicate that career development is a circular, or at least non-linear, process that lasts throughout our entire lives. We often learn about careers from important adults in our lives during our childhood. As we go through our primary and secondary education, we learn about additional careers from our educational experiences, and we ourselves may start working in part-time jobs and/or volunteer in our communities.
If we continue on for post-secondary education, we learn more about careers from our instructors and perhaps required internships or other experiential learning activities. Some may enter the military and learn about careers in our armed forces. Others may move into full-time employment rather than post-secondary education. On all of those paths, we are exposed to events and opportunities. How we react to those events and opportunities plays a large part in our career development process. We learn about the work world and ourselves in all of our experiences, and that learning influences career choices. If you’d like to explore specific career development theories, please see the references and recommendations for further reading at the end of this chapter.


8.1: Introduction
Learning Objectives
- Apply the nursing process to clients with bipolar disorders
- Describe nursing assessments related to bipolar disorders
- Identify common nursing problems/diagnoses related to bipolar disorders
- Establish a safe environment
- Apply evidence-based practice when planning and implementing nursing care
- Describe common treatments for bipolar disorders
- Identify appropriate referrals to community resources
- Evaluate the effectiveness of interventions
- Provide patient education to clients and their family members
Bipolar disorder is a mental illness that causes dramatic shifts in a person’s mood, energy, and ability to think clearly. Moods shift from abnormally elevated moods called manic episodes to abnormal low moods of depression. See Figure 8.1 [1] for a depiction of the shifts in mood that occur with bipolar disorder. Severe bipolar episodes of mania can also include hallucinations or delusions, which can be confused with symptoms of schizophrenia. [2] This chapter will discuss the signs, symptoms, and treatments for bipolar disorder and explain how to apply the nursing process when caring for clients with bipolar disorder.
- “ P_culture.svg ” by he:משתמש:נעמה מ is licensed under CC BY-SA 3.0 ↵
- National Alliance on Mental Illness. (2017, August). Bipolar disorder. https://www.nami.org/About-Mental-Illness/Mental-Health-Conditions/Bipolar-Disorder ↵


Definition of statically equivalent systems. Outline of how to determine if sets of forces are statically equivalent, and how to convert multiple forces into a single statically equivalent force.
The process for resolving a force applied at one point on a body into a force and a moment acting on another point, using the concept of couples. Includes several worked examples.
Definition of the equivalent point force and methods of calculating it in two and three dimensions: integration; using the centroid or center of volume. Includes several worked samples.
Procedure to find the location of the centroid/center of mass of a shape by the Method of Composite Parts: breaking the shape down into simpler components for individual analysis.


1.10: Exercises
Exercises
Exercise \(\PageIndex{1}\)
In Sec. 1.2, we derived the eigenstates of a particle in an empty infinite space by considering a box of length \(L\) on each side, applying periodic boundary conditions, and taking \(L \rightarrow \infty\) . Suppose we instead use Dirichlet boundary conditions (i.e., the wavefunction vanishes on the walls of the box). Show that this gives rise to the same set of momentum eigenstates in the \(L \rightarrow \infty\) limit.
Exercise \(\PageIndex{2}\)
Using the results for the 1D delta-function scattering problem described in Section 1.3, calculate the probability current
\[J(x) = \frac{\hbar}{2mi}\left(\psi^*\frac{d\psi}{dx} - \psi\frac{d\psi^*}{dx}\right),\]
where \(\psi(x)\) is the total (incident + scattered) wavefunction. Explain the relationship between the values of \(J\) on the left and right side of the scatterer.
Exercise \(\PageIndex{3}\)
Derive the Green’s function for a free particle in 1D space:
\[\langle x|\hat{G}_0|x'\rangle = \frac{2m}{\hbar^2} \cdot \frac{1}{2ik_i} \exp\left(ik_i|x-x'|\right).\]
Exercise \(\PageIndex{4}\)
In Section 1.8, the scattering amplitude \(f(\mathrm{k}\rightarrow\mathrm{k}')\) for the 3D scattering problem was derived using the Born series. Derive the corresponding expressions for 1D and 2D.
Further Reading
[1] Bransden & Joachain, §13.1—13.3 and §13.5—13.6.
[2] Sakurai, §7.1–7.3, 7.5–7.6


1: Defining Open
Origins of OER
If you are reading this, then perhaps you, too, believe that Open Educational Resources (OER) benefits teaching and learning. OERs not only help mitigate the rising course material costs for students, they also provide faculty with a text
unbound
by publishers.
Unbound
means that they are largely self-published, openly and frequently updated, and they are re-mixable, easily providing students with course materials without having to use costly new editions.
The "Open" part of OER, relies on educators sharing their materials freely with students in a course, but sharing them freely in general. Allowing other educators to use them in their course, in part or in whole, so that they can then share their work and so on. This open ecosystem allows educators to quickly iterate, providing the most up-to-date course materials to any course, all while students are netting the benefit from zero course material costs.
-
- 1.1: Freely Open and Accessible
- With the advent of digital materials and learning management systems being integrated in grounded and online classes, course materials and resources are more accessible (in terms of access) and accessible (in terms of WACAG standards) than ever before. The terms 'free' and 'open licensed' as they relate to educational materials are often used interchangeably or in association with OER. That is because the intent of OER materials, by design, is to be open for use and adaption freely.
-
- 1.2: Open Licensing and Creative Commons Primer
- This resource is a condensed adaptation of the Open Educational Resources: OER 101 Tutorials for Designers. Providing explanations behind what Open Educational Resources (OER) are, how Creative Commons impacts it, and the meaning and ideology of "Open".
-
- 1.3: OERs in Teaching
- Open educational resources are somewhat different from open learning, in that they are primarily content, while open learning includes both content and educational services, such as specially designed online materials, in-built learner support and assessment. Open educational resources cover a wide range of online formats, including online textbooks, video recorded lectures, YouTube clips, web-based textual materials designed for independent study, animations and simulations, digital diagrams a
When did OER become a thing?
While the sharing of information and knowledge isn't a new concept in higher education, the shift from a business model to a more open-source model can be seen as relatively new. One of the earliest references that helped coin the term OER can be see within a report in 2002 UNESCO Forum on Open Courseware. The definition and the notions behind OERs were further expanded upon five years later by the William and Flora Hewlett Foundation .
Why Open Education Matters from Blink Tower on Vimeo .
In 2018, the EDUCAUSE Learning Initiative, or ELI, published a definition of aspects and policies around OERs, stating that,
Open educational resources (OER) are t eaching, learning, and research materials in any medium that reside in the public domain or that have been released under an open license that permits no-cost access, use, adaptation, and redistribution by others . OER confer significant dollar savings while also giving learners ready access to a wide range of high-quality, highly flexible educational materials. Open content offers faculty a means to customize curriculum to better align with learner needs and interests and to collaborate in new ways with peers worldwide. ( 7 Things You Should Know About Open Education )
Redistribution is aided by the Creative Commons movement, as it provided the avenue for creators to share works that could be reused by others while retaining attribution. From these three examples, we see another term develop at universities and other institutions: the Zero-Textbook course or ZTC . These are courses where the faculty have designed their course around OERs and other digital materials freely available to students and thus removing the additional cost burden of the expense of textbooks and other course-related material fees. Today, more and more institutions are working to bring OER options to students. Whether that is sourcing OER texts that are already available or providing faculty opportunities to author their own.
Slow but steady
The shift to more widely developed and available OERs has felt like a slow process in the eye of many instructional designers and faculty. As rising course costs increase, student demand could increase adoption. Although OERs can be written using any kind of word processing software, there are a lot of additional benefits of having an OER created through a web-based platform solution. The ability to update content easily, send out notifications to users of the text of changes, and provide accessibility features are just a few of the benefits of using a web-published host. It also provides more flexible options for remixing, or the ability to use whole or parts of the works in another OER. Later in this OER, we will take a look at LibreText and its Remixer tool that is built into their platform that was designed to leverage those features.
Attribution:
Background and Introduction is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.


1.2: Use the Language of Algebra
By the end of this section, you will be able to:
- Find factors, prime factorizations, and least common multiples
- Use variables and algebraic symbols
- Simplify expressions using the order of operations
- Evaluate an expression
- Identify and combine like terms
- Translate an English phrase to an algebraic expression
This chapter is intended to be a brief review of concepts that will be needed in an Intermediate Algebra course. A more thorough introduction to the topics covered in this chapter can be found in the Elementary Algebra chapter, Foundations.
Find Factors, Prime Factorizations, and Least Common Multiples
The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.
Similarly, a multiple of 3 would be the product of a counting number and 3.
