1: Electronic Structure of Atoms
In this chapter, we describe how electrons are arranged in atoms and how the spatial arrangements of electrons are related to their energies. We also explain how knowing the arrangement of electrons in an atom enables chemists to predict and explain the chemistry of an element. As you study the material presented in this chapter, you will discover how the shape of the periodic table reflects the electronic arrangements of elements. In this and subsequent chapters, we build on this information to explain why certain chemical changes occur and others do not. After reading this chapter, you will know enough about the theory of the electronic structure of atoms to explain what causes the characteristic colors of neon signs, how laser beams are created, and why gemstones and fireworks have such brilliant colors. In later chapters, we will develop the concepts introduced here to explain why the only compound formed by sodium and chlorine is NaCl, an ionic compound, whereas neon and argon do not form any stable compounds, and why carbon and hydrogen combine to form an almost endless array of covalent compounds, such as CH 4 , C 2 H 2 , C 2 H 4 , and C 2 H 6 . You will discover that knowing how to use the periodic table is the single most important skill you can acquire to understand the incredible chemical diversity of the elements.
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- 1.1: The Wave Nature of Light
- Understanding the electronic structure of atoms requires an understanding of the properties of waves and electromagnetic radiation. A basic knowledge of the electronic structure of atoms requires an understanding of the properties of waves and electromagnetic radiation. A wave is a periodic oscillation by which energy is transmitted through space. All waves are periodic, repeating regularly in both space and time. Waves are characterized by several interrelated properties.
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- 1.2: Quantized Energy and Photons
- Blackbody radiation is the radiation emitted by hot objects and could not be explained with classical physics. Max Planck postulated that energy was quantized and may be emitted or absorbed only in integral multiples of a small unit of energy, known as a quantum. The energy of a quantum is proportional to the frequency of the radiation; the proportionality constant h is a fundamental constant (Planck’s constant). Albert Einstein used the quantization of energy to explain the photoelectric effect
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- 1.3: Line Spectra and the Bohr Model
- There is an intimate connection between the atomic structure of an atom and its spectral characteristics. Most light is polychromatic and contains light of many wavelengths. Light that has only a single wavelength is monochromatic and is produced by devices called lasers, which use transitions between two atomic energy levels to produce light in a very narrow range of wavelengths. Atoms can also absorb light of certain energies, resulting in a transition from the ground state or a lower-energy e
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- 1.4: The Wave Behavior of Matter
- An electron possesses both particle and wave properties. Louis de Broglie showed that the wavelength of a particle is equal to Planck’s constant divided by the mass times the velocity of the particle. The electron in Bohr’s circular orbits could thus be described as a standing wave, one that does not move through space. Werner Heisenberg’s uncertainty principle states that it is impossible to precisely describe both the location and the speed of particles that exhibit wavelike behavior.
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- 1.5: Quantum Mechanics and Atomic Orbitals
- There is a relationship between the motions of electrons in atoms and molecules and their energies that is described by quantum mechanics. Because of wave–particle duality, scientists must deal with the probability of an electron being at a particular point in space. To do so required the development of quantum mechanics, which uses wavefunctions to describe the mathematical relationship between the motion of electrons in atoms and molecules and their energies.
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- 1.6: 3D Representation of Orbitals
- Orbitals with l = 0 are s orbitals and are spherically symmetrical, with the greatest probability of finding the electron occurring at the nucleus. Orbitals with values of n > 1 and l = 0 contain one or more nodes. Orbitals with l = 1 are p orbitals and contain a nodal plane that includes the nucleus, giving rise to a dumbbell shape. Orbitals with l = 2 are d orbitals and have more complex shapes with at least two nodal surfaces. l = 3 orbitals are f orbitals, which are still more complex.
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- 1.7: Many-Electron Atoms
- In addition to the three quantum numbers (n, l, ml) dictated by quantum mechanics, a fourth quantum number is required to explain certain properties of atoms. This is the electron spin quantum number (ms), which can have values of +½ or −½ for any electron, corresponding to the two possible orientations of an electron in a magnetic field. This is important for chemistry because the Pauli exclusion principle implies that no orbital can contain more than two electrons (with opposite spin).
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- 1.8: Electron Configurations
- Based on the Pauli principle and a knowledge of orbital energies obtained using hydrogen-like orbitals, it is possible to construct the periodic table by filling up the available orbitals beginning with the lowest-energy orbitals (the aufbau principle), which gives rise to a particular arrangement of electrons for each element (its electron configuration). Hund’s rule says that the lowest-energy arrangement of electrons is the one that places them in degenerate orbitals with parallel spins.
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- 1.9: Electron Configurations and the Periodic Table
- The arrangement of atoms in the periodic table results in blocks corresponding to filling of the ns, np, nd, and nf orbitals to produce the distinctive chemical properties of the elements in the s block, p block, d block, and f block, respectively.
Thumbnail: Electron shell diagram for Sodium , the 19th element in the periodic table of elements. (CC BY-SA; 2.5; Pumbaa )


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10.1
Points, Lines, and Planes
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line segment
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plane
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union
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intersection
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parallel
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perpendicular
10.2
Angles
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vertex
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right angle
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acute angle
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obtuse angle
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straight angle
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complementary
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supplementary
10.3
Triangles
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acute
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obtuse
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isosceles
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equilateral
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hypotenuse
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congruence
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similarity
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scaling factor
10.4
Polygons, Perimeter, and Circumference
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perimeter
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polygon
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pentagon
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hexagon
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heptagon
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octagon
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quadrilateral
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trapezoid
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parallelogram
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circumference
10.5
Tessellations
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tessellation
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translation
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reflection
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rotation
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glide reflection
10.6
Area
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triangle
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square
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rectangle
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rhombus
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apothem
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radius
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circle
10.7
Volume and Surface Area
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surface area
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volume
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right prism
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right cylinder
10.8
Right Triangle Trigonometry
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right triangle
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sine
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cosine
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tangent


27.5: Single Slit Diffraction
Learning Objectives
By the end of this section, you will be able to:
- Discuss the single slit diffraction pattern.
Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings. Figure 1 shows a single slit diffraction pattern. Note that the central maximum is larger than those on either side, and that the intensity decreases rapidly on either side. In contrast, a diffraction grating produces evenly spaced lines that dim slowly on either side of center.
The analysis of single slit diffraction is illustrated in Figure 2. Here we consider light coming from different parts of the same slit. According to Huygens’s principle, every part of the wavefront in the slit emits wavelets. These are like rays that start out in phase and head in all directions. (Each ray is perpendicular to the wavefront of a wavelet.) Assuming the screen is very far away compared with the size of the slit, rays heading toward a common destination are nearly parallel. When they travel straight ahead, as in Figure 2a, they remain in phase, and a central maximum is obtained. However, when rays travel at an angle \(\theta\) relative to the original direction of the beam, each travels a different distance to a common location, and they can arrive in or out of phase. In Figure 2b, the ray from the bottom travels a distance of one wavelength \(\lambda\) farther than the ray from the top. Thus a ray from the center travels a distance \(\lambda / 2\) farther than the one on the left, arrives out of phase, and interferes destructively. A ray from slightly above the center and one from slightly above the bottom will also cancel one another. In fact, each ray from the slit will have another to interfere destructively, and a minimum in intensity will occur at this angle. There will be another minimum at the same angle to the right of the incident direction of the light.
At the larger angle shown in Figure 2c, the path lengths differ by \(3 \lambda / 2\) for rays from the top and bottom of the slit. One ray travels a distance \(\lambda\) different from the ray from the bottom and arrives in phase, interfering constructively. Two rays, each from slightly above those two, will also add constructively. Most rays from the slit will have another to interfere with constructively, and a maximum in intensity will occur at this angle. However, all rays do not interfere constructively for this situation, and so the maximum is not as intense as the central maximum. Finally, in Figure 2d, the angle shown is large enough to produce a second minimum. As seen in the figure, the difference in path length for rays from either side of the slit is \(D \sin{\theta}\), and we see that a destructive minimum is obtained when this distance is an integral multiple of the wavelength.
Thus, to obtain destructive interference for a single slit , \[D \sin{\theta} = m \lambda,~for~m = 1, -1, 2, -2, 3,... \left(destructive\right), \label{27.6.1}\] where \(D\) is the slit width, \(\lambda\) is the light's wavelength, \(\theta\) is the angle relative to the original direction of the light, and \(m\) is the order of the minimum. Figure 3 shows a graph of intensity for single slit interference, and it is apparent that the maxima on either side of the central maximum are much less intense and not as wide. This is consistent with the illustration in Figure 1b.
Example \(\PageIndex{1}\): Calculating Single Slit Diffraction
Visible light of wavelength 550 nm falls on a single slit and produces its second diffraction minimum at an angle of \(45.0^{\circ}\) relative to the incident direction of the light.
- What is the width of the slit?
- At what angle is the first minimum produced?
Strategy:
From the given information, and assuming the screen is far away from the slit, we can use the equation \(D \sin{\theta} = m \lambda\) first to find \(D\), and again to find the angle for the first minimum \(\theta_{1}\).
Solution (a):
We are given that \(\lambda = 550 nm\), \(m =2\), and \(\theta_{2} = 45.0^{\circ}\). Solving the equation \(D = \sin{\theta} = m \lambda\) for \(D\) and substituting known values gives \[D = \frac{m \lambda}{\sin{\theta_{2}}} = \frac{2\left(550 nm\right)}{\sin{45.0^{\circ}}} \label{27.6.2}\] \[= \frac{1100 \times 10^{-9}}{0.707}\] \[=1.56 \times 10^{-6}.\]
Solution (b):
Solving the equation \(D = \sin{\theta} = m \lambda\) for \(\sin{\theta_{1}}\) and substituting known values gives \[\sin_{\theta_{1}} = \frac{m \lambda}{D} = \frac{1 \left(550 \times 10^{-9} m \right)}{1.56 \times 10^{-6}}. \label{27.6.3}\] Thus the angle \(\theta_{1}\) is \[\theta_{1} = \sin{0.354}^{-1} = 20.7^{\circ} \label{27.6.4}\]
Discussion:
We see that the slit is narrow (it is only a few times greater than the wavelength of light). This is consistent with the fact that light must interact with an object comparable in size to its wavelength in order to exhibit significant wave effects such as this single slit diffraction pattern. We also see that the central maximum extends \(20.7^{\circ}\) on either side of the original beam, for a width of about \(41^{\circ}\). The angle between the first and second minima is only about \(24^{\circ} \left(45.0^{\circ} - 20.7^{\circ}\right)\). Thus the second maximum is only about half as wide as the central maximum.
