<IP_ADDRESS>.2: Polypodiopsida
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- Tom Buckley
- University of California, Davis
Learning Objectives
- Differentiate between ferns, horsetails, and lycophytes.
- Identify features of vegetative and reproductive shoots of Equisetum .
- Identify features and phases of the fern life cycle; know their ploidy.
- Label a fern gametophyte and sporophyte.
The Polypodiopsida includes the horsetails and ferns. These plants produce leaves with branching veins of vascular tissue called megaphylls. Like the lycophytes, asexual reproduction of the sporophyte can be accomplished via rhizomes. Spores produced in this group develop into gametophytes that can produce both antheridia and archegonia.
Characteristics
- Megaphylls . Leaves have branching veins of vascular tissue. Megaphylls are thought to have evolved from branching stems. Webbing filled in the spaces between the branches, forming the flat blade of the leaf. The arrangement of the leaf veins (vascular bundles) reflects the original branching pattern of the stems.
- Rhizomes . Asexual propogation of the sporophyte through underground stems.
- Homosporous . Haploid spores grow into bisexual gametophytes that produce both antheridia and archegonia, or are capable of producing one or the other, dependent upon conditions.
Equisteum (subclass Equisetidae)
Horsetails are a small group with a single extant genus, Equisetum , which has about 30 different herbaceous species that typically live in moist habitats. The common name comes from the characteristic pattern of branching: whorls or rings of branches arising from an above-ground shoot. The leaves of these plants have been reduced to scales, and instead the segmented stems are photosynthetic. If you look closely at the nodes of a green vegetative shoot , you will see that branches and leaves have not only switched roles, they have also switched places, with the photosynthetic branches emerging below the papery, non-photosynthetic leaves.
Horsetails often grow in sandy places and incorporate silica in their stem epidermis, which gives it an abrasive surface. Because of this, American pioneers would use this plant to scour pots and pans. This is how it received the nickname “scouring rush.” The stem has multiple canals, an analogous characteristic to stems of grasses. The sporangia are associated with hexangular stalked sporangiophores produced on terminal strobili. Within the sporangia, there are elaters that are not separate cells but parts of the spore walls. Gametophytes are typically minute and dioecious, but the plants themselves are homosporous: smaller suppressed gametophytes develop only antheridia while larger gametophytes develop only archegonia.
Gametophyte Morphology
Horsetail gametophytes are reduced and thalloid (figure \(\PageIndex{1}\)). Gametophytes grow from homospores and can produce both antheridia and archegonia.
Sporophyte Morphology
In some Equisetum species, there are two different types of shoots produced by the sporophyte: vegetative shoots that perform photosynthesis and reproductive shoots that form strobili and undergo meiosis. In other species, the strobilus is formed at the apex of a photosynthetic shoot.
Vegetative Shoots
On the vegetative shoot, the leaves are dark, papery and non-photosynthetic. Branches are photosynthetic and produced in whorls. Branches and leaves emerge at nodes , separated by regions of the main stem called internodes . Unlike most plants, the branches emerge below the leaves in the node (Figure \(\PageIndex{2-3}\)). The epidermis of the stems contain silica , which has an abrasive texture.
Reproductive Shoots
Sporangia are produced in a terminal strobilus on the reproductive shoot (figure \(\PageIndex{4-5}\)). In some species, this reproductive shoot lacks chlorophyll and is instead fed through the rhizome of connected vegetative shoots. Spores are photosynthetic and have four hygroscopic arms called elaters.
Video \(\PageIndex{1}\): This video shows how the elaters of Equisetum spores respond to changes in humidity. Retrieved from YouTube .
Ferns (subclass Polypodiidae)
Some 15,000 species of ferns live on earth today. Many of these are found in the tropics where some — the "tree ferns" — may grow to heights of 40 ft (13 m) or more. The ferns of temperate regions are smaller. They are usually found in damp, shady locations. They produce perennial rhizomes that can overwinter. Their leaves, called fronds due to apical growth, emerge from the rhizome each spring as coiled fiddleheads (Figure \(\PageIndex{6}\)). These fiddleheads open through a process called circinate vernation , where the growing tissues are protected at the center of the coil and emerge last.
True ferns are megaphyllous : their leaves originated from flattened branches and have branching veins of vascular tissue. True ferns have unique sporangia: leptosporangia (Figure \(\PageIndex{6}\)). Leptosporangia originate from a single cell in a leaf, they have long, thin stalks, and the wall of one cell layer. They also open actively: when sporangium matures (dries), a row of cells with thickened walls on the outside of the sporangium (called an annulus ) will shrink slower than surrounding cells and finally would break and release all spores at once. Leptosporangia are grouped in clusters called sori which are often covered with umbrella- or pocket-like indusia . Gametophytes are minute and grow aboveground. While most ferns are homosporous, some genera of true ferns (like the water fern Azolla , water shamrock Marsilea and several others) are heterosporous (Figure \(\PageIndex{7}\)).
True ferns are highly competitive even to angiosperms. In spite of their “primitive” life cycle, they have multiple advantages: abilities to photosynthesize in deep shade (they are not obliged to grow fast), to survive high humidity, and to make billions of reproductive units (spores). Ferns do not need to spend their resources on flowers and fruits, and are also less vulnerable to vertebrate herbivores and insect pests, probably because they do not employ them as pollinators and, therefore, can poison tissues against all animals.
Gametophyte Morphology
Fern gametophytes are reduced, thalloid, and heart-shaped (Figure \(\PageIndex{8}\)). They are often referred to as a prothallus or prothallium . Rhizoids are produced from the underside of the thallus, just like in the bryophytes. Similar to the horsetails, whether a gametophyte produces antheridia or archegonia can be regulated by environmental cues (Figure \(\PageIndex{9}\)). Each antheridium produces many swimming sperm, but archegonia produce only a single egg (Figure \(\PageIndex{10}\)).