We could find the multiples of any number by continuing this process.
| Counting Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Multiples of 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| Multiples of 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| Multiples of 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| Multiples of 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| Multiples of 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| Multiples of 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| Multiples of 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| Multiples of 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
A number is a multiple of \(n\) if it is the product of a counting number and \(n\).
Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, \(15÷3\) is \(5\), so \(15\) is \(5⋅3\).
If a number \(m\) is a multiple of \(n\), then \(m\) is divisible by \(n\).
If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:
A number is divisible by:
- 2 if the last digit is 0, 2, 4, 6, or 8.
- 3 if the sum of the digits is divisible by 3.
- 5 if the last digit is 5 or 0.
- 6 if it is divisible by both 2 and 3.
- 10 if it ends with 0.
Is 5,625 divisible by
- 2?
- 3?
- 5 or 10?
- 6?
- Answer
-
a.
\(\text{Is 5,625 divisible by 2?}\)
\( \begin{array}{ll} \text{Does it end in 0, 2, 4, 6 or 8?} & {\text{No.} \\ \text{5,625 is not divisible by 2.}} \end{array}\) -
b.
\(\text{5,625 divisible by 3?}\)
\(\begin{array}{ll} {\text{What is the sum of the digits?} \\ \text{Is the sum divisible by 3?}} & {5+6+2+5=18 \\ \text{Yes.} \\ \text{5,625 is divisible by 3.}}\end{array}\) -
c.
\(\text{Is 5,625 divisible by 5 or 10?}\)
\(\begin{array}{ll} \text{What is the last digit? It is 5.} & \text{5,625 is divisible by 5 but not by 10.} \end{array}\) d.\(\text{Is 5,625 divisible by 6?}\)
\(\begin{array}{ll}\text{Is it divisible by both 2 and 3?} & {\text{No, 5,625 is not divisible by 2, so 5,625 is} \\ \text{not divisible by 6.}} \end{array}\)
Is 4,962 divisible by a. 2? b. 3? c. 5? d. 6? e. 10?
- Answer
-
a. yes b. yes c. no d. yes e. no
Is 3,765 divisible by a. 2? b. 3? c. 5? d. 6? e. 10?
- Answer
-
a. no b. yes c. yes d. no e. no
In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if \(m\) is a multiple of \(n\), we can say that \(m\) is divisible by \(n\). For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.
Since \(8·9=72\), we say that 8 and 9 are factors of 72. When we write \(72=8·9\), we say we have factored 72.
Other ways to factor \(72\) are \(1·72, \; 2·36, \; 3·24, \; 4·18,\) and \(6⋅12\). The number 72 has many factors: \(1,\,2,\,3,\,4,\,6,\,8,\,9,\,12,\,18,\,24,\,36,\) and \(72\).
If \(a\) and \(b\) are counting numbers, and \(a·b=m\), then \(a\) and \(b\) are factors of \(m\).
Some numbers, such as 72, have many factors. Other numbers have only two factors.
A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.
A composite number is a counting number that is not prime. A composite number has factors other than 1 and the number itself.
The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.
A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.
The prime factorization of a number is the product of prime numbers that equals the number.
To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.
If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.
Factor 48.
- Answer
-
We say \(2⋅2⋅2⋅2⋅3\) is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.\(2⋅2⋅2⋅2⋅3\) is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.
If we first factored 48 in a different way, for example as \(6·8\), the result would still be the same. Finish the prime factorization and verify this for yourself.
Find the prime factorization of \(80\).
- Answer
-
\(2⋅2⋅2⋅2⋅5\)
Find the prime factorization of \(60\).
- Answer
-
\(2⋅2⋅3⋅5\)
- Find two factors whose product is the given number, and use these numbers to create two branches.
- If a factor is prime, that branch is complete. Circle the prime, like a leaf on the tree.
- If a factor is not prime, write it as the product of two factors and continue the process.
- Write the composite number as the product of all the circled primes.
One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.
Find the least common multiple (LCM) of 12 and 18 using the prime factors method.
- Answer
-
Notice that the prime factors of 12 \((2·2·3)\) and the prime factors of 18 \((2⋅3⋅3)\) are included in the LCM \((2·2·3·3)\). So 36 is the least common multiple of 12 and 18.
By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.
Find the LCM of 9 and 12 using the Prime Factors Method.
- Answer
-
36
Find the LCM of 18 and 24 using the Prime Factors Method.
- Answer
-
72
- Write each number as a product of primes.
- List the primes of each number. Match primes vertically when possible.
- Bring down the columns.
- Multiply the factors.
Use Variables and Algebraic Symbols
In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are \(x,\,y,\,a,\,b,\) and \(c.\)
A variable is a letter that represents a number whose value may change.
A number whose value always remains the same is called a constant .
A constant is a number whose value always stays the same.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below.
| Operation | Notation | Say: | The result is… |
|---|---|---|---|
| Addition | \(a+b\) | \(a\) plus \(b\) | the sum of \(a\) and \(b\) |
| Subtraction | \(a−b\) | \(a\) minus \(b\) | the difference of \(a\) and \(b\) |
| Multiplication | \(a⋅b,\,ab,\,(a)(b),\,(a)b,\,a(b)\) | \(a\) times \(b\) | the product of \(a\) and \(b\) |
| Division | \(a÷b,\,\space a/b,\,\space\frac{a}{b},\,\space b \overline{\smash{)}a}\) | \(a\) divided by \(b\) |
the quotient of \(a\) and \(b\);
\(a\) is called the dividend, and \(b\) is called the divisor |
When two quantities have the same value, we say they are equal and connect them with an equal sign.
\(a=b\) is read "\(a\) is equal to \(b\)."
The symbol “\(=\)” is called the equal sign.
On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “\(<\)” and “\(>\)”.
The expressions \(a<b\) or \(a>b\) can be read from left to right or right to left, though in English we usually read from left to right. In general,
\[a<b \text{ is equivalent to }b>a. \text{For example, } 7<11 \text{ is equivalent to }11>7.\]
\[a>b \text{ is equivalent to }b<a. \text{For example, } 17>4 \text{ is equivalent to }4<17.\]
| Inequality Symbols | Words |
|---|---|
| \(a\neq b\) | \(a\) is not equal to \(b\). |
| \(a<b\) | \(a\) is less than \(b\). |
| \(a\leq b\) | \(a\) is less than or equal to \(b\). |
| \(a>b\) | \(a\) is greater than \(b\). |
| \(a\geq b\) | \(a\) is greater than or equal to \(b\). |
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.
\[\begin{array}{lc} \text{Parentheses} & \mathrm{()} \\ \text{Brackets} & \mathrm{[]} \\ \text{Braces} & \mathrm{ \{ \} } \end{array}\]
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
\[8(14−8) \qquad 21−3[2+4(9−8)] \qquad 24÷ \{13−2[1(6−5)+4]\}\]
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have expressions and equations .
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
\[\begin{array}{lll} \textbf{Expression} & \textbf{Words} & \textbf{English Phrase} \\ \mathrm{3+5} & \text{3 plus 5} & \text{the sum of three and five} \\ \mathrm{n−1} & n\text{ minus one} & \text{the difference of } n \text{ and one} \\ \mathrm{6·7} & \text{6 times 7} & \text{the product of six and seven} \\ \frac{x}{y} & x \text{ divided by }y & \text{the quotient of }x \text{ and }y \end{array} \]
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.
An equation is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
An equation is two expressions connected by an equal sign.
\[\begin{array}{ll} \textbf{Equation} & \textbf{English Sentence} \\ 3+5=8 & \text{The sum of three and five is equal to eight.} \\ n−1=14 & n \text{ minus one equals fourteen.} \\ 6·7=42 & \text{The product of six and seven is equal to forty-two.} \\ x=53 & x \text{ is equal to fifty-three.} \\ y+9=2y−3 & y \text{ plus nine is equal to two } y \text{ minus three.} \end{array}\]
Suppose we need to multiply 2 nine times. We could write this as \(2·2·2·2·2·2·2·2·2\). This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write \(2·2·2\) as \(\mathrm{2^3}\) and \(2·2·2·2·2·2·2·2·2\) as \(2^9\). In expressions such as \(2^3\), the 2 is called the base and the 3 is called the exponent . The exponent tells us how many times we need to multiply the base .
We say \(2^3\) is in exponential notation and \(2·2·2\) is in expanded notation .
\(a^n\) means multiply \(n\) factors of the number \(a\).
The expression \(a^n\) is read \(a\) to the \(n^{th}\) power.
While we read \(a^n\) as \(“a\) to the \(n^{th}\) power”, we usually read:
\[\begin{array}{cc} a^2 & “a \text{ squared}” \\ a^3 & “a \text{ cubed}” \end{array}\]
We’ll see later why \(a^2\) and \(a^3\) have special names.