Summary
- A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides.
- There is destructive interference for a single slit when \(D \sin{\theta} = m \lambda,~ \left(for~m = 1, -1, 2, -2, 3, ...\right)\) where \(D\) is the slit width, \(\lambda\) is the light's wavelength, \(\theta\) is the angle relative to the original direction of the light, and \(m\) is the order of the minimum. Note that there is no \(m = 0\) minimum.
Glossary
- destructive interference for a single slit
- occurs when \(D \sin{\theta} = m \lambda, \left(for~m = 1, -1, 2, -2, 3, ...\right)\), where \(D\) is the slit width, \(\lambda\) is the light's wavelength, \(\theta\) is the angle relative to the original direction of the light, and \(m\) is the order of the minimum


6.10: Case Study 5- Adolescent Client
Initial Assessment Data
- Biographical data: Adolescent client (age 17)
- Reason for seeking care: Post-op surgery
- History of health/illness: Healthy
- Past history: No medications, no illnesses
- Client had abdominal surgery and was transferred from the recovery room to the surgical floor. Lethargic and oriented x 3. Awakes when name is called and responds to questions appropriately. Client rates pain 2/10. The healthcare provider assesses client’s vital signs every 30 minutes x 2 hours.
Vital Sign Record
Think about the client data documented on the vital sign record and try to answer the following questions.
- Review the vital sign record and identify the trends that you see.
- What should the healthcare provider’s next action be?
Go to the next page to check your answers.


7.8: Patterns of Organization and Methods of Development
Patterns of organization can help readers follow the ideas within your writing and also work as methods of development to help you recognize and further develop ideas and relationships in your writing.
General to specific or specific to general
It might be useful to organize your topic like a triangle. In Figure 7.5 , the arrow on the left side of the triangle represents starting with the most general, big-picture information first, and then later moving to more detailed and more personal information. The arrow on the right side of the triangle represents starting with the specific information first and then moving to the more global, big-picture stuff. The triangle is a general guide, and you can move around within it. For example, each paragraph might be its own triangle, starting with the general or specific and moving out or in. However, if you begin your paper very broadly, it might be effective to end it in a more specific, personal way. If you begin your paper with a personal story, consider ending by touching on the global impact and importance of your topic.
Figure 7.5: Organizing topic like a triangle
Other ways to organize ideas
Instead of organizing ideas based on whether they are broad or specific, you could organize them using one of the following ways:
- Chronological order (the order in time that events take place).
- Compare and contrast (ideas are organized together because of their relationship to each other).
- Most important information first (consider what you want readers to focus on first).
Note that order of importance is a particularly good choice when you are trying to persuade and convince the reader. Most papers move from the least to the most important point, and the paragraphs are arranged in an effort to build the essay’s strength. Sometimes, however, it is necessary to begin with your most important supporting point, such as in an essay that contains a thesis that is highly debatable or when writing persuasively.
Attribution statement
Content from this page was remixed with our own original content, and adapted, with editorial and organizational changes, from:
The Word on College Reading and Writing by Carol Burnell, Jaime Wood, Monique Babin, Susan Pesznecker, and Nicole Rosevear, licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.Download for free at: https://openoregon.pressbooks.pub/wrd/
Content from the last paragraph was adapted from:
Writing for Success 1st Canadian Edition by Tara Horkoff is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted. Download for free at: https://opentextbc.ca/writingforsuccess/


3.5.E: Problems on Linear Spaces (Exercises)
Prove that \(F^{n}\) in Example \((\mathrm{b})\) is a vector space, i.e., that it satisfies all laws stated in Theorem 1 in §§1-3; similarly for \(W\) in Example (d).
Verify that dot products in \(C^{n}\) obey the laws \((\mathrm{i})-\left(\mathrm{v}^{\prime}\right) .\) Which of these laws would fail if these products were defined by
\[
x \cdot y=\sum_{k=1}^{n} x_{k} y_{k} \text { instead of } x \cdot y=\sum_{k=1}^{n} x_{k} \overline{y}_{k} ?
\]
How would this affect the properties of absolute values given in \(\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right) ?\)
Complete the proof of formulas \(\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right)\) for Euclidean spaces. What change would result if property (ii) of dot products were restated as
\[
" x \cdot x \geq 0 \text { and } \overrightarrow{0} \cdot \overrightarrow{0}=0^{\prime \prime} ?
\]
Define orthogonality, parallelism and angles in a general Euclidean space following the pattern of §§1-3 (text and Problem 7 there). Show that \(u=\overrightarrow{0}\) iff \(u\) is orthogonal to all vectors of the space.
Define the basic unit vectors \(e_{k}\) in \(C^{n}\) exactly as in \(E^{n},\) and prove
Theorem 2 in §§1-3 for \(C^{n}\left(\text { replacing } E^{1} \text { by } C\right).\) Also, do Problem 5\((\mathrm{a})\) of §§1-3 for \(C^{n}\).
Define hyperplanes in \(C^{n}\) as in Definition 3 of §§4-6, and prove Theorem 1 stated there, for \(C^{n} .\) Do also Problems \(4-6\) there for \(C^{n}\) (replacing \(E^{1}\) by \(C )\) and Problem 4 there for vector spaces in general (replacing \(E^{1}\) by the scalar field \(F ) .\)
Do Problem 3 of §§4-6 for general Euclidean spaces (real or complex). Note: Do not replace \(E^{1}\) by \(C\) in the definition of a line and a line segment.
A finite set of vectors \(B=\left\{x_{1}, \ldots, x_{m}\right\}\) in a linear space \(V\) over \(F\) is said to be independent iff
\[
\left(\forall a_{1}, a_{2}, \ldots, a_{m} \in F\right) \quad\left(\sum_{i=1}^{m} a_{i} x_{i}=\overrightarrow{0} \Longrightarrow a_{1}=a_{2}=\cdots=a_{m}=0\right).
\]
Prove that if \(B\) is independent, then
(i) \(\overrightarrow{0} \notin B\);
(ii) each subset of \(B\) is independent \((\emptyset \text { counts as independent }) ;\) and
(iii) if for some scalars \(a_{i}, b_{i} \in F\),
\[
\sum_{i=1}^{m} a_{i} x_{i}=\sum_{i=1}^{m} b_{i} x_{i},
\]
then \(a_{i}=b_{i}, i=1,2, \ldots, m\).
Let \(V\) be a vector space over \(F\) and let \(A \subseteq V .\) By the span of \(A\) in \(V\), denoted \(\operatorname{span}(A),\) is meant the set of all "linear combinations" of vectors from \(A,\) i.e., all vectors of the form
\[
\sum_{i=1}^{m} a_{i} x_{i}, \quad a_{i} \in F, x_{i} \in A, m \in N.
\]
Show that \(\operatorname{span}(A)\) is itself a vector space \(V^{\prime} \subseteq V\) (a subspace of \(V )\) over the same field \(F,\) with the operations defined in \(V .\) (We say that A spans \(V^{\prime} .\) Show that in \(E^{n}\) and \(C^{n},\) the basic unit vectors span the entire space.


21.6: Chapter Summary
Race, ethnicity, culture, and nationality are related concepts that are often used interchangeably but have distinct meanings. A person’s race and ethnicity can shape their cultural experiences and traditions, while their nationality can influence their legal status and access to resources. Understanding the complexities and nuances of these concepts is important for promoting diversity, equity, and inclusion in society.
Culture is a broad term that refers to the shared beliefs, values, customs, behaviors, and artifacts that characterize a particular group or society. Culture encompasses many distinct aspects of the human experience, including language, religion, ceremonies, celebrations, art, music, food, clothing, social habits, and more. It is transmitted from generation to generation through socialization, education, and other means of communication, and it shapes how an individual thinks, behaves, and interacts with the world. Culture can be specific to a particular region, country, or ethnic group or shared across national and ethnic boundaries, and it evolves over time. Nurses must understand and appreciate cultural differences to provide client-centered care.
The definition of health has evolved over time and varies across cultures depending on cultural beliefs, values, and practices. Among many cultures, the causes of illness are believed to include both physical and spiritual factors. For example, some cultures attribute illness to a lack of spiritual balance, while others see it as the result of social or emotional stressors. Understanding an individual’s perception of health can help nurses develop plans of care that are appropriate and effective for the individual client.
Culture plays a significant role in shaping health beliefs and practices. Health is viewed and experienced differently across cultures, and beliefs about appropriate treatments for conditions and diseases can vary widely. Cultural beliefs can influence health behaviors and decision-making, such as seeking medical care or adhering to treatment plans. By understanding and respecting cultural beliefs and practices related to health, nurses can provide more effective client-centered care.
Culture is a social determinant of health because cultural factors affect an individual’s health and well-being. These cultural factors include beliefs, values, traditions, and social norms, which can all influence an individual’s health behaviors and outcomes. Intersectionality acknowledges that individuals may experience multiple forms of discrimination or privilege based on their various identities and that these experiences cannot be understood by looking at each identity in isolation. In health care, an intersectional approach can help nurses address health disparities by recognizing and addressing the complex and intersecting factors that contribute to health outcomes. Nurses have a primary responsibility to understand the role culture plays in shaping not only health beliefs and practices but also implications for health outcomes.


12.2: Historical Perspective
By the end of this section, you should be able to:
- Describe the historical origins of epidemiology.
- Identify key figures influential in the field of epidemiology.
- Examine the contribution of nursing to the field of epidemiology.
The origin of epidemiology has been traced to physician Hippocrates ’ “On Airs, Waters, and Places.” This medical textbook, written in ancient Greece circa 400 BCE, details Hippocrates’ belief that disease was caused not by the supernatural but instead by environmental factors and his recommendations that healers observe their patients’ lifestyle choices (Kleisiaris et al., 2014). Several important figures followed Hippocrates in the evolution of epidemiology; this chapter will highlight four of the most influential: Edward Jenner, Ignaz Semmelweis, John Snow, and Florence Nightingale.
Edward Jenner (1749–1823) developed an immunization against smallpox that ultimately resulted in its eradication (Riedel, 2005). Smallpox affected all levels of society in 18th-century Europe, killing 400,000 people each year. Many of those who survived went blind (Louten, 2022; Riedel, 2005). The case-fatality rate in London was estimated to be as high as 80 percent in infants and 60 percent in children and adults. Edward Jenner had heard that dairy maids seemed to be protected from smallpox after having contracted cowpox. He theorized that inoculating a person with matter from a cowpox lesion would provide them with protection from smallpox. To test his theory, Jenner inoculated a young boy with matter from the cowpox lesions of a dairy maid and then inoculated the child 6 weeks later with matter from a fresh smallpox pustule. Jenner’s experiment seemed to work: the child did not contract the disease. This is the first recorded scientific attempt to control an infectious disease by the deliberate use of vaccination , a method that eventually saved millions of humans from death and disability worldwide (Celentano & Szklo, 2019). On May 8, 1980, the World Health Assembly recommended ceasing smallpox vaccination as the world was smallpox-free (Riedel, 2005). Jenner operated solely on observational data that provided the basis for a preventative intervention.
In the early 19th century, puerperal fever (childbed fever) had mortality rates as high as 25 percent among women shortly after childbirth (Celentano & Szklo, 2019). Theories about its causes included solar and magnetic influences and air. In 1846, Ignaz Semmelweis (1818–1865), a physician specializing in obstetrics, was put in charge of the General Hospital in Vienna where there were two obstetrical clinics, the First and Second. The First Clinic was staffed by physicians and medical students and the Second Clinic by midwives. Semmelweis noticed the mortality rates in the two clinics were very different; in 1842, the rate in the First Clinic was more than twice as high as the rate in the Second Clinic (Celentano & Szklo, 2019). Knowing that physicians and medical students began their days performing autopsies before providing care in the First Clinic and that midwives did not perform autopsies, Semmelweis suggested mortality was higher in the First Clinic because the hands of physicians and students were transmitting disease from the cadavers to the women in labor.
In 1847, Semmelweis’s colleague contracted an infection after being accidentally punctured while performing an autopsy on a woman who had died of childbed fever. An autopsy on Semmelweis’s colleague demonstrated similar pathology to the women dying from childbed fever, confirming Semmelweis’s suspicions regarding the transmission of the disease (Celentano & Szklo, 2019). Based on these causal findings, Semmelweis implemented a handwashing policy for physicians and students in the First Clinic. In 1848, mortality in the First Clinic dropped to rates comparable to those in the Second Clinic. Reinforcing this causal relationship, when Semmelweis was replaced by another obstetrician, the handwashing policy was eliminated, and mortality rates rose again in the First Clinic. Semmelweis’s findings and recommendations had a global impact on the practice of medicine. His observations and interventions came before there was any knowledge of germ theory , proving it is possible and may be effective to implement a prevention strategy without knowledge of the exact cause of disease (Celentano & Szklo, 2019).
British physician John Snow (1813–1858) is often considered the “father of field epidemiology” (CDC, 2012) for his work translating epidemiologic observations into public policy during the 1850s cholera epidemic in London. His work illuminates the sequence from descriptive epidemiology to hypothesis generation to analytic epidemiology (hypothesis testing) to application. Snow approached his investigation of the source and transmission of the disease by examining where the people affected with cholera lived and worked. He marked each residence on a map of the area. This type of map is called a spot map , and it shows the geographic distribution of cases of illness or disease. On this map, Snow also marked the location of water pumps to look for a relationship between the residences with cases of cholera and the location of pumps. He noticed a pattern of affected residences clustered around Pump A, more so than around Pumps B or C. He concluded that Pump A was the primary source of water and most likely the source of cholera infection. Snow gathered information on where those affected with cholera obtained their water and found that, confirming his suspicions, Pump A was the one common factor. After presenting these findings, the handle of the pump was removed, and the outbreak ceased (CDC, 2012). Visit this site to see what a spot map looks like.
Florence Nightingale (1820–1910), long considered the founder of professional nursing, is also considered the first nurse epidemiologist. Nightingale devoted her life to the prevention of illness and death and used statistical methods to visualize data and bring about health reforms (Bradshaw, 2020; Fee & Garofalo, 2010; McDonald, 2014). In 1854 she joined a group of nurses aiding troops in Crimea (Figure 12.2). She saw the suffering in the hospital barracks—not due to battle injuries, but due to the rodent-infested buildings, filthy environment, and overflowing sewers. Soldiers suffered from wounds, dysentery, malnutrition, cholera, typhus, and scurvy, with a high mortality rate (Bradshaw, 2020; Fee & Garofalo, 2010; McDonald, 2014). Nightingale gathered data and documented the results of her sanitation reforms to illustrate the unnecessary deaths in military hospitals. After the war, she continued documenting negative health outcomes associated with poor sanitation. She eventually convinced the government to implement her sanitary reforms and continued to track the data (Bradshaw, 2020; McDonald, 2014). Nightingale monitored disease mortality rates, used a research framework to study the distribution and patterns of disease in a population, used applied statistical methods to visualize the data, and published statistical reports to gain the support of politicians and powerful people to bring about public health reforms that ultimately created changes in hygiene and treatment of clients. As one of the first to apply statistics to health care, Florence Nightingale was a pioneer who brought the field of public health to international attention and became the first woman to be awarded Britain’s Order of Merit in 1907 (Fee & Garofalo, 2010).


5.12A: Substrates for Biosysnthesis
- Describe the importance of substrates for biosynthesis
Microorganisms have numerous pathways and processes in place to ensure both energy and nutrient production. These pathways are necessary for survival and cellular function. The major metabolic pathways require substrates to be acted upon for the formation of larger, more complex products. Biosynthetic processes are defined by the production of more complex products that are required for growth and maintenance of life. These processes require pathways that are often multi-step. There are various components deemed necessary for biosynthetic processes to occur, including: precursor compounds, chemical energy, and carious catalytic enzymes.
TCA Cycle
The citric acid cycle, commonly referred to as the Krebs cycle, is characterized by the production of energy through the oxidation of acetate derived from carbohydrates, fats, and proteins into carbon dioxide. The cycle is one of the major metabolic processes utilized to generate energy. The citric acid cycle, comprised of a series of chemical reactions, provides precursors for additional biochemical pathways. These precursors are used as substrates for the biogenesis of large complex products. The precursors include amino acids and reducing agents such as NADH. Additional pathways that require precursors formed by the TCA include amino acid and nucleotide synthesis.
Glycolysis
An additional central metabolic pathway includes glycolysis. Glycolysis is characterized by a series of reactions that results in the conversion of glucose into pyruvate. This process is characterized by the production of various intermediates and molecules that function as substrates in additional pathways. Additional pathways that require substrates or metabolites produced by the glycolytic pathway include: gluconeogenesis, lipid metabolism, the pentose phosphate pathway, and the TCA.
Key Points
- Biogenesis or anabolism, requires substrates to be acted upon that result in the formation of larger more complex molecules.
- A central metabolic pathway that produces precursors and substrates used in biosynthetic processes is the TCA cycle.
- A central metabolic pathway that produces precursors and substrates used in biosynthetic processes is glycolysis.
Key Terms
- reducing agent : A substance that functions in reducing or donating electrons to another substance until that specific substance becomes oxidized.
- oxidation : A reaction in which the atoms of an element lose electrons and the valence of the element increases.


3.14: Capabilities and human rights
Several capability theories are closely related to accounts of human rights. Within the capability literature, some scholars have developed capability theories that they regard as a human rights theory. In the human rights literature, scholars have examined to what extent the capability approach can help to develop stronger theories of human rights. The same topics (e.g. provision of or right to basic health or basic education) are defended based on both approaches, or are defended appealing both to human rights and capabilities (e.g. Osmani 2000). Amartya Sen has in several of his publications analysed the relationship between human rights and capabilities (e.g. Sen 2004b, 2005). In addition, Martha Nussbaum has claimed that her capabilities theory is a version of human rights theory, which has drawn much attention to the question about the relationship between capabilities and human rights.
It should not be surprising that there are so many scholars and practitioners interested in both the human rights framework and the capability approach, since they share some important aspects. First, they are both widely endorsed ethical frameworks. Second, they seem to share an underlying motivation, namely to protect and enhance people’s freedoms. Third, they are both used for global as well as domestic questions. Fourth, both frameworks want to build strong links between theory and practice: they are studied and used by scholars but also used by practitioners (political parties, activists, policy makers, etc.). Finally, both discourses are strongly interdisciplinary in nature.
All this raises some questions. What is the relationship between human rights and capabilities? Can we say that capabilities are the objects of human rights? If so, do human rights theories and analyses have something to gain by developing capabilities-based human rights theories? Can the capability approach deliver all that is important in human rights theories? And what should we make of the alleged disadvantages of using the capability approach in thinking about human rights?
3.14.1 What are human rights?
Human rights are rights each human being is entitled to in order to protect her from severe harms that could be inflicted by others — either by deliberate actions, or else by the failure to protect human rights caused by institutional design. They are norms aimed at protecting people from severe social, political and legal abuse (Nickel 2014, 1). Examples of human rights are the right to life, the right to food, the right to freedom of assembly, the right to freedom of religion, the right to a fair trial when charged with a crime, the right not to be tortured, and the right to privacy.