Table shows how we read some expressions with exponents.
| Expression | In Words | |
|---|---|---|
| 7 2 | 7 to the second power or | 7 squared |
| 5 3 | 5 to the third power or | 5 cubed |
| 9 4 | 9 to the fourth power | |
| 12 5 | 12 to the fifth power |
Simplify Expressions Using the Order of Operations
To simplify an expression means to do all the math possible. For example, to simplify \(\mathrm{4·2+1}\) we would first multiply \(\mathrm{4⋅2}\) to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
\[ 4⋅2+1 \\ 8+1 \\ 9\]
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
To simplify an expression , do all operations in the expression.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations . Otherwise, expressions may have different meanings, and they may result in different values.
For example, consider the expression \(4+3⋅7\). Some students simplify this getting 49, by adding \(4+3\) and then multiplying that result by 7. Others get 25, by multiplying \(3·7\) first and then adding 4.
The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.
-
Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
-
Exponents
- Simplify all expressions with exponents.
-
Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
-
Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase “Please Excuse My Dear Aunt Sally”.
\[\begin{array}{ll} \text{Parentheses} & \text{Please} \\ \text{Exponents} & \text{Excuse} \\ \text{Multiplication Division} & \text{My Dear} \\ \text{Addition Subtraction} & \text{Aunt Sally} \end{array}\]
It’s good that “ M y D ear” goes together, as this reminds us that m ultiplication and d ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “ A unt S ally” goes together and so reminds us that a ddition and s ubtraction also have equal priority and we do them in order from left to right.
Simplify: \(18÷6+4(5−2)\).
- Answer
-
Parentheses? Yes, subtract first. Exponents? No. Multiplication or division? Yes. Divide first because we multiply and divide left to right. Any other multiplication or division? Yes. Multiply. Any other multiplication of division? No. Any addition or subtraction? Yes. Add.
Simplify: \(30÷5+10(3−2).\)
- Answer
-
16
Simplify: \(70÷10+4(6−2).\)
- Answer
-
23
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
Simplify: \(5+2^3+3[6−3(4−2)].\)
- Answer
-
Are there any parentheses (or other grouping symbols)? Yes. Focus on the parentheses that are inside the brackets. Subtract. Continue inside the brackets and multiply. Continue inside the brackets and subtract. The expression inside the brackets requires no further simplification. Are there any exponents? Yes. Simplify exponents. Is there any multiplication or division? Yes. Multiply. Is there any addition of subtraction? Yes. Add. Add.
Simplify: \(9+5^3−[4(9+3)].\)
- Answer
-
86
Simplify: \(7^2−2[4(5+1)].\)
- Answer
-
1
Evaluate an Expression
In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.
Evaluate when \(x=4\): a. \(x^2\) b. \(3^x\) c. \(2x^2+3x+8\).
- Answer
-
a.
b.Use definition of exponent. Simplify.
c.Use definition of exponent. Simplify. Follow the order of operations.
Evaluate when \(x=3\), a. \(x^2\) b. \(4^x\) c. \(3x^2+4x+1\).
- Answer
-
a. 9
b. 64
c. 40
Evaluate when \(x=6\), a. \(x^3\) b. \(2^x\) c. \(6x^2−4x−7\).
- Answer
-
a. 216
b. 64
c. 185
Identify and Combine Like Terms
Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.
A term is a constant or the product of a constant and one or more variables.
Examples of terms are \(7,\,y,\,5x^2,\,9a,\) and \(b^5\).
The constant that multiplies the variable is called the coefficient .
The coefficient of a term is the constant that multiplies the variable in a term.
Think of the coefficient as the number in front of the variable. The coefficient of the term \(3x\) is 3. When we write \(x\), the coefficient is 1, since \(x=1⋅x\).
Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms .
Look at the following 6 terms. Which ones seem to have traits in common?
\[5x \quad 7 \quad n^2 \quad 4 \quad 3x \quad 9n^2\]
We say,
\(7\) and \(4\) are like terms.
\(5x\) and \(3x\) are like terms.
\(n^2\) and \(9n^2\) are like terms.
Terms that are either constants or have the same variables raised to the same powers are called like terms.
If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.
\[\begin{array}{lc} \text{Simplify.} & 4x+7x+x \\ \text{Add the coefficients.} & 12x \end{array}\]
Simplify: \(2x^2+3x+7+x^2+4x+5\).
- Answer
-
Simplify: \(3x^2+7x+9+7x^2+9x+8\).
- Answer
-
\(10x^2+16x+17\)
Simplify: \(4y^2+5y+2+8y^2+4y+5.\)
- Answer
-
\(12y^2+9y+7\)
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add or subtract the coefficients and keep the same variable for each group of like terms.
Translate an English Phrase to an Algebraic Expression
We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table summarizes them.
| Operation | Phrase | Expression |
|---|---|---|
| Addition |
\(a\) plus \(b\)
the sum of \(a\) and \(b\) \(a\) increased by \(b\) \(b\) more than \(a\) the total of \(a\) and \(b\) \(b\) added to \(a\) |
\(a+b\) |
| Subtraction |
\(a\) minus \(b\)
the difference of \(a\) and \(b\) \(a\) decreased by \(b\) \(b\) less than \(a\) \(b\) subtracted from \(a\) |
\(a−b\) |
| Multiplication |
\(a\) times \(b\)
the product of \(a\) and \(b\) twice \(a\) |
\(a·b,\,ab,\,a(b),\,(a)(b)\)
\(2a\) |
| Division |
\(a\) divided by \(b\)
the quotient of \(a\) and \(b\) the ratio of \(a\) and \(b\) \(b\) divided into \(a\) |
\(a÷b,\,a/b,\,\frac{a}{b},\,b \overline{\smash{)}a}\) |
Look closely at these phrases using the four operations:
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
Translate each English phrase into an algebraic expression:
a. the difference of \(14x\) and \(9\)
b. the quotient of \(8y^2\) and \(3\)c. twelve more than \(y\)
d. seven less than \(49x^2\)
- Answer
-
a. The key word is difference , which tells us the operation is subtraction. Look for the words of and and t o find the numbers to subtract.
b. The key word is quotient , which tells us the operation is division.
c. The key words are more than. They tell us the operation is addition. More than means “added to.”
\[\text{twelve more than }y \\ \text{twelve added to }y \\ y+12\]
d. The key words are less than . They tell us to subtract. Less than means “subtracted from.”
\[\text{seven less than }49x^2 \\ \text{seven subtracted from }49x^2 \\ 49x^2−7\]
Translate the English phrase into an algebraic expression:
a. the difference of \(14x^2\) and \(13\)
b. the quotient of \(12x\) and \(2\)
c. \(13\) more than \(z\)
d. \(18\) less than \(8x\)
- Answer
-
a. \(14x^2−13\) b. \(12x÷2\)
c. \(z+13\) d. \(8x−18\)
Translate the English phrase into an algebraic expression:
a. the sum of \(17y^2\) and \(19\)
b. the product of \(7\) and \(y\)
c. Eleven more than \(x\)
d. Fourteen less than \(11a\)
- Answer
-
a. \(17y^2+19\) b. \(7y\)
c. \(x+11\) d. \(11a−14\)
We look carefully at the words to help us distinguish between multiplying a sum and adding a product.
Translate the English phrase into an algebraic expression:
a. eight times the sum of \(x\) and \(y\)
b. the sum of eight times \(x\) and \(y\)
- Answer
-
There are two operation words— times tells us to multiply and sum tells us to add.
a. Because we are multiplying \(8\) times the sum, we need parentheses around the sum of \(x\) and \(y\), \((x+y)\). This forces us to determine the sum first. (Remember the order of operations.)
\[\text{eight times the sum of }x \text{ and }y \\ 8(x+y)\]
b. To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of eight times \(x\) and \(y\).
Translate the English phrase into an algebraic expression:
a. four times the sum of \(p\) and \(q\)
b. the sum of four times \(p\) and \(q\)
- Answer
-
a. \(4(p+q)\) b. \(4p+q\)
Translate the English phrase into an algebraic expression:
a. the difference of two times \(x\) and \(8\)
b. two times the difference of \(x\) and \(8\)
- Answer
-
a. \(2x−8\) b. \(2(x−8)\)
Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.
The length of a rectangle is 14 less than the width. Let \(w\) represent the width of the rectangle. Write an expression for the length of the rectangle.
- Answer
-
\[\begin{array}{lc} \text{Write a phrase about the length of the rectangle.} & \text{14 less than the width} \\ \text{Substitute }w \text{ for “the width.”} & w \\ \text{Rewrite less than as subtracted from.} & \text{14 subtracted from } w \\ \text{Translate the phrase into algebra.} & w−14 \end{array}\]
The length of a rectangle is 7 less than the width. Let \(w\) represent the width of the rectangle. Write an expression for the length of the rectangle.