Human rights are not all the rights that people have. As Sen (2004b, 329) writes, “there have to be some ‘threshold conditions’ of (i) importance and (ii) social influenceability for a freedom to figure within the interpersonal and interactive spectrum of human rights”. Take the ‘importance threshold’ first. Here is an example of a right that is not a human right, because it does not meet the threshold condition of importance: the right to parental leave. In many European countries, parents who are employed have a right to paid parental leave upon the birth of their child or when they adopt a child. While many have argued that such a right would help meet our duties towards children and parents as well as advance gender justice (e.g. Gheaus and Robeyns 2011), it is not at all plausible to argue that this should be seen as a human right. It has a much weaker moral urgency than the right to a fair trial, let alone the right to life. Human rights thus correspond to a subset of the domain of justice, and focus on those questions that are of utter importance, the protection of which should have a greater urgency than the support of other normative claims. The second threshold — social influenceability — implies that even if something valuable is hugely important, as long as there is no or very limited social influenceability, its protection cannot be a human right. For example, it makes no sense to speak of a human right to be protected from volcano eruptions, or a human right to be protected from cancer. However, one can say that there is a human right to be warned about volcano eruptions if the government has the relevant information. To the extent that there is more social influenceability, the scope to speak coherently of human rights increases.
Human rights have corresponding duties. But on whom do those duties fall and what kind of duties are they? A broad, inclusive account of duties is given by Pablo Gilabert (2009, 673) who writes:
[Human rights impose] a duty of the highest priority for individuals and governments to identify ways to protect certain important interests through (a) specific rights and entitlements, but also, when these are insufficient or not presently feasible, through (b) urgent goals of institution-building.
Note that this definition does not limit the duty to protect human rights to governments only, and that it does include institution-building as an important path towards protecting human rights. 28
3.14.2 The interdisciplinary scholarship on human rights
The human rights literature is, just like the capability approach, deeply multidisciplinary and interdisciplinary. The philosophy of human rights “addresses questions about the existence, content, nature, universality, justification, and legal status of human rights” (Nickel 2014, 1). How can human rights exist in the first place? What should be the content of human rights, that is, what kinds of harms or abuses should they protect us from? What kind of rights are human rights — are they moral claims, or legal claims, or political claims, or something else? The question of justification asks: on what grounds can we say that people have human rights? Is it because humans have rational capacities or agency? If so, does that mean that newborn babies do not have human rights? All these questions are studied in the vast philosophical literature on human rights.
Note that while the relationship between normative political philosophy, justice and human rights is not entirely disputed, the dominant view in the contemporary literature is that the domain of human rights is a subset of the domain of justice, which in turn is a subset of the domain of morality. The reason is that “[n]ot everything that is desirable to be realized in politics is a matter of human rights, and not everything that is a matter of justice is a matter of human rights.
Legal scholars are interested in questions related to the treaties and constitutions in which human rights are codified. The idea of human rights gained momentum with the 1948 adoption of the Universal Declaration of Human Rights (UDHR), which over time received a canonical status in legal and political debates. The UDHR subsequently served as a template for human rights instruments that are legally binding, such as the International Covenant on Civil and Political Rights, the International Covenant on Economic, Social and Cultural Rights, the European Convention on Human Rights, the American Convention on Human Rights and the African Charter of Human and People’s Rights. One question this raises is to what extent national constitutions are consistent with those legally binding treaties, or with the UDHR. Another question frequently asked by legal scholars is to what extent national jurisprudence can be in tension with — and violate — a human right that is part of an international treaty to which that particular nation signed up. For example, in the famous ‘Lautsi case’, the question emerged whether the Italian state’s policy to have a compulsory crucifix in the classroom of public schools was in violation of the human right to the freedom of religion as codified in the European Convention on Human Rights (Weiler 2010; Pierik 2012; Pierik and van der Burg 2011).
In the social sciences and international relations, questions are asked about what role human rights play in politics. How do countries differ in the degree in which they protect human rights? What are the effective instruments that support human rights in countries in which they are not violated? Is it effective to condemn human rights violations in other countries, or is silent support for grassroots human rights activists a more effective strategy?
Of course, human rights are hugely important for human right activists, who are working on actual human rights protections. Other activists, such as those focusing on the empowerment of disadvantaged groups, often take a more instrumental attitude towards human rights, and ask whether such rights are effective instruments to reach their goals of inclusion, development and combatting forms of injustice and oppression.
3.14.3 Why a capability-based account of human rights?
After this brief sketch of the huge literature on human rights, we can now explore the relation between human rights and capabilities. The first question that needs to be asked is: why would we be at all interested in a capabilities-based theory of human rights? What could be gained by theorizing human rights, or trying to protect human rights, by referring to capabilities?
The first reason is philosophical, and concerns the justification of human rights. Human rights are norms or instruments to protect certain valuable things (which are called ‘the objects of human rights’). But every time we claim an object is so important its protection must be enshrined as a human right, we must argue why that object has this special importance. Part of the philosophical literature on capabilities does precisely that — to justify why we need to protect certain valuable personal states. Both Sen (2004b, 2005) and Nussbaum (1997, 2011a) have argued that human rights can be seen as entitlements to certain capabilities. However, Sen’s views are more qualified, since he has argued that the object of some but not necessarily all human rights can be viewed as capabilities. There are plausibly also other objects of human rights, such as process freedoms and liberties.
The second reason builds on the first. If some human rights can be understood in terms of capabilities, and poverty can also be conceptualized in terms of the denial of capabilities, then poverty can be conceptualized as a human rights violation (Osmani 2005). This is important for various reasons, including the strong rhetorical force that a human rights violation has in comparison with other claims, and also because socio-economic human rights have sometimes been regarded as more in need of conceptual foundations, in comparison with the civic and political human rights whose status as human rights has been less contested.
This relates to the third reason why it can be helpful to conceptualise human rights in terms of capabilities, which is the often-stated worry that the protection of human rights, especially social and economic rights, is infeasible (Gilabert 2009). To counter that pessimism, we need greater clarity on the chain of steps that are involved in socio-economic human rights protection. Capabilities are the objects of our rights, and we know, from our understanding of how capabilities relate to resources and social structures, which parameters can influence the capabilities that people enjoy (see the figure in section 2.12). Hence, if we want to protect human rights, in particular socio-economic rights, which sceptics believe cannot effectively be protected, the capability approach helps us see that “promoting socioeconomic rights may require attention to specific parameters that affect the capabilities of people” (Gilabert 2009, 666). In sum, the language of the capability approach helps us to respond and address the feasibility worry of socio-economic rights.
Fourthly, we are unsure whether some things we want to protect meet the threshold condition of importance that human rights should meet. If one is unsure about whether a certain freedom should be a right, let alone a human right, one could already start to protect or enhance it if one sees it as a capability. One does not need to wait until the discussion about the threshold is settled before one starts to protect something that everyone agrees is in any case important.
A final reason is more practical or political. In some countries, the terminology of ‘human rights’ is regarded with suspicion, as it is seen as stemming from a colonial era, and, as a consequence, is regarded as an instrument of western domination. This makes it hard for both local and global advocates of human rights to advance their cause. By using the terminology of capabilities, which is not linked to a particular colonial era or western power, instead of the language of human rights, these same valuable rights can be argued for.
3.14.4 Are capabilities sufficient to construct a theory of human rights?
There is quite a lot of interest among capability scholars and those working in the human development paradigm to try to bring the best of the capability approach and the human rights approach together (e.g. Vizard 2006, 2007; Vizard, Fukuda‐Parr and Elson 2011; Fukuda-Parr 2011; Gilabert 2009, 2013). One important question, though, is how much the capability approach can offer if one is interested in constructing a powerful human rights theory. Is the notion of capabilities sufficient for such a theory?
The answer clearly must be negative. A theory of human rights needs other elements, such as a discussion of the scope of, and, importantly, the justification for, human rights. Yet, by making use of the distinction between the capability approach and capability theories that was introduced in section 2.3, we can see that it is not at all an embarrassment for the capability approach that, by itself, it cannot deliver a theory of human rights. Instead, that should be the task of a specific capability theory, for which, in the various modules, additional elements that are needed for a human rights theory can be added.
This raises the next question: what would have to be added, then? One important thing that may need to be added (in A6 — other dimensions of ultimate value) are process freedoms. Sen (2004b) argues that we should make a distinction between freedoms as substantive opportunities and the process aspect of freedom (procedural aspects). Both are, in his view, relevant when thinking about human rights, but only the opportunity aspect of freedom is captured by the notion of ‘capabilities’. Linda Barclay (2016) makes a similar point, by saying that rights that concern equitable processes are very important for human rights, and cannot be captured by the notion of capabilities. Clearly, procedural characteristics, for example those that guarantee a free trial, may not necessarily best be understood as capabilities, but perhaps rather more as elements of institutional design. Yet as proposition A6 emphasizes, not everything that is of crucial importance is a capability. Recall that proposition A6 allows us to include other elements of ultimate value, and this could incorporate what Sen calls process freedoms. 29 In short, by seeing a capability-based human rights theory as a capability theory , for which various theoretical additions and choices are possible, it becomes clear that more is needed than the mere reference to ‘capabilities’.
Note also that we can, of course, ask the question the other way around — what is needed to make a capability theory? — and use the human rights framework as the theory of value that is used to make the selection of capabilities. This route has been developed by Polly Vizard, and has led to the “human rights based capability set” (Vizard 2006, 2007).
3.14.5 The disadvantages
Finally, we need to ask whether there are any disadvantages in using the capability approach to further our thinking, policy making and activism on human rights, and — ultimately — in letting a capabilities-based human rights theory compete with the existing human rights accounts.
The first thing to note is that there is a long-standing human rights discourse that is used by activists all over the world, and often very effectively so. Clearly there are costs involved for these activists to become familiarized with the capability language. If the human rights discourse delivers to them what they need, why would we change it?
Second, there is a worry about legitimacy. The current human rights declarations and treaties have been the result of actual political processes, and the treaties were drafted by a large number of people, drawn from all over the world. For the capability approach, this is different. Given the prominence of Amartya Sen and Martha Nussbaum, and also the many publications that (wrongly) reduce the capability approach to the work by, primarily, Sen and Nussbaum, 30 the capability approach is much more associated with specific individuals. A capability-based human rights theory that is the work of one thinker can never have the political leverage that the existing human rights framework has. For the various practices in which human rights are used (creating laws, making policy and activism) a capability-based human rights framework can therefore never replace the existing human rights framework.