- Answer
-
\(w−7\)
The width of a rectangle is \(6\) less than the length. Let \(l\) represent the length of the rectangle. Write an expression for the width of the rectangle.
- Answer
-
\(l−6\)
The expressions in the next example will be used in the typical coin mixture problems we will see soon.
June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes.
- Answer
-
\[\begin{array}{lc} \text{Write a phrase about the number of dimes.} & \text{7 less than 4 times }q \\ \text{Translate 4 times }q. & \text{7 less than 4}q \\ \text{Translate the phrase into algebra.} & 4q−7 \end{array}\]
Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes.
- Answer
-
\(4q−8\)
Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let \(n\) represent the number of nickels. Write an expression for the number of dimes.
- Answer
-
\(7n+3\)
Key Concepts
-
Divisibility Tests
A number is divisible by:
2 if the last digit is 0, 2, 4, 6, or 8.
3 if the sum of the digits is divisible by 3.
5 if the last digit is 5 or 0.
6 if it is divisible by both 2 and 3.
10 if it ends with 0. -
How to find the prime factorization of a composite number.
- Find two factors whose product is the given number, and use these numbers to create two branches.
- If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
- If a factor is not prime, write it as the product of two factors and continue the process.
- Write the composite number as the product of all the circled primes.
-
How To Find the least common multiple using the prime factors method.
- Write each number as a product of primes.
- List the primes of each number. Match primes vertically when possible.
- Bring down the columns.
- Multiply the factors.
-
Equality Symbol
\(a=b\) is read “ \(a\) is equal to \(b\).” The symbol “=” is called the equal sign. -
Inequality Symbols
Inequality Symbols Words \(a≠b\) \(a\) is not equal to \(b\). \(a<b\) \(a\) is less than \(b\). \(a≤b\) \(a\) is less than or equal to \(b\). \(a>b\) \(a\) is greater than \(b\). \(a≥b\) \(a\) is greater than or equal to \(b\). - Grouping Symbols \(\begin{array}{lc} \text{Parentheses} & \mathrm{()} \\ \text{Brackets} & \mathrm{[]} \\ \text{Braces} & \mathrm{ \{ \} } \end{array}\)
- Exponential Notation \(a^n\) means multiply \(a\) by itself, \(n\) times. The expression an is read \(a\) to the \(n^{th}\) power.
-
Simplify an Expression
To simplify an expression, do all operations in the expression. -
How to use the order of operations.
-
Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
-
Exponents
- Simplify all expressions with exponents.
-
Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
-
Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
-
Parentheses and Other Grouping Symbols
-
How to combine like terms.
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add or subtract the coefficients and keep the same variable for each group of like terms.
Operation Phrase Expression Addition \(a\) plus \(b\)
the sum of \(a\) and \(b\)
\(a\) increased by \(b\)
\(b\) more than \(a\)
the total of \(a\) and \(b\)
\(b\) added to \(a\)\(a+b\) Subtraction \(a\) minus \(b\)
the difference of \(a\) and \(b\)
\(a\) decreased by \(b\)
\(b\) less than \(a\)
\(b\) subtracted from \(a\)\(a−b\) Multiplication \(a\) times \(b\)
the product of \(a\) and \(b\)
twice \(a\)\(a·b,\,ab,\,a(b),\,(a)(b)\)
\(2a\)
Division \(a\) divided by \(b\)
the quotient of \(a\) and \(b\)
the ratio of \(a\) and \(b\)
\(b\) divided into \(a\)\(a÷b,\,a/b,\,\frac{a}{b},\,b \overline{\smash{)}a}\)
Glossary
- coefficient
- The coefficient of a term is the constant that multiplies the variable in a term.
- composite number
- A composite number is a counting number that is not prime. It has factors other than 1 and the number itself.
- constant
- A constant is a number whose value always stays the same.
- divisible by a number
- If a number \(m\) is a multiple of \(n\), then \(m\) is divisible by \(n\).
- equation
- An equation is two expressions connected by an equal sign.
- evaluate an expression
- To evaluate an expression means to find the value of the expression when the variables are replaced by a given number.
- expression
- An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
- factors
- If \(a·b=m\), then \(a\) and \(b\) are factors of \(m\).
- least common multiple
- The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
- like terms
- Terms that are either constants or have the same variables raised to the same powers are called like terms.
- multiple of a number
- A number is a multiple of \(n\) if it is the product of a counting number and \(n\).
- order of operations
- The order of operations are established guidelines for simplifying an expression.
- prime factorization
- The prime factorization of a number is the product of prime numbers that equals the number.
- prime number
- A prime number is a counting number greater than 1 whose only factors are 1 and the number itself.
- simplify an expression
- To simplify an expression means to do all the math possible.
- term
- A term is a constant, or the product of a constant and one or more variables.
- variable
- A variable is a letter that represents a number whose value may change.


8.12: Symphony No. 9, From the New World
Dvorak’s “New World Symphony” is one of his best-known works. It was an immediate success in its day and has been widely performed since. Please pay special attention to section titled “Influences,” as it explores his use of American folk elements in the piece.
Introduction
The Symphony No. 9 in E minor, “From the New World,” Op. 95, B. 178 (Czech: Symfonie č. 9 e moll „Z nového světa“ ), popularly known as the New World Symphony , was composed by Antonín Dvořák in 1893 while he was the director of the National Conservatory of Music of America from 1892 to 1895. It is by far his most popular symphony, and one of the most popular of all symphonies. In older literature and recordings, this symphony was often numbered as Symphony No. 5. Neil Armstrong took a recording of the New World Symphony to the Moon during the Apollo 11 mission, the first Moon landing, in 1969.
Instrumentation
This symphony is scored for an orchestra of at least the following:
- 2 flutes (one doubling piccolo)
- 2 oboes (one doubling English horn)
- 2 clarinets in A
- 2 bassoons
- 4 horns in E, C and F
- 2 trumpets in E, C and E ♭
- 2 tenor trombones
- Bass trombone
- Tuba (second movement only)
- Timpani
- Triangle (third movement only)
- Cymbals (fourth movement only)
- Strings
Listen: Movements
The piece has four movements. Please listen to each of the movements to hear the entire symphony.
Adagio, 4/8—Allegro molto, 2/4, E minor
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=332
Largo, common time, D-flat major, then later C-sharp minor
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=332
Scherzo: Molto vivace—Poco sostenuto, 3/4, E minor
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=332
Allegro con fuoco, common time, E minor, ends in E major
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=332
Influences
Dvořák was interested in Native American music and the African-American spirituals he heard in America. As director of the National Conservatory he encountered an African-American student, Harry T. Burleigh, later a composer himself, who sang traditional spirituals to him and said that Dvořák had absorbed their ‘spirit’ before writing his own melodies. Dvořák stated:
I am convinced that the future music of this country must be founded on what are called Negro melodies. These can be the foundation of a serious and original school of composition, to be developed in the United States. These beautiful and varied themes are the product of the soil. They are the folk songs of America and your composers must turn to them.
The symphony was commissioned by the New York Philharmonic, and premiered on December 16, 1893, at Carnegie Hall conducted by Anton Seidl. A day earlier, in an article published in the New York Herald on December 15, 1893, Dvořák further explained how Native American music had been an influence on this symphony:
I have not actually used any of the [Native American] melodies. I have simply written original themes embodying the peculiarities of the Indian music, and, using these themes as subjects, have developed them with all the resources of modern rhythms, counterpoint, and orchestral colour.
In the same article, Dvořák stated that he regarded the symphony’s second movement as a “sketch or study for a later work, either a cantata or opera . . . which will be based upon Longfellow’s [ The Song of ] Hiawatha ” (Dvořák never actually wrote such a piece). He also wrote that the third movement scherzo was “suggested by the scene at the feast in Hiawatha where the Indians dance.”
In 1893, a newspaper interview quoted Dvořák as saying “I found that the music of the negroes and of the Indians was practically identical”, and that “the music of the two races bore a remarkable similarity to the music of Scotland.” Most historians agree that Dvořák is referring to the pentatonic scale, which is typical of each of these musical traditions.
In a 2008 article in The Chronicle of Higher Education , prominent musicologist Joseph Horowitz asserts that African-American spirituals were a major influence on Dvořák’s music written in America, quoting him from an 1893 interview in the New York Herald as saying, “In the negro melodies of America I discover all that is needed for a great and noble school of music.” Dvořák did, it seems, borrow rhythms from the music of his native Bohemia, as notably in his Slavonic Dances, and the pentatonic scale in some of his music written in America from African-American and/or Native American sources. Statements that he borrowed melodies are often made but seldom supported by specifics. One verified example is the song of the Scarlet Tanager in the Quartet. Michael Steinberg writes that a flute solo theme in the first movement of the symphony resembles the spiritual “Swing Low, Sweet Chariot.” Leonard Bernstein averred that the symphony was truly multinational in its foundations.