However, there is, of course, a more fruitful relationship possible, and that is to see the two frameworks as complementary rather than competitive (Nussbaum 2011a). Note, however, that any merging of the two frameworks has to be between a particular capability theory and human rights thinking, rather than between the general capability approach and human rights thinking. 31 A good example of such practical work is the UK’s Equality and Human Rights Framework. Tania Burchardt and Polly Vizard (2011) used insights from both the capability approach and the existing work on human rights to create a framework that is used for the monitoring undertaken by the Equality and Human Rights Commission in order to meet its legal mandate. 32
28 This relates to a complex discussion in legal and political philosophy on whether human rights can be protected by so-called ‘imperfect duties’ or ‘imperfect obligations’ which is beyond the scope of this book. See, amongst others, Polly Vizard (2006, 84–91) and Frances Kamm (2011) for further discussion.
29 The example that Barclay gives is the right not to be discriminated against. Barclay believes that “to be protected from discrimination” is a very important human right, but cannot plausibly be conceptualized as a capability, since one does not have a choice to be discriminated against or not. However, not being discriminated against is a functioning, and it is a mistake to think that the capability approach holds that we should only focus on capabilities and never on functionings, as was argued in section 3.4.
30 Unfortunately, Nussbaum’s (2011c) account of the capability approach only adds to that reductive and misleading portrayal of the capability literature (Robeyns 2011, 2016b; Unterhalter 2013).
31 Hence, when Nussbaum (2011a, 24) writes “the CA is a type of human rights approach”, we should read this as “Martha Nussbaum’s capability theory is a type of human rights approach”. Many other capability theories are, evidently, not human rights approaches, and hence the capability approach, as the overarching framework, cannot be either.


11.4: Temperature and Microbial Growth
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Illustrate and briefly describe minimum, optimum, and maximum temperature requirements for growth
- Identify and describe different categories of microbes with temperature requirements for growth: psychrophile, psychrotrophs, mesophile, thermophile, hyperthermophile
- Give examples of microorganisms in each category of temperature tolerance
When the exploration of Lake Whillans started in Antarctica, researchers did not expect to find much life. Constant subzero temperatures and lack of obvious sources of nutrients did not seem to be conditions that would support a thriving ecosystem. To their surprise, the samples retrieved from the lake showed abundant microbial life. In a different but equally harsh setting, bacteria grow at the bottom of the ocean in sea vents (Figure \(\PageIndex{1}\)), where temperatures can reach 340 °C (700 °F).
Microbes can be roughly classified according to the range of temperature at which they can grow. The growth rates are the highest at the optimum growth temperature for the organism. The lowest temperature at which the organism can survive and replicate is its minimum growth temperature. The highest temperature at which growth can occur is its maximum growth temperature. The following ranges of permissive growth temperatures are approximate only and can vary according to other environmental factors.
Organisms categorized as mesophiles (“middle loving”) are adapted to moderate temperatures, with optimal growth temperatures ranging from room temperature (about 20 °C) to about 45 °C. As would be expected from the core temperature of the human body, 37 °C (98.6 °F), normal human microbiota and pathogens (e.g., E. coli , Salmonella spp., and Lactobacillus spp.) are mesophiles.
Organisms called psychrotrophs, also known as psychrotolerant, prefer cooler environments, from a high temperature of 25 °C to refrigeration temperature about 4 °C. They are found in many natural environments in temperate climates. They are also responsible for the spoilage of refrigerated food.
Clinical Focus: Resolution
The presence of Listeria in Jeni’s blood suggests that her symptoms are due to listeriosis, an infection caused by L. monocytogenes . Listeriosis is a serious infection with a 20% mortality rate and is a particular risk to Jeni’s fetus. A sample from the amniotic fluid cultured for the presence of Listeria gave negative results. Because the absence of organisms does not rule out the possibility of infection, a molecular test based on the nucleic acid amplification of the 16S ribosomal RNA of Listeria was performed to confirm that no bacteria crossed the placenta. Fortunately, the results from the molecular test were also negative.
Jeni was admitted to the hospital for treatment and recovery. She received a high dose of two antibiotics intravenously for 2 weeks. The preferred drugs for the treatment of listeriosis are ampicillin or penicillin G with an aminoglycoside antibiotic. Resistance to common antibiotics is still rare in Listeria and antibiotic treatment is usually successful. She was released to home care after a week and fully recovered from her infection.
L. monocytogenes is a gram-positive short rod found in soil, water, and food. It is classified as a psychrophile and is halotolerant. Its ability to multiply at refrigeration temperatures (4–10 °C) and its tolerance for high concentrations of salt (up to 10% sodium chloride [NaCl]) make it a frequent source of food poisoning. Because Listeria can infect animals, it often contaminates food such as meat, fish, or dairy products. Contamination of commercial foods can often be traced to persistent biofilms that form on manufacturing equipment that is not sufficiently cleaned.
Listeria infection is relatively common among pregnant women because the elevated levels of progesterone downregulate the immune system, making them more vulnerable to infection. The pathogen can cross the placenta and infect the fetus, often resulting in miscarriage, stillbirth, or fatal neonatal infection. Pregnant women are thus advised to avoid consumption of soft cheeses, refrigerated cold cuts, smoked seafood, and unpasteurized dairy products. Because Listeria bacteria can easily be confused with diphtheroids, another common group of gram-positive rods, it is important to alert the laboratory when listeriosis is suspected.
The organisms retrieved from arctic lakes such as Lake Whillans are considered extreme psychrophiles (cold loving). Psychrophiles are microorganisms that can grow at 0 °C and below, have an optimum growth temperature close to 15 °C, and usually do not survive at temperatures above 20 °C. They are found in permanently cold environments such as the deep waters of the oceans. Because they are active at low temperature, psychrophiles and psychrotrophs are important decomposers in cold climates.
Organisms that grow at optimum temperatures of 50 °C to a maximum of 80 °C are called thermophiles (“heat loving”). They do not multiply at room temperature. Thermophiles are widely distributed in hot springs, geothermal soils, and manmade environments such as garden compost piles where the microbes break down kitchen scraps and vegetal material. Examples of thermophiles include Thermus aquaticus and Geobacillus spp. Higher up on the extreme temperature scale we find the hyperthermophiles, which are characterized by growth ranges from 80 °C to a maximum of 110 °C, with some extreme examples that survive temperatures above 121 °C, the average temperature of an autoclave. The hydrothermal vents at the bottom of the ocean are a prime example of extreme environments, with temperatures reaching an estimated 340 °C (Figure \(\PageIndex{1}\)). Microbes isolated from the vents achieve optimal growth at temperatures higher than 100 °C. Noteworthy examples are Pyrobolus and Pyrodictium , archaea that grow at 105 °C and survive autoclaving. Figure \(\PageIndex{2}\) shows the typical skewed curves of temperature-dependent growth for the categories of microorganisms we have discussed.
Life in extreme environments raises fascinating questions about the adaptation of macromolecules and metabolic processes. Very low temperatures affect cells in many ways. Membranes lose their fluidity and are damaged by ice crystal formation. Chemical reactions and diffusion slow considerably. Proteins become too rigid to catalyze reactions and may undergo denaturation. At the opposite end of the temperature spectrum, heat denatures proteins and nucleic acids. Increased fluidity impairs metabolic processes in membranes. Some of the practical applications of the destructive effects of heat on microbes are sterilization by steam, pasteurization, and incineration of inoculating loops. Proteins in psychrophiles are, in general, rich in hydrophobic residues, display an increase in flexibility, and have a lower number of secondary stabilizing bonds when compared with homologous proteins from mesophiles. Antifreeze proteins and solutes that decrease the freezing temperature of the cytoplasm are common. The lipids in the membranes tend to be unsaturated to increase fluidity. Growth rates are much slower than those encountered at moderate temperatures. Under appropriate conditions, mesophiles and even thermophiles can survive freezing. Liquid cultures of bacteria are mixed with sterile glycerol solutions and frozen to −80 °C for long-term storage as stocks. Cultures can withstand freeze drying (lyophilization) and then be stored as powders in sealed ampules to be reconstituted with broth when needed.
Macromolecules in thermophiles and hyperthermophiles show some notable structural differences from what is observed in the mesophiles. The ratio of saturated to polyunsaturated lipids increases to limit the fluidity of the cell membranes. Their DNA sequences show a higher proportion of guanine–cytosine nitrogenous bases, which are held together by three hydrogen bonds in contrast to adenine and thymine, which are connected in the double helix by two hydrogen bonds. Additional secondary ionic and covalent bonds, as well as the replacement of key amino acids to stabilize folding, contribute to the resistance of proteins to denaturation. The so-called thermoenzymes purified from thermophiles have important practical applications. For example, amplification of nucleic acids in the polymerase chain reaction (PCR) depends on the thermal stability of Taq polymerase, an enzyme isolated from T. aquaticus . Degradation enzymes from thermophiles are added as ingredients in hot-water detergents, increasing their effectiveness.
Exercise \(\PageIndex{1}\)
- What temperature requirements do most bacterial human pathogens have?
- What DNA adaptation do thermophiles exhibit?
Feeding the World…and the World’s Algae
Artificial fertilizers have become an important tool in food production around the world. They are responsible for many of the gains of the so-called green revolution of the 20th century, which has allowed the planet to feed many of its more than 7 billion people. Artificial fertilizers provide nitrogen and phosphorus, key limiting nutrients, to crop plants, removing the normal barriers that would otherwise limit the rate of growth. Thus, fertilized crops grow much faster, and farms that use fertilizer produce higher crop yields.
However, careless use and overuse of artificial fertilizers have been demonstrated to have significant negative impacts on aquatic ecosystems, both freshwater and marine. Fertilizers that are applied at inappropriate times or in too-large quantities allow nitrogen and phosphorus compounds to escape use by crop plants and enter drainage systems. Inappropriate use of fertilizers in residential settings can also contribute to nutrient loads, which find their way to lakes and coastal marine ecosystems. As water warms and nutrients are plentiful, microscopic algae bloom, often changing the color of the water because of the high cell density.
Most algal blooms are not directly harmful to humans or wildlife; however, they can cause harm indirectly. As the algal population expands and then dies, it provides a large increase in organic matter to the bacteria that live in deep water. With this large supply of nutrients, the population of nonphotosynthetic microorganisms explodes, consuming available oxygen and creating “dead zones” where animal life has virtually disappeared.