Dvořák was influenced not only by music he had heard, but by what he had seen, in America. He wrote that he would not have composed his American pieces as he had, if he had not seen America. It has been said that Dvořák was inspired by the American “wide open spaces” such as prairies he may have seen on his trip to Iowa in the summer of 1893. Notices about several performances of the symphony include the phrase “wide open spaces” about what inspired the symphony and/or about the feelings it conveys to listeners.
Dvořák was also influenced by the style and techniques used by earlier classical composers including Beethoven and Schubert. The falling fourths and timpani strokes in the New World Symphony ’s Scherzo movement evokes the Scherzo of Beethoven’s Choral Symphony. In his fourth movement, Dvořák’s use of flashbacks to prior movements is reminiscent of Beethoven quoting prior movements as part of the opening Presto of the last movement.
Reception
At the premiere in Carnegie Hall, the end of every movement was met with thunderous clapping and Dvořák felt obliged to stand up and bow. This was one of the greatest public triumphs of Dvořák’s career. When the symphony was published, several European orchestras soon performed it. Alexander Mackenzie conducted the London Philharmonic Society in the European premiere on June 21, 1894. Clapham says the symphony became “one of the most popular of all time” and at a time when the composer’s main works were being welcomed in no more than ten countries, this symphony reached the rest of the musical world and has become a “universal favorite.” It is performed [as of 1978] more often “than any other symphony at the Royal Festival Hall, London” and is in “tremendous demand in Japan.”
Several themes from the symphony have been used widely in films, TV shows, anime, video games, and advertisements.
The Song “Goin’ Home”
The theme from the Largo was adapted into the spiritual-like song “Goin’ Home,” often mistakenly considered a folk song or traditional spiritual, by Dvořák’s pupil William Arms Fisher, who wrote the lyrics in 1922.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Symphony No. 9 (Dvorak). Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Symphony_No._9_(Dvo%C5%99%C3%A1k) . License : CC BY-SA: Attribution-ShareAlike


8.7: Conversational Implicature
Now that we've talked a lot about logical implication, let's take a second to talk about a perhaps more important phenomenon: conversational implicature.
If I go into the kitchen, find the main course missing from the counter, and ask "what happened to the roast?" You might, noticing that the dog looks satisfied and full, say "well, the dog certainly looks happy." Logically, you haven't answered the question. We can say a lot about the logical implications of what you've said, but that won't tell us what you've accomplished by saying what you said. Instead, we need the concept of a conversational implicature: a way we can communicate more than is logically implied by what we've said by relying on the rules of conversation.
In this example, there's a rule in conversations that says "don't say things that aren't relevant to the topic at hand." I'll therefore assume that you mean "the dog looks happy" to be a relevant piece of information to my question. So it seems like what you're implicating (even though you're not logically implying it) is that the dog ate the roast.
To highlight some of the awkwardness that can happen when we accidentally implicate things, here is a list of funny implicatures and the rules they violate which give them the unintended meaning:
-
I have a brother, I also have another brother
- This is called "scalar implicature": it's where you say something and everyone assumes that you've said the most informative quantitative statement you could have. Here, it's assumed that you would have made the strongest claim available: I have two brothers
- Mitch Hedberg has a great one liner that's also a sort of scalar implicature: "I used to do drugs. I still do, but I used to, too."
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Another Mitch Hedberg gem (he's a master of implicatures): "I want to hang a map of the world in my house. Then I'm gonna put pins into all the locations I've traveled to, but first I'm gonna have to travel to the top two corners of the map, so it won't fall down."
- The rule is something like "don't say you're going to do something like hang something on your wall if you don't mean to say that you're going to do everything required to do it in a normal way." By ignoring this rule, Mitch makes us assume that he is going to hang the map normally and then plays with that assumption.
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More Mitch Hedberg: "I wrote a script and I gave it to a guy who reads scripts. And he read it and he said he really likes it, but he thinks I need to re-write it. I said "Screw that, I'll just make a copy.""
- Of course, we're assuming that the editor in question meant that the script needed serious revision because of a rule that says something like "don't say something needs to be rewritten when you just mean it needs to be copied." or more generally "make appropriate requests given the technology of the time." Mitch is playing on that assumption by acting like he interpreted the editor the opposite way from the way he intended.
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Mitch Hedberg again: This one commercial said "Forget everything you know about slip covers," so I did. And it was a load off my mind. Then the commercial tried to sell slip covers, but I didn't know what they were.
- We all know that when someone says "forget everything you know about x" they aren't intending the phrase literally and instead, by convention, mean something like "here's a paradigm-shattering new product," or "I'm about to show you something which defies your previous assumptions about slip covers." Mitch is instead interpreting it literally, which isn't what convention dictates, even if Logic might dictate that we do so (in this case, it's unclear what logic would tell us to do)
To take a more relevant and controversial example: If my friend says, "Black people matter," and I respond with, "All people matter", then I'm saying something which logically implies that "black people matter." But even though my statement has this logical implication, there's a rule of conversation that says something like "don't make a broader statement when you mean to communicate something more specific." In breaking this rule, the conversational implicature is that people shouldn't be claiming that "black people matter" and instead should be claiming that "all people matter." The implicature is that there's something somehow wrong with saying "Black people matter" and that they should be saying something more general—that the person affirming the worth of black people has made some sort of mistake. In saying "all people matter" in response to someone saying, "black people matter", therefore, I'm doing something equivalent to the following:
"I love Oak trees"
"Well, I love all trees"
The implicature is that there's something wrong with saying one loves oak trees. That there's something mistaken in making the more specific claim. Again, it's like the following:
"We need to cure cancer as soon as possible."
"We need to cure all diseases as soon as possible."
The second speaker, because they are "zooming out" to a less specific claim, is implicating that there's a mistake in the first claim—that it’s too specific. But presumably there’s no problem with saying we need to cure cancer as soon as possible. This is partially because cancer is a particularly widespread and deadly disease.
Conversely, some people read the saying "black lives matter" as having the implicature that "black lives matter and other lives do not," or "black lives matter more than other lives." The idea here, I take it, is that in making this claim we are singling out a specific group when what we mean to be doing is affirming the importance of all lives and so we are making a mistake: we're implicating that black lives exclusively or especially matter. I don’t think there is a mistake in saying “black lives matter,” but the goal here is to understand how confusions begin (apart, perhaps, from problematic racial attitudes).
Both of these examples show how communication—particularly about sensitive and important issues—breaks down easily. When we're not all in agreement on the rules of the conversation and the common ground stock of knowledge we share, we can easily make claims that get misinterpreted. We all need to be charitable, take care and take our time, and be humble in our verbal interactions with one another. As always, try to focus more on your own mistakes than those of others.


6.1: Forest Site Productivity
6.1 Forest Site Productivity
Just as a farmer might wish to know how “good” his ground might be for various crops, so a forester will wish to know how “good” his forest land is. Since forests are dominated by trees, this generally translates to predicting how well trees will grow. Decisions about what species to grow, how intensely to manage the trees, or whether or not trees are the best crop for a particularly piece of ground, are all tied to how plants grow on a site. For example, temperatures may dictate a shift from Douglas-fir to noble fir ( Abies procera ) once a certain elevation is reached. Trees growing on a very productive site may be thinned more frequently over a rotation than trees growing on a less productive site. Marginal pine forestland may be more suitably managed for wildlife habitat than for timber.
Site quality refers to the inherent ability of a forest to produce biomass (or grow trees). It is the composite expression of a variety of physical and chemical attributes of a forested area, including its soil, topography and climate. Site characteristics such as:
- soil depth, texture, and fertility;
- slope, aspect, and elevation; and
- precipitation, temperature and length of growing season;
all combine to influence how well trees grow (Figure 6.1).
(Note: many people use the terms site quality and site productivity interchangeably. Purists, however, prefer “site quality” as a baseline indicator, as the productivity of a site can be altered by fertilizing, irrigating, mulching or altering the soil makeup.)
To obtain a measure of site quality, one might first think of examining these site variables and correlating them to tree growth. There have been some attempts at doing just that, but the amount of work and expense required to get meaningful data are generally too great for the range of conditions found in most forest ownerships. Some larger companies, such as Weyerhaeuser, have inventoried and classified their soils, and the Forest Service has developed plant associations to indicate site conditions, but neither may be available for the majority of land managers.
Intuitively, if one is interested in tree growth, then one solution is simply to measure how trees grow on the site. When measuring the trees, correlations are not required; the summation of all variables that influence tree growth is expressed in the biomass itself. The “proof is in the pudding” so to speak. And although the causal reasons for the productivity are not identified, as would be the case if all the influential variables were measured, a reliable indicator for tree growth on the site can be obtained. The question then becomes, “What is the best way to measure tree growth?”
Many measurable tree growth attributes are strongly influenced by stand density. If few trees are present on a site, the individual trees will have large crowns, and thus large diameters and wide growth rings. Conversely, trees of the same age in a denser stand will have narrower crowns, smaller diameters, and tighter growth rings (Figure 6.2).
Since tree volume is a function of tree diameter and height, volume is also tied directly to stand density. Therefore, diameter, crown volume, tree volume or tree ring growth do not make good measures of overall forest productivity. Average tree height, on the other hand, is not confounded in this manner except at extreme densities. Tree height is relatively independent of tree density for most forest tree species.  Therefore, tree height is a more reliable measure of the site’s inherent productivity than most other measures. It is also a quick and easy measurement to take in the field, unlike parameters such as soil fertility or microclimate.


2.7: Atomic Electron Configurations
Skills to Develop
Make sure you thoroughly understand the following essential ideas:
- State the principle feature of that distinguishes the energies of the excited states of a single-electron atom from atoms containing more than one electron.
- Explain why the first ionization energy of the helium atom is smaller than twice the first ionization of the hydrogen atom.
- Be able to write a plausible electron configuration for any atom having an atomic number less than 90.
In the previous section you learned that an electron standing-wave pattern characterized by the quantum numbers ( n,l,m ) is called an orbital . According to the Pauli exclusion principle , no two electrons in the same atom can have the same set of quantum numbers ( n,l,m,s ). This limits the number of electrons in a given orbital to two ( s = ±1), and it requires that atom containing more then two electrons must place them in standing wave patterns corresponding to higher principal quantum numbers n , which means that these electrons will be farther from the nucleus and less tightly bound by it.
In this chapter, we will see how the Pauli restrictions on the allowable quantum numbers of electrons in an atom affect the electronic configuration of the different elements, and, by influencing their chemical behavior, governs the structure of the periodic table.
One-electron atoms
Let us begin with atoms that contain only a single electron. Hydrogen is of course the only electrically neutral species of this kind, but by removing electrons from heavier elements we can obtain one-electron ions such as \(He^+\) and \(Li^{2+}\), etc. Each has a ground state configuration of 1 s 1 , meaning that its single electron exhibits a standing wave pattern governed by the quantum numbers n =1, m =0 and l =0, with the spin quantum number s undefined because there is no other electron to compare it with. All have simple emission spectra whose major features were adequately explained by Bohr's model.
The most important feature of a single-electron atom is that the energy of the electron depends only on the principal quantum number n . As the above diagram shows, the quantum numbers l and m have no effect on the energy; we say that all orbitals having a given value of n are degenerate . Thus the emission spectrum produced by exciting the electron to the n =2 level consists of a single line, not four lines. The wavelength of this emission line for the atoms H, He + and Li 2 + will diminish with atomic number because the greater nuclear charge will lower the energies of the various n levels. For the same reason, the energies required to remove an electron from these species increases rapidly as the nuclear charge increases, because the increasing attraction pulls the electron closer to the nucleus, thus producing an even greater attractive force.
Electron-Electron Repulsion
It takes 1312 kJ of energy to remove the electron from a mole of hydrogen atoms. What might we expect this value to be for helium? Helium contains two electrons, but its nucleus contains two protons; each electron "sees" both protons, so we might expect that the electrons of helium would be bound twice as strongly as the electron of hydrogen. The ionization energy of helium should therefore be twice 1312 kJ/mol, or 2612 kJ/mol. However, if one looks at the spectrum of helium, the continuum is seen to begin at a wavelength corresponding to an ionization energy of 2372 kJ/mol, or about 90% of the predicted value.
Why are the electrons in helium bound less tightly than the +2 nuclear charge would lead us to expect? The answer is that there is another effect to consider: the repulsion between the two electrons; the resulting electron-electron repulsion subtracts from the force holding the electron to the nucleus, reducing the local binding of each.
Electron-electron repulsion is a major factor in both the spectra and chemical behavior of the elements heavier than hydrogen. In particular, it acts to "break the degeneracy" (split the energies) of orbitals having the same value of n but different l .
Figure \(\PageIndex{1}\): The numbers on this diagram show the energies of some orbitals in the atoms of H, He, and Li, relative to the 2s-2p orbitals of H. The energies of the 1s orbitals fall rapidly as Z increases. The fall in energies of the 2s orbitals is smaller owing to electron-electron repulsion. The 2p orbitals have a node at the nucleus and are less affected by Z but are more affected by electron repulsion, so they remain higher than the 2s orbitals.
The diagram below shows how the energies of the s - and p -orbitals of different principal quantum numbers get split as the result of electron-electron repulsion. Notice the contrast with the similar diagram for one-electron atoms near the top of this page. The fact that electrons preferentially fill the lowest-energy empty orbitals is the basis of the rules for determining the electron configuration of the elements and of the structure of the periodic table.
The Aufbau rules
The German word Aufbau means "building up", and this term has traditionally been used to describe the manner in which electrons are assigned to orbitals as we carry out the imaginary task of constructing the atoms of elements having successively larger atomic numbers. In doing so, we are effectively "building up" the periodic table of the elements, as we shall shortly see.
- Electrons occupy the lowest-energy available orbitals; lower-energy orbitals are filled before the higher ones.
- No more than two electrons can occupy any orbital.
- For the lighter elements, electrons will fill orbitals of the same type only one electron at a time, so that their spins are all unpaired. They will begin to pair up only after all the orbitals are half-filled. This principle, which is a consequence of the electrostatic repulsion between electrons, is known as Hund's rule .
- For the first 18 elements, up to the point where the 3 s and 3 p levels are completely filled, this scheme is entirely straightforward and leads to electronic configurations which you are expected to be able to work out for each of these elements.
The preceding diagram illustrates the main idea here. Each orbital is represented as a little box that can hold up to two electrons having opposite spins, which we designated by upward- or downward-pointing arrows. Electrons fill the lowest-energy boxes first, so that additional electrons are forced into wave-patterns corresponding to higher (less negative) energies. Thus in the above diagram, the "third" electron of lithium goes into the higher-energy 2s orbital, giving this element an electron configuration which we write 1s 2 2s 1 .
Example \(\PageIndex{1}\): Phosphorus
What is the electron configuration of the atom of phosphorus, atomic number 15?
Solution:
The number of electrons filling the lowest-energy orbitals are:
1 s : 2 electrons, 2 s : 2 electrons; 2 p : 6 electrons, 3 s : 2 electrons. This adds up to 12 electrons. The remaining three electrons go into the 3 p orbital, so the complete electron configuration of P is 1 s 2 2 s 2 2 p 6 3 s 2 3 p 3 .
Energies of the highest occupied orbitals of the elements: This diagram illustrates the Aufbau rules as they are applied to all the elements. Note especially how the energies of the nd orbitals fall between the ( n –1) s and ( n –1) p orbitals so, for example the 3 d orbitals begin to fill after the 4 s orbital is filled, but before electrons populate the 4 p orbitals. A similar relation exists with d - and f -orbitals.
It is very important that you understand this diagram and how it follows from the Pauli exclusion principle: You should be able to reproduce it from memory up to the 6 s level, because it forms the fundamental basis of the periodic table of the elements.
Bending the rules
Inspection of a table of electron configurations of the elements reveals a few apparent non-uniformities in the filling of the orbitals, as is illustrated here for the elements of the so-called first transition series in which the 3 d orbitals are being populated. These anomalies are a consequence of the very small energy differences between some of the orbitals, and of the reduced electron-electron repulsion when electrons remain unpaired (Hund's rule), as is evident in chromium, which contains six unpaired electrons.
The other anomaly here is copper, which "should" have the outer-shell configuration 3 d 9 4 s 2 . The actual configuration of the Cu atom appears to be 3 d 10 4 s 1 . Although the 4 s orbital is normally slightly below the 3 d orbital energy, the two are so close that interactions between the two when one is empty and the other is not can lead to a reversal. Detailed calculations in which the shapes and densities of the charge distributions are considered predict that the relative energies of many orbitals can reverse in this way. It gets even worse when f -orbitals begin to fill!
Because these relative energies can vary even for the same atom in different chemical environments, most instructors will not expect you to memorize them.
This diagram shows how the atomic orbitals corresponding to different principal quantum numbers become interspersed with one another at higher values of n . The actual situation is more complicated than this; calculations show that the energies of d and f orbitals vary with the atomic number of the element.
The Periodic Table
The relative orbital energies illustrated above and the Pauli exclusion principle constitute the fundamental basis of the periodic table of the elements which was of course worked out empirically late in the 19th century, long before electrons had been heard of.