Depletion of oxygen in the water is not the only damaging consequence of some algal blooms. The algae that produce red tides in the Gulf of Mexico, Karenia brevis , secrete potent toxins that can kill fish and other organisms and also accumulate in shellfish. Consumption of contaminated shellfish can cause severe neurological and gastrointestinal symptoms in humans. Shellfish beds must be regularly monitored for the presence of the toxins, and harvests are often shut down when it is present, incurring economic costs to the fishery. Cyanobacteria, which can form blooms in marine and freshwater ecosystems, produce toxins called microcystins, which can cause allergic reactions and liver damage when ingested in drinking water or during swimming. Recurring cyanobacterial algal blooms in Lake Erie (Figure \(\PageIndex{3}\)) have forced municipalities to issue drinking water bans for days at a time because of unacceptable toxin levels.
This is just a small sampling of the negative consequences of algal blooms, red tides, and dead zones. Yet the benefits of crop fertilizer—the main cause of such blooms—are difficult to dispute. There is no easy solution to this dilemma, as a ban on fertilizers is not politically or economically feasible. In lieu of this, we must advocate for responsible use and regulation in agricultural and residential contexts, as well as the restoration of wetlands, which can absorb excess fertilizers before they reach lakes and oceans.
Key Concepts and Summary
- Microorganisms thrive at a wide range of temperatures; they have colonized different natural environments and have adapted to extreme temperatures. Both extreme cold and hot temperatures require evolutionary adjustments to macromolecules and biological processes.
- Psychrophiles grow best in the temperature range of 0–15 °C whereas psychrotrophs thrive between 4 °C and 25 °C.
- Mesophiles grow best at moderate temperatures in the range of 20 °C to about 45 °C. Pathogens are usually mesophiles.
- Thermophiles and hyperthemophiles are adapted to life at temperatures above 50 °C.
- Adaptations to cold and hot temperatures require changes in the composition of membrane lipids and proteins.


9.5: The Spread of Agriculture and Great Zimbabwe
Most of the African indigenous languages belong to one of the major language groups shown in Map \(\PageIndex{1}\). Over the past several decades, historians have used comparisons of core vocabulary words in related languages to examine the spread of ancient technologies and the interaction between peoples. Using linguistics (the study of languages), information found in other sources, like oral traditions of dynastic origins and archaeological findings, can be corroborated.
During European colonization 150 years ago, anthropologists grouped Africans into “tribes” based on presumed physical, cultural, and linguistic similarities. As result of this study, they found that people in most of Sub-Saharan Africa spoke languages that used the root – ntu to refer to person, with the prefix ba - added in the plural. Combining the root and the plural prefix, 19th Century colonial anthropologists referred to people in these communities as Bantu and later traced Bantu languages back to a root, a mother language spoken in parts of Cameroon and Nigeria.
European scholars hypothesized that about 2,000 years ago there was a Bantu Migration , a massive departure of thousands of Bantu speakers from the Bantu homeland. These Bantu-speakers imposed iron technology and traditions of agriculture on the peoples they encountered in eastern and southern Africa. Influenced by their own conceptions of colonization, nineteenth-century anthropologists portrayed the Bantu Migration as a rapid conquest of Sub-Saharan hunter-gatherer societies by the technologically advanced, Iron Age Bantu speakers.
Since the 1990s, some historians have agreed that the movement of Bantu speakers was more of a slow diffusion of languages and technologies that lasted about 4500 years, from roughly 3000 BCE to 1500 CE. Bantu speakers took multiple routes, and traveled sometimes as a single-family as opposed to a mass of thousands. From the linguistic evidence, historians also suspect there was mutual “teaching and learning from one another” and between groups. 14
The current view of the Bantu expansions is much more complex as it recognizes the give and take between Bantu newcomers and indigenous populations. For example, some indigenous populations rejected Bantu languages, while others repackaged Bantu technologies incorporating their own innovations. There was no Bantu migratory conquest of indigenous communities. Instead, the study of linguistics seems to confirm that Bantu languages, iron-working, and agriculture slowly spread through eastern and southern Africa in the early centuries CE.
In the colonial era, European scholars sometimes jumped to misleading conclusions when they encountered evidence of early African states. For example, in 1871, German geographer Carl Maunch concluded that the ruins of Great Zimbabwe came from the people of Yemen. Biased by nineteenth-century racism, he assumed that Africans were incapable of statehood and skilled masonry techniques. Subsequent Europeans reached similar conclusions upon viewing the site, attributing the civilization to Phoenicians and Arabs.
In the meantime, a number of scholars had confirmed the African origins of Great Zimbabwe. Archaeologists showed that Great Zimbabwe had features, like stone masonry and rituals involving cattle, found in nearby African kingdoms. Historians used oral tradition and linguistics to track African state formation in the region and show that Great Zimbabwe was a Bantu civilization. From approximately 1200 to 1450 CE, Great Zimbabwe was the thriving commercial and political center of a rich southern African state.
During the Middle Ages, a prosperous elite based in Great Zimbabwe ruled over about 300 settlements on the Zimbabwe Plateau. Great Zimbabwe and the linked settlements had similarly constructed walled enclosures, practiced mixed farming (they grew crops and kept livestock), and used iron, copper, and bronze. The settlements paid tribute in the form of ivory, gold, cattle, and crops to the rulers in Great Zimbabwe. Gold and ivory were exported to cities like Sofala and Kilwa Kisanwani on the East African coast. From the coast, these goods were carried to the Persian Gulf, India, and China. In exchange, the city's elite imported luxury items like stoneware, colored glass beads, and cotton. Out of these imports, artisans made jewelry, ornaments, and cloth for elite consumption.
Great Zimbabwe’s social hierarchies is one of the most dramatic elements of the site’s ruins. Covering three square miles, the ruins consist of many clusters of stone buildings constructed with local granite and stacked without mortar. The most famous structures are the Hill Complex (Figure \(\PageIndex{2}\)) and the Great Enclosure (Figure \(\PageIndex{1}\)). Scholars hypothesize that the ruling elite resided and performed ceremonies on the Hill Complex, symbolically demonstrating their authority with the height and separation of the complex. From about 1300 CE, more than 15,000 people lived in the valley below in small, circular homes with thatched roofs and walls made of clay and gravel.
The Hill Complex overlooked a number of other structures. With its stone walls up to thirty-five feet tall, the Great Enclosure was the largest structure in precolonial sub-Saharan Africa. It was a ceremonial site, perhaps used by religious leaders or as a site for the initiation of youth. Scholars disagree about its exact function, but suggest that the Great Enclosure further demonstrated the status and wealth of the capital city and the ruling classes.
In the 15th Century, Great Zimbabwe declined and was abandoned by 1450 CE. Some scholars suggest that the site supporting up to 30,000 people became too crowded, deforested, and stripped bare of resources through overuse. Surrounding gold mines may have also been depleted. In any case, trade shifted to support the rise of two new kingdoms, Batua to the west and Mutapa to the east. Both kingdoms built stone walls like those seen in Great Zimbabwe and practiced mixed agriculture, using cattle for ceremonies and as symbols of the ruling elite’s power. From the fifteenth through seventeenth centuries, the kingdoms also faced the Portuguese and the influx of other African populations. The Mutapa Kingdom lasted the longest, enduring until 1760.
Overall, this rewritten history of southern African statehood acknowledges the significance of the Bantu expansions that brought agriculture and iron to many regions. It also celebrates the African origins of great civilizations and demonstrates how Africans shared technologies and cultural practices across the Zimbabwean plateau.
14 As quoted in Brizuela-Garcia, Esperanza and Trevor Getz, African Histories: New Sources and New Techniques for Studying African Pasts . Boston: Prentice Hall, 2012: 43.


2.13: Simplifying, Multiplying, and Dividing Rational Expressions
You have learned about the properties of fractions and their operations. Rational numbers are just fractions where the numerators and denominators are integers. We will now work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression .
A rational expression is an expression of the form \(\dfrac{p}{q}\), where p and q are polynomials and \(q\neq 0\).
Here are some examples of rational expressions:
\[−\dfrac{24}{56} \qquad \dfrac{5x}{12y} \qquad \dfrac{4x+1}{x^2−9} \qquad \dfrac{4x^2+3x−1}{2x−8}\nonumber\]
Notice that the first rational expression listed above, \(−\dfrac{24}{56}\), is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.
We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.
Determining the Values for Which a Rational Expression is Undefined
If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.
When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.
So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.
Determine the value for which each rational expression is undefined.
- \(\dfrac{8a^2b}{3c}\)
- \(\dfrac{4b−3}{2b+5}\)
- \(\dfrac{x+4}{x^2+5x+6}\)
- Solution
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The expression will be undefined when the denominator is zero.
- \(\begin{array} {ll} &\dfrac{8a^2b}{3c} \\ \begin{array} {l} \text{Set the denominator equal to zero and solve} \\ \text{for the variable.} \end{array} &3c=0 \\ &c=0 \\ &\dfrac{8a^2b}{3c}\text{ is undefined for }c=0 \end{array} \)
- \(\begin{array} {ll} &\dfrac{4b-3}{2b+5} \\ \begin{array} {l} \text{Set the denominator equal to zero and solve} \\ \text{for the variable.} \end{array} &2b+5=0 \\ &2b=-5 \\ &b=-\dfrac{5}{2} \\ & \\ &\dfrac{4b-3}{2b+5} \text{ is undefined for }b=-\dfrac{5}{2} \end{array} \)
- \(\begin{array} {ll} &\dfrac{x+4}{x^2 + 5x + 6} \\ \begin{array} {l} \text{Set the denominator equal to zero and solve } \\ \text{for the variable.} \end{array} &x^2+5x+6=0 \\ &(x+2)(x+3)=0 \\ &x+2=0\text{ or }x+3=0 \\ &x=-2\text{ or }x=-3 \\ & \\ &\dfrac{x+4}{x^2+5x+6}\text{ is undefined for }x=-2\text{ or }x=-3 \end{array} \)
Determine the value for which each rational expression is undefined.
- \(\dfrac{4p}{5q}\)
- \(\dfrac{y−1}{3y+2}\)
- \(\dfrac{m−5}{m^2+m−6}\)
- Answer
-
- \(q=0\)
- \(y=−\dfrac{2}{3}\)
- \(m=2,m=−3\)
Simplifying Rational Expressions
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.
A rational expression is considered simplified if there are no common factors in its numerator and denominator.
For example,
\[ \begin{array} {l} \dfrac{x+2}{x+3} \text{ is simplified because there are no common factors of } x+2 \text{ and }x+3. \\ \dfrac{2x}{3x} \text{ is not simplified because x is a common factor of }2x\text{ and }3x. \\ \end{array} \nonumber\]
We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.
If a , b , and c are numbers where \(b\neq 0,c\neq 0,\)
\[\text {then } \dfrac{a}{b}=\dfrac{a·c}{b·c} \text{ and } \dfrac{a·c}{b·c}=\dfrac{a}{b}\nonumber\]
Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see \(b\neq 0,c\neq 0\) clearly stated.
To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.
Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.
Removing the x ’s from \(\dfrac{x+5}{x}\) would be like cancelling the 2’s in the fraction \(\dfrac{2+5}{2}!\)
How to Simplify a Rational Expression
Simplify: \(\dfrac{x^2+5x+6}{x^2+8x+12}\)
- Solution
-
Simplify: \(\dfrac{a^2−3a−10}{a^2+a−2}\)
- Answer
-
\(\dfrac{a−5}{a−1},a\neq −2,a\neq 1\)
We now summarize the steps you should follow to simplify rational expressions.
- Factor the numerator and denominator completely.
- Simplify by dividing out common factors.
Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors.
We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples.
Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.
Simplify: \(\dfrac{3a^2−12ab+12b^2}{6a^2−24b^2}\)
- Solution
-
\(\begin{array} {ll} &\dfrac{3a^2−12ab+12b^2}{6a^2−24b^2} \\ & \\ & \\ \begin{array} {l} \text{Factor the numerator and denominator,} \\ \text{first factoring out the GCF.} \end{array} &\dfrac{3(a^2−4ab+4b^2)}{6(a^2−4b^2)} \\ & \\ &\dfrac{3(a−2b)(a−2b)}{6(a+2b)(a−2b)} \\ & \\ \text{Remove the common factors of }a−2b\text{ and }3. &\dfrac{\cancel{3}(a−2b)\cancel{(a−2b)}}{\cancel{3}·2(a+2b)\cancel{(a−2b)}} \\ &\dfrac{a−2b}{2(a+2b)} \end{array} \)
Simplify: \(\dfrac{2x^2−12xy+18y^2}{3x^2−27y^2}\)
- Answer
-
\(\dfrac{2(x−3y)}{3(x+3y)}\)
Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is \(−a\) and \(−a=−1·a\).
The numerical fraction, say \(\dfrac{7}{−7}\) simplifies to \(−1\). We also recognize that the numerator and denominator are opposites.
The fraction \(\dfrac{a}{−a}\), whose numerator and denominator are opposites also simplifies to \(−1\).
\[\begin{array} {ll} \text{Let’s look at the expression }b−a. &b−a \\ \text{Rewrite.} &−a+b \\ \text{Factor out }–1. &−1(a−b) \nonumber\end{array} \]
This tells us that \(b−a\) is the opposite of \(a−b\).
In general, we could write the opposite of \(a−b\) as \(b−a\). So the rational expression \(\dfrac{a−b}{b−a}\) simplifies to \(−1\).
The opposite of \(a−b\) is \(b−a\).
\[\dfrac{a−b}{b−a}=−1 \quad a\neq b\nonumber\]
An expression and its opposite divide to \(−1\).
We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat \(a+b\) and \(b+a\) as opposites. Recall that in addition, order doesn’t matter so \(a+b=b+a\). So if \(a\neq −b\), then \(\dfrac{a+b}{b+a}=1\).
Simplify: \(\dfrac{x^2−4x−32}{64−x^2}\)
- Solution
-
Factor the numerator and the denominator. Recognize the factors that are opposites. Simplify.
Simplify: \(\dfrac{b^2+b−2}{1−b^2}\)
- Answer
-
\(−\dfrac{b+2}{b+1}\)
Multiplying Rational Expressions
To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.
If p , q , r , and s are polynomials where \(q\neq 0\), \(s\neq 0\), then
\[\dfrac{p}{q}·\dfrac{r}{s}=\dfrac{pr}{qs}\nonumber\]
To multiply rational expressions, multiply the numerators and multiply the denominators.
Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, x\neq 0,x\neq 0,x\neq 3,x\neq 3, and x\neq 4.x\neq 4.
Simplify: \(\dfrac{2x}{x^2−7x+12}·\dfrac{x^2−9}{6x^2}\)
- Solution
-
Simplify: \(\dfrac{5x}{x^2+5x+6}·\dfrac{x^2−4}{10x}\)
- Answer
-
\(\dfrac{x−2}{2(x+3)}\)
- Factor each numerator and denominator completely.
- Multiply the numerators and denominators.
- Simplify by dividing out common factors.
Multiply: \(\dfrac{3a^2−8a−3}{a^2−25}·\dfrac{a^2+10a+25}{3a^2−14a−5}\)
- Solution
-
\(\begin{array} {ll} &\dfrac{3a^2−8a−3}{a^2−25}·\dfrac{a^2+10a+25}{3a^2−14a−5} \\ & \\ \begin{array} {ll} \text{Factor the numerators and denominators} \\ \text{and then multiply.} \end{array} &\dfrac{(3a+1)(a−3)(a+5)(a+5)}{(a−5)(a+5)(3a+1)(a−5)} \\ & \\ \begin{array} {l} \text{Simplify by dividing out} \\ \text{common factors.} \end{array} &\dfrac{\cancel{(3a+1)}(a−3)\cancel{(a+5)}(a+5)}{(a−5)\cancel{(a+5)}\cancel{(3a+1)}(a−5)} \\ & \\ \text{Simplify.} &\dfrac{(a−3)(a+5)}{(a−5)(a−5)} \\ & \\ \text{We can also rewrite }(a−5)(a−5)\text{ using an exponent.} &\dfrac{(a−3)(a+5)}{(a−5)^2} \end{array}\)
Simplify: \(\dfrac{4b^2+7b−2}{1−b^2}·\dfrac{b^2−2b+1}{4b^2+15b−4}\)
- Answer
-
\(−\dfrac{(b+2)(b−1)}{(1+b)(b+4)}\)
Dividing Rational Expressions
Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.
If p , q , r, and s are polynomials where \(q\neq 0\), \(r\neq 0\), \(s\neq 0\), then
\[\dfrac{p}{q}÷\dfrac{r}{s}=\dfrac{p}{q}·\dfrac{s}{r}\nonumber\]
To divide rational expressions, multiply the first fraction by the reciprocal of the second.
Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.
Divide: \(\dfrac{p^3+q^3}{2p^2+2pq+2q^2}÷\dfrac{p^2−q^2}{6}\)
- Solution
-
Simplify: \(\dfrac{2z^2}{z^2−1}÷\dfrac{z^3−z^2+z}{z^3+1}\)
- Answer
-
\(\dfrac{2z}{z−1}\)
- Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Factor the numerators and denominators completely.
- Multiply the numerators and denominators together.
- Simplify by dividing out common factors.
A complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
Divide: \(\dfrac{\dfrac{6x^2−7x+2}{4x−8}}{\dfrac{2x^2−7x+3}{x^2−5x+6}}\)
- Solution
-
\(\begin{array} {ll} &\dfrac{\dfrac{6x^2−7x+2}{4x−8}}{\dfrac{2x^2−7x+3}{x^2−5x+6}} \\ & \\ \text{Rewrite with a division sign.} &\dfrac{6x^2−7x+2}{4x−8}÷\dfrac{2x^2−7x+3}{x^2−5x+6} \\ & \\ \begin{array} {l} \text{Rewrite as product of first times reciprocal} \\ \text{of second.} \end{array} &\dfrac{6x^2−7x+2}{4x−8}·\dfrac{x^2−5x+6}{2x^2−7x+3} \\ & \\ \begin{array} {l} \text{Factor the numerators and the} \\ \text{denominators, and then multiply.} \end{array} &\dfrac{(2x−1)(3x−2)(x−2)(x−3)}{4(x−2)(2x−1)(x−3)} \\ & \\ \text{Simplify by dividing out common factors.} &\dfrac{\cancel{(2x−1)}(3x−2)\cancel{(x−2)}\cancel{(x−3)}}{4\cancel{(x−2)}\cancel{(2x−1)}\cancel{(x−3)}} \\ \text{Simplify.} &\dfrac{3x−2}{4} \end{array}\)
Simplify: \(\dfrac{\dfrac{y^2−36}{2y^2+11y−6}}{\dfrac{2y^2−2y−60}{8y−4}}\)
- Answer
-
\(\dfrac{2}{y+5}\)
If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.
Perform the indicated operations: \(\dfrac{3x−6}{4x−4}·\dfrac{x^2+2x−3}{x^2−3x−10}÷\dfrac{2x+12}{8x+16}\)
- Solution
-
Rewrite the division as multiplication
by the reciprocal.Factor the numerators and the denominators. Multiply the fractions. Bringing the constants to
the front will help when removing common factors.Simplify by dividing out common factors. Simplify.
Perform the indicated operations: \(\dfrac{2n^2+10n}{n−1}÷\dfrac{n^2+10n+24}{n^2+8n−9}·\dfrac{n+4}{8n^2+12n}\)
- Answer
-
\(\dfrac{(n+5)(n+9)}{2(n+6)(2n+3)}\)


2.4: Principal Ideals and Euclidean Domains
In this section, we'll seek to answer the questions:
- What are principal ideals, and what are principal ideal domains?
- What are Euclidean domains, and how are they related to PIDs?
One of the ways in which mathematicians study the structure of an abstract object is by considering how it interacts with other (related) objects. This is especially true of its sub objects. Thus, in linear algebra, we are often concerned with subspaces of a vector space as a means of understanding the vector space, or even submatrices as a way of understanding a matrix (see, e.g., the cofactor expansion formula for the determinant). In real analysis and topology, the important subobjects are usually open sets, or subsequences, and the study of a graph's subgraphs is an important approach to many questions in graph theory.