Figure \(\PageIndex{2}\) : Structure of the periodic table
The periodic table of the elements is conventionally divided into sections called blocks , each of which designates the type of "sub-orbital" ( s, p, d, f ) which contains the highest-energy electrons in any particular element. Note especially that
- The non-metallic elements occur only in the p -block;
- The d -block elements contain the so-called transition elements;
- The f -block elements go in between Groups 3 and 4 of the d -block.
The above diagram illustrates the link between the electron configurations of the elements and the layout of the periodic table. Each row, also known as a period , commences with two s-block elements and continues through the p block. At the end of the rows corresponding to n >1 is an element having a p 6 configuration, a so-called noble gas element . At n values of 2 and 3, d - and f -block element sequences are added.
The table shown above is called the long form of the periodic table; for many purposes, we can use a "short form" table in which the d -block is shown below the s - and p - block "representative elements" and the f -block does not appear at all. Note that the "long form" would be even longer if the f -block elements were shown where they actually belong, between La-Hf and Ac-Db.


12.1: Cumulative Review
Answers to the Cumulative Review can be found in the Supplemental Resources. Please visit http://openstax college.org to view an updated list of the Learning Resources for this title and how to access them.
Chapter 1 - Whole Numbers
No exercises.
Chapter 2 - The Language of Algebra
Simplify :
- 5(3 + 2 · 6) − 8 2
Solve :
- 17 = y − 13
- p + 14 = 23
Translate into an algebraic expression.
- 11 less than the product of 7 and x.
Translate into an algebraic equation and solve.
- Twice the difference of y and 7 gives 84.
- Find all the factors of 72.
- Find the prime factorization of 132.
- Find the least common multiple of 12 and 20.
Chapter 3 - Integers
Simplify :
- |8 − 9| − |3 − 8|
- −2 + 4(−3 + 7)
- 27 − (−4 − 7)
- 28 ÷ (−4) − 7
Translate into an algebraic expression or equation.
- The sum of −5 and 13, increased by 11.
- The product of −11 and 8.
- The quotient of 7 and the sum of −4 and m.
- The product of −3 and is −51.
Solve :
- −6r = 24
Chapter 4 - Fractions
- Locate the numbers on a number line. \(\dfrac{7}{8}, \dfrac{5}{3}, 3\dfrac{1}{4}, 5.
Simplify:
- \(\dfrac{21p}{57q}\)
- \(\dfrac{3}{7} \cdot \left(− \dfrac{28}{45}\right)\)
- \(−6 \dfrac{3}{4} \div \dfrac{9}{2}\)
- \(−3 \dfrac{3}{5} \div 6\)
- \(−4 \dfrac{2}{3} \left(− \dfrac{6}{7}\right)\)
- \(\dfrac{−2 \dfrac{1}{4}}{− \dfrac{3}{8}}\)
- \(\dfrac{7 \cdot 8 + 4(7 − 12)}{9 \cdot 6 − 2 \cdot 9}\)
- \(− \dfrac{23}{36} + \dfrac{17}{20}\)
- \(\dfrac{\dfrac{1}{2} + \dfrac{1}{3}}{\dfrac{3}{4} − \dfrac{1}{3}}\)
- \(3 \dfrac{5}{8} − 2 \dfrac{1}{2}\)
- \(− \dfrac{2}{3}\)r = 24
Chapter 5 - Decimals
Simplify :
- 24.76 − 7.28
- 12.9 + 15.633
- (−5.6)(0.25)
- $6.29 ÷ 12
- 3 4 (13.44 − 9.6)
- \(\sqrt{64} + \sqrt{225}\)
- \(\sqrt{121x^{2} y^{2}}\)
- Write in order from smallest to largest: \(\dfrac{5}{8}\), 0.75, \(\dfrac{8}{15}\)
Solve :
- −8.6x = 34.4
- Using 3.14 as the estimate for pi, approximate the (a) circumference and (b) area of a circle whose radius is 8 inches.
- Find the mean of the numbers, 18, 16, 20, 12
- Find the median of the numbers, 24, 29, 27, 28, 30
- Identify the mode of the numbers, 6, 4, 4, 5, 6, 6, 4, 4, 4, 3, 5
- Find the unit price of one t-shirt if they are sold at 3 for $28.97.
Chapter 6 - Percents
- Convert 14.7% to (a) a fraction and (b) a decimal.
Translate and solve.
- 63 is 35% of what number?
- The nutrition label on a package of granola bars says that each granola bar has 180 calories, and 81 calories are from fat. What percent of the total calories is from fat?
- Elliot received $510 commission when he sold a $3,400 painting at the art gallery where he works. What was the rate of commission?
- Nandita bought a set of towels on sale for $67.50. The original price of the towels was $90. What was the discount rate?
- Alan invested $23,000 in a friend’s business. In 5 years the friend paid him the $23,000 plus $9,200 interest. What was the rate of interest?
Solve :
- \(\dfrac{9}{p} = \dfrac{-6}{14}\)
Chapter 7 - The Properties of Real Numbers
- List the (a) whole numbers, (b) integers, (c) r ational numbers, (d) irrational numbers, (e) real numbers −5, \(−2 \dfrac{1}{4}, − \sqrt{4}, 0.2\bar{5}, \dfrac{13}{5}\), 4
Simplify :
- \(\left(\dfrac{8}{15} + \dfrac{4}{7}\right) + \dfrac{3}{7}\)
- 3(y + 3) − 8(y − 4)
- \(\dfrac{8}{17} \cdot 49 \cdot \dfrac{17}{8}\)
- A playground is 55 feet wide. Convert the width to yards.
- Every day last week Amit recorded the number of minutes he spent reading. The recorded number of minutes he read each day was 48, 26, 81, 54, 43, 62, 106. How many hours did Amit spend reading last week?
- June walked 2.8 kilometers. Convert this length to miles knowing 1 mile is 1.61 kilometer.
Chapter 8 - Solve Linear Equations
Solve :
- y + 13 = −8
- p + \(\dfrac{2}{5} = \dfrac{8}{5}\)
- 48 = \(\dfrac{2}{3}\)x
- 4(a − 3) − 6a = −18
- 7q + 14 = −35
- 4v − 27 = 7v
- \(\dfrac{7}{8}\)y − 6 = \(\dfrac{3}{8}\)y − 8
- 26 − 4(z − 2) = 6
- \(\dfrac{3}{4} x − \dfrac{2}{3} = \dfrac{1}{2} x − \dfrac{5}{6}\)
- 0.7y + 4.8 = 0.84y − 5.3
Translate and solve.
- Four less than n is 13.
Chapter 9 - Math Models and Geometry
- One number is 8 less than another. Their sum is negative twenty-two. Find the numbers.
- The sum of two consecutive integers is −95. Find the numbers.
- Wilma has $3.65 in dimes and quarters. The number of dimes is 2 less than the number of quarters. How many of each coin does she have?
- Two angles are supplementary. The larger angle is 24° more than the smaller angle. Find the measurements of both angles.
- One angle of a triangle is 20° more than the smallest angle. The largest angle is the sum of the other angles. Find the measurements of all three angles.
- Erik needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is 12 feet tall and Erik has 15 feet of wire. How far from the base of the antenna can he attach the wire?
- The width of a rectangle is 4 less than the length. The perimeter is 96 inches. Find the length and the width.
- Find the (a) volume and (b) surface area of a rectangular carton with length 24 inches, width 18 inches, and height 6 inches.
Chapter 10 - Polynomials
Simplify :
- (8m 2 + 12m − 5) − (2m 2 − 7m − 1)
- p 3 • p 10
- (y 4 ) 3
- (3a 5 ) 3
- (x 3 ) 5 (x 2 )3
- \(\left(\dfrac{2}{3} m^{3} n^{6}\right) \left(\dfrac{1}{6} m^{4} n^{4}\right)\)
- (y − 4)(y + 12)
- (3c + 1)(9c − 4)
- (x − 1)(x 2 − 3x − 2)
- (8x) 0
- \(\dfrac{(x^{3})^{5}}{(x^{2})^{4}}\)
- \(\dfrac{32a^{7} b^{2}}{12a^{3} b^{6}}\)
- (ab −3 )(a −3 b 6 )
- Write in scientific notation: (a) 4,800,000 (b) 0.00637
Factor the greatest common factor from the polynomial.
- 3x 4 − 6x 3 − 18x 2
Chapter 11 - Graphs
Graph :
- y = 4x − 3
- y = −3x
- y = \(\dfrac{1}{2}\)x + 3
- x − y = 6
- y = −2
- Find the intercepts. 2x + 3y = 12
Graph using the intercepts.
- 2x − 4y = 8
- Find the slope of the line shown.