In this section, we begin a set-theoretic structural exploration of the notion of ring by considering a particularly important class of subring which will be integral to our understanding of factorization.
These subrings are called ideals . They arose in the work of Kummer and Dedekind as a way of trying to recover some notion of unique factorization in rings that do not have properties like the fundamental theorem of arithmetic in \(\mathbb{Z}\text{.}\)
A subset \(I\) of a (not necessarily commutative) ring \(R\) is called an ideal if:
- \(\displaystyle 0\in I\)
- for all \(x,y\in I\text{,}\) \(x+y\in I\text{;}\) and,
- for all \(x\in I\) and for all \(r\in R\text{,}\) \(xr\in I\) and \(rx\in I\text{.}\)
Observe that the third requirement for a set \(I\) to be an ideal of \(R\) is simplified slightly if \(R\) is commutative (which, we recall, all of our rings are).
There are many important examples and types of ideals, but there are also some trivial ideals contained in every ring.
Let \(R\) be a ring. Then \(R\) and \(\{0\}\) are ideals of \(R\text{.}\)
All ideals are subrings.
The following theorem provides a useful characterization of when an ideal \(I\) is in fact the whole ring.
Let \(R\) be a ring and \(I\) an ideal of \(R\text{.}\) Then \(I = R\) if and only \(I\) contains a unit of \(R\text{.}\)
The most important type of ideals (for our work, at least), are those which are the sets of all multiples of a single element in the ring. Such ideals are called principal ideals .
Let \(R\) be commutative with identity and let \(a\in R\text{.}\) The set
\begin{equation*} \langle a \rangle= \{ra : r\in R\} \end{equation*}
is an ideal (called the principal ideal generated by \(a\) ).
The element \(a\) in the theorem is known as a generator of \(\langle a \rangle\text{.}\)
Let \(R\) be commutative with identity, and let \(x,y,z\in R\text{.}\) Give necessary and sufficient conditions for \(z\in \langle x \rangle\) and, separately, \(\langle x \rangle \subseteq \langle y \rangle\text{.}\)
That is, fill in the blanks: “\(z\in \langle x \rangle \Leftrightarrow\) _________” and “\(\langle x \rangle\subseteq \langle y \rangle \Leftrightarrow\) _________.”
Justify your answers.
Principal ideals may have more than one generator.
Let \(R\) be a ring and \(a\in R\text{.}\) Then \(\langle a \rangle= \langle ua \rangle\text{,}\) where \(u\) is any unit of \(R\text{.}\)
In \(R = \mathbb{Z}\text{,}\) describe the principal ideals generated by
- 2
- \(\displaystyle -9\)
- 9
- 0
- 27
- 3
Determine the subset relations among the above ideals.
It is the case in many familiar settings that all ideals are principal. Such domains are given a special name.
An integral domain \(R\) in which every ideal is principal is known as a principal ideal domain (PID) .
The ring \(\mathbb{Z}\) is a principal ideal domain.
- Hint
-
Use properties specific to \(\mathbb{Z} \text{,}\) perhaps from Section 1 .
Find an integer \(d\) such that \(I = \langle d \rangle\subseteq \mathbb{Z}\text{,}\) if
- \(\displaystyle I = \{4x+10y: x,y\in\mathbb{Z}\}\)
- \(\displaystyle I = \{6s+7t : s,t\in\mathbb{Z}\}\)
- \(\displaystyle I = \{9w+12z : w,z\in\mathbb{Z}\}\)
- \(\displaystyle I = \{am+bn : m,n\in\mathbb{Z}\}\)
You do not need to prove that each of the sets above are ideals (though you should make sure you can do it).
Let \(R\) be a principal ideal domain and \(x,y\in R\) be not both zero. Let \(I = \{xm+yn: m,n\in R\}\text{.}\) Then:
- \(I\) is an ideal, and
- \(I = \langle d \rangle\text{,}\) where \(d\) is any greatest common divisor of \(x\) and \(y\text{.}\)
We conclude that there exist \(s,t\in R\) such that \(d = xs + yt\text{.}\)
We have so far abstracted and axiomatized several important algebraic properties of \(\mathbb{Z}\) that we discussed in § 1 . In particular, we have our usual operations of addition and multiplication, and their interactions; we have notions of divisibility/factorization, irreducibility, and primality; we also have cancellation and greatest common divisors.
Our last major abstraction from \(\mathbb{Z}\) is the division algorithm. The main obstacle to postulating domains with a division algorithm is a clear notion of comparison relations. That is, if \(R\) is an arbitrary domain with \(r,s\in R\text{,}\) is it possible to clearly and sensibly say which of \(r\) or \(s\) is “bigger” ? (Recall that this was a requirement for the division algorithm with nonzero remainders.) However, if there is a way to relate elements of a domain \(R\) to \(\mathbb{N}_0\text{,}\) we can sensibly define a division algorithm.
Let \(R\) be an integral domain. We call \(R\) a Euclidean Domain if there is a function \(\delta : R\setminus \{0\} \to \mathbb{N}_0\) such that:
- If \(a,b\in R\setminus \{0\}\text{,}\) then \(\delta(a) \le \delta(ab)\text{.}\)
- If \(a,b\in R\text{,}\) \(b\ne 0\text{,}\) then there exist \(q,r\in R\) such that \(a = bq+r\text{,}\) where either \(r = 0\) or \(\delta(r) \lt \delta(b)\text{.}\)
We call the function \(\delta\) a norm for \(R\text{.}\)
\(\delta\): This is the lowercase Greek letter delta .
Thus, a Euclidean domain is an integral domain with a division algorithm that behaves in a familiar way. In the remainder of this section, we will investigate the properties of Euclidean domains. First, we consider some examples.
The field \(\mathbb{Q}\) is a Euclidean domain under ordinary addition and multiplication, with \(\delta(x) = 0\) for all \(x\in \mathbb{Q}\text{.}\)
Is \(\mathbb{Z}\) a Euclidean domain? If so, what is the norm function \(\delta\text{,}\) and why does this function have the required properties of a norm?
Let \(F\) be a field and \(S\subseteq F[x]\) a set containing a nonzero polynomial. Prove that \(S\) contains a polynomial \(f\) such that \(\deg(f) \le \deg(g)\) for all nonzero \(g\in S\text{.}\)
Let \(F\) be a field and \(f(x),g(x)\in F[x]\) with \(g(x)\ne 0\text{.}\) If \(\deg f(x) \ge \deg g(x) > 0\text{,}\) and \(f(x) = a_0 + a_1 x + \cdots + a_m x^m\) and \(g(x) = b_0 + b_1 x + \cdots + b_n x^n\text{,}\) then \(h(x) = f(x) - a_m b_n^{-1} x^{m-n} g(x)\) has degree strictly less than \(\deg f(x)\text{.}\)
Let \(F\) be a field and \(f(x),g(x)\in F[x]\) with \(g(x)\ne 0\text{.}\) Then there exist unique \(q(x), r(x) \in F[x]\) such that
\begin{equation*} f(x) = g(x) q(x) + r(x)\text{,} \end{equation*}
where \(\deg(r(x)) \lt \deg g(x)\text{.}\)
- Hint
-
For existence, consider three cases: \(f(x) = 0\text{;}\) \(f(x) \ne 0\) and \(\deg f \lt \deg g\text{;}\) \(f(x) \ne 0\) and \(\deg f \ge \deg g\text{.}\) In the last case, use induction on \(m = \deg f(x)\text{.}\) For uniqueness, mimic the uniqueness proof of Theorem 1.2.4 .
Let \(F\) be a field. Then the ring \(F[x]\) is a principal ideal domain.
- Hint
-
Mimic the proof of Theorem 2.4.6 and use Lemma 2.4.2 !
Is \(F[x]\) a Euclidean domain for all fields \(F\text{?}\) If so, what is the norm function \(\delta\text{,}\) and why does this function have the required properties of a norm? If not, why not? Prove your answer.
In fact, every Euclidean domain is a PID.
Every Euclidean domain is a principal ideal domain.
- Hint
-
Mimic the proof of Theorem 2.4.6 .
Where do Euclidean domains and PIDs fit in the hierarchy of abstraction found in Exploration 2.3.5 ?


12.3: Hubble’s Law
Hubble’s Law
Two years later, in 1929, Hubble confirmed the Universe is expanding. Hubble also was able to infer the recessional velocities of a number of objects from the spectral redshifts he observed.
Hubble’s Law states that an object’s recessional velocity is proportional to the distance from the observer. In equation form, Hubble’s Law is described by:
v = H o d
Where:
- v is the velocity of the object, in km/s
- d is the distance to the object, in megaparsecs, Mpc, where 1 Mpc = 1 million parsecs; and
- H o , the Hubble constant or Hubble parameter, a proportionality between d and v ; also known as the rate of expansion, in (km/s)/Mpc or simply km/s/Mpc.
What does the km/s/Mpc unit mean? The km/s is a velocity, kilometers per second. Most of us are used to speeds in miles per hour, or mi/hr. A km/s velocity is much faster velocity than we are used to; 1 km/s is approximately 2,237 mi/hr. A megaparsec, Mpc, is a specific distance, about 3.26 million light years, 30.86 × 10 18 kilometers, or 1.92 × 10 19 miles. The km/s/Mpc is a unit of velocity per distance, velocity per megaparsec or about 3.26 million light years.
The Hubble’s Law equation can be rearranged to solve for an object’s distance:
v = H o d
d = v/H o
With Hubble’s Law arranged in this form, one can determine the distance to an object like a distant galaxy or quasar by determining the object’s recessional velocity from the object’s spectral red shift and knowing the Hubble parameter.
With an understanding of the relationship between an object’s redshift, its receding velocity, and distance, the work turned to determining the Hubble parameter value, H o . A correct, or at least close, Hubble parameter would lead astronomers and cosmologists to not only determining the distances to these galaxies, but the age of the observable Universe itself. Hubble’s Law applies to galaxies and objects that are extremely far away; specifically, more than 10 megaparsecs from the observer.