- Use the slope formula to find the slope of the line between the points (−5, −2), (3, 2).
- Graph the line passing through the point (−3, 4) and with slope m = \(− \dfrac{1}{3}\).
Contributors and Attributions
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Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


24.5: Importance of Fungi in Human Life
Skills to Develop
- Describe the importance of fungi to the balance of the environment
- Summarize the role of fungi in food and beverage preparation
- Describe the importance of fungi in the chemical and pharmaceutical industries
- Discuss the role of fungi as model organisms
Although we often think of fungi as organisms that cause disease and rot food, fungi are important to human life on many levels. As we have seen, they influence the well-being of human populations on a large scale because they are part of the nutrient cycle in ecosystems. They have other ecosystem roles as well. As animal pathogens, fungi help to control the population of damaging pests. These fungi are very specific to the insects they attack, and do not infect animals or plants. Fungi are currently under investigation as potential microbial insecticides, with several already on the market. For example, the fungus Beauveria bassiana is a pesticide being tested as a possible biological control agent for the recent spread of emerald ash borer. It has been released in Michigan, Illinois, Indiana, Ohio, West Virginia and Maryland (Figure \(\PageIndex{1}\)).
The mycorrhizal relationship between fungi and plant roots is essential for the productivity of farm land. Without the fungal partner in root systems, 80–90 percent of trees and grasses would not survive. Mycorrhizal fungal inoculants are available as soil amendments from gardening supply stores and are promoted by supporters of organic agriculture.
We also eat some types of fungi. Mushrooms figure prominently in the human diet. Morels, shiitake mushrooms, chanterelles, and truffles are considered delicacies (Figure \(\PageIndex{2}\)). The humble meadow mushroom, Agaricus campestris , appears in many dishes. Molds of the genus Penicillium ripen many cheeses. They originate in the natural environment such as the caves of Roquefort, France, where wheels of sheep milk cheese are stacked in order to capture the molds responsible for the blue veins and pungent taste of the cheese.
Fermentation—of grains to produce beer, and of fruits to produce wine—is an ancient art that humans in most cultures have practiced for millennia. Wild yeasts are acquired from the environment and used to ferment sugars into CO 2 and ethyl alcohol under anaerobic conditions. It is now possible to purchase isolated strains of wild yeasts from different wine-making regions. Louis Pasteur was instrumental in developing a reliable strain of brewer’s yeast, Saccharomyces cerevisiae , for the French brewing industry in the late 1850s. This was one of the first examples of biotechnology patenting.
Many secondary metabolites of fungi are of great commercial importance. Antibiotics are naturally produced by fungi to kill or inhibit the growth of bacteria, limiting their competition in the natural environment. Important antibiotics, such as penicillin and the cephalosporins, are isolated from fungi. Valuable drugs isolated from fungi include the immunosuppressant drug cyclosporine (which reduces the risk of rejection after organ transplant), the precursors of steroid hormones, and ergot alkaloids used to stop bleeding. Psilocybin is a compound found in fungi such as Psilocybe semilanceata and Gymnopilus junonius, which have been used for their hallucinogenic properties by various cultures for thousands of years.
As simple eukaryotic organisms, fungi are important model research organisms. Many advances in modern genetics were achieved by the use of the red bread mold Neurospora crassa . Additionally, many important genes originally discovered in S. cerevisiae served as a starting point in discovering analogous human genes. As a eukaryotic organism, the yeast cell produces and modifies proteins in a manner similar to human cells, as opposed to the bacterium Escherichia coli, which lacks the internal membrane structures and enzymes to tag proteins for export. This makes yeast a much better organism for use in recombinant DNA technology experiments. Like bacteria, yeasts grow easily in culture, have a short generation time, and are amenable to genetic modification.
Summary
Fungi are important to everyday human life. Fungi are important decomposers in most ecosystems. Mycorrhizal fungi are essential for the growth of most plants. Fungi, as food, play a role in human nutrition in the form of mushrooms, and also as agents of fermentation in the production of bread, cheeses, alcoholic beverages, and numerous other food preparations. Secondary metabolites of fungi are used as medicines, such as antibiotics and anticoagulants. Fungi are model organisms for the study of eukaryotic genetics and metabolism.


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2.1: Tool Choices and Applications
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Although there are numerous advantages to choosing tools based on many factors, safety should be the first factor to be considered.
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2.2: Measuring, Marking, Leveling and Layout Tools
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Measuring and marking tools are common to multiple trades and ensure accuracy and quality craftsmanship in the building and construction process. Measuring devices are available in fractional and metric, with classic western construction practices adopting the fractional inch configuration. While some measuring tools have the actual fractions printed (1/2″, 1/4″, and 1/8″) next to each corresponding mark on the scale, many do not.
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2.3: Nails, Hammers and Pneumatic Nailers
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2.4: Threaded Fasteners, Drivers, Pliers and Wrenches
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2.5: Saws
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Saws should be chosen based on the material type to be cut and the particular task being performed. Each type of saw has it’s own purpose, may be available in a variety of sizes, and offer safety and convenience features which vary by manufacturer, style, and application.
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2.6: Drills and Accessories
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2.7: Grinders, Sanders, and Accessories
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4.1.8: End of Chapter Discussion
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Discussion
- Why is the concept of culture difficult to define? What do you think are the most important elements of culture?
- In the twenty-first century, people have much greater contact with members of other cultures than they did in the past. Which topics or concerns should be priorities for future studies in culture?
- COVID-19 had an impact on all cultures around the world and there were many short-term changes in behavior that we all had to incorporate into our daily lives. Since culture is integrated, what were some of the short-term changes to other aspects of our culture? What are some long-term cultural changes that we may see as a result of the pandemic?
GLOSSARY
Acculturation: An ongoing exchange of cultural traits between groups that have continuous first-hand contact; both groups experience change while remaining two distinct groups.
Adaptation: Any alteration in the structure or functioning of an organism (or group of organisms) that improves its ability to survive and reproduce in its environment.
Beliefs: All the mental aspects of culture including values, norms, philosophies, worldview, knowledge, and so forth.
Biological plasticity : An ability to adapt biologically in response to the environment.
Colonialism: The political, social, economic, and cultural domination of a territory and its people by a foreign power for an extended time.
Core Values: The key, basic, or central values that integrate culture and help distinguish it from others.
Cultural adaptation: The knowledge or behavior that enables humans or groups to adjust, survive and thrive in their environment.
Cultural generalities: Patterns or traits that are found in several, but not all, societies.
Cultural particularity: A distinct trait or feature that is confined to a single place, culture, or society.
Cultural relativism : The idea that we should seek to understand another person’s beliefs and behaviors from the perspective of their culture rather than our own.
Cultural universals: Patterns or traits that are found in all cultures.
Culture: A set of beliefs, practices, and symbols that are learned and shared. Together, they form an all-encompassing, integrated whole that binds people together and shapes their worldview and lifeways.
Culture lag: The time that elapses between the introduction of a new item of material culture and its acceptance as part of the nonmaterial culture.
Custom: A widely accepted way of doing something, specific to a particular society that has developed through repetition over a long period of time.
Diffusion: The borrowing of cultural traits between cultures, either directly or through intermediaries.
Enculturation: The process of learning culture.
Ethnocentrism : The tendency to view one’s own culture as most important and correct and as a stick by which to measure all other cultures.
Folkways: A loose collection of usual or customary ways in which the members of a particular cultural community behave.
Globalization: A series of processes that work trans-nationally to promote change in a world in which nations and people are increasingly interlinked and mutually dependent.
Independent invention: The process by which humans innovate, creatively finding solutions to problems.
International culture: Cultural traits that extend beyond national boundaries.
Laws: Explicitly stated rules that are enforced.
Mores: The standards of moral conduct and ethical behavior that the people in a cultural community expect of one another; what a community considers morally or ethically right or wrong.
National culture: The beliefs, behavior patterns, values, cultural traits and institutions shared within a country.
Norms: The expectations or rules, formal or informal, about what is considered appropriate or inappropriate behavior in a particular society.
Practices: Behaviors and actions that may be motivated by belief or performed without reflection as part of everyday routines.
Popular culture: The pattern of cultural experiences and attitudes that exist in mainstream society.
Social sanctions: Ways of communicating disapproval or putting pressure on people who violate a community’s mores.
Society: Organized life in groups.
Stressors: Something that causes strain or tension.
Subcultures: Smaller groups of people who share cultural traits and patterns within the same country.
Symbol : Something, verbal or non-verbal, that stands for something else, often without an obvious or natural connection.
Tradition: A set of practices, a constellation of beliefs, or mode of thinking that exists in the present, but was inherited from the past.
Values: The symbolic, abstract concepts or standards that represent the ideals of a group.