Sporophyte Morphology
Fern sporophytes are composed of megaphylls, often pinnately compound fronds that emerging as coiled fiddleheads in the spring. Sporangia are produced in clusters called sori (sorus, singular) on the fronds (Figure \(\PageIndex{11}\)).
Circinate vernation is a term used to describe the development of the fern fiddlehead (Figure \(\PageIndex{12}\)) into a frond. Because plants grow apically, it is important to protect the apical meristems in growing organs (as we have seen in both axillary and terminal buds with the protective bud scales). The fiddlehead is essentially a structure that tucks away the growing tips of the fronds. As the frond develops, it gradually unfurls, releasing the tips last.
A sorus (plural, sori) is a cluster of sporangia, often protected by an umbrella-like structure called the indusium as the spores mature (Figure \(\PageIndex{13}\)). Some sori are protected by an extension of the leaf called a false indusium (Figure \(\PageIndex{14}\)), while others lack any protective covering. Each sporangium is lined by an inflated strip of cells called an annulus . When the spores have matured, the cells in the annulus begin to dry out, causing the cells to collapse and pull the sporangium open, releasing the spores.
Full Life Cycle Diagram
Ferns rely on water for dispersal of the sperm, which must swim into an archegonium to fertilize an egg (Figure \(\PageIndex{15}\)). If moisture is plentiful, the sperm swim to archegonia - usually on another prothallus because the two kinds of sex organs generally do not mature at the same time on a single prothallus.
Another method for promoting cross-fertilization: The first spores to germinate develop into prothallia with archegonia. These prothallia secrete a gibberellin into their surroundings. This is absorbed by younger prothallia and causes them to produce antheridia exclusively.
Fertilization restores the diploid number and begins a new sporophyte generation. The embryo sporophyte develops a foot that penetrates the tissue of the prothallus and enables the sporophyte to secure nourishment until it becomes self-sufficient. Although it is tiny, the haploid fern prothallus is a fully-independent, autotrophic plant. Soon, the sporophyte is nutritionally independent. It is the larger, longer-lived stage of the life cycle. To reproduce, many sori are formed on the undersides of the fronds. Within each sporangium of the sorus, the spore mother cells undergo meiosis producing four haploid spores each.
When the humidity drops, the thin-walled lip cells of each sporangium separate, the annulus slowly straightens out, then the annulus snaps forward expelling the spores. Each of these homospores can then grow into a gametophyte capable of producing antheridia and archegonia.
Attributions
- Curated and authored by Maria Morrow using 16.3C Fern Life Cycle and 19.1.5 Diversity and Evolutionary Relationships of the Plants Biology by John. W. Kimball (licensed CC-BY )
- Megaphyll evolution and Salvinia natans information added by Melissa Ha


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HGTC BIO 225: Microbiology Lab Manual
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5: The Gram Stain
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11.5: Fossil Fuels and Greenhouse Gases
Fossil fuels are made up mainly of hydrogen and carbon. When burned, the carbon combines with oxygen to create carbon dioxide (CO 2 ). The amount of CO 2 produced depends on the carbon content of the fuel. For example, for the same amount of energy produced, natural gas produces about half and petroleum produces about three-fourths of the amount of CO 2 produced by coal. Energy-related CO 2 emissions, resulting from the combustion of coal, petroleum, and natural gas, account for about 80% of total U.S. human-caused (anthropogenic) greenhouse gas (GHG) emissions. There are many sources of non-energy CO 2 emissions, but those emissions account for a relatively small share of total GHG emissions. See chapter 7 for a discussion of the results of GHG emissions.
Energy use is largely driven by economic growth and by weather patterns that affect heating and cooling needs. The fuels used in electricity generation also have an impact on the amount of GHG emissions. In the United States, most of the electricity generated comes from coal power plants and consequently, the majority of the carbon dioxide emission resulting from electricity generation is from coal combustion (Figure \(\PageIndex{1}\)). Although the industrial sector is the largest consumer of energy (including direct fuel use and purchased electricity), the transportation sector emits more carbon dioxide because of its near complete dependence on petroleum fuels. The residential and commercial sectors have lower emission levels (most of which comes from fossil energy combustion to produce electricity) than the transportation and industry sectors.


6.17: How to Write an Informal Report
- Discuss how to write an informal report
Writing informal reports follows the same steps of any other writing task. First is the plan. Second is the writing. Third is the revising.
Planning Your Informal Report
When asked to create an informal report, first check to see if your organization has a form or template that should be used. Then verify your understanding of the report’s purpose.
For example, say you are a shift manager at a grocery store, and there has been an increase in customer complaints about fruit that seems to spoil more quickly than it used to. Your store manager has asked you to create a report on this issue. You need to determine whether your manager wants to know causes of fruit spoilage (including items such as time each type of fruit stays fresh from date picked, types of shipping containers, or temperature of storage units), or if your manager wants to know what is happening in the store after the fruit is received (how the fruit is handled, how much fruit can sit on top of other fruit, or temperature in the various storage units). The purpose of a report will impact the amount and type of research to be done.
Next you’ll complete any data gathering needed; by the end of the project, you should have more data and knowledge than you started with (and possibly more than you need for the report itself). You’ll use that data to create the report’s outline. Writers must take care to provide only what is needed for the purpose of the report: avoid wandering to interesting side issues or presenting everything you learned whether or not it’s relevant.
In the process of writing a report, or almost any business writing, the planning step should take at minimum 25–30 percent of the time or effort of the full report.
Writing Your Informal Report
With the detailed outline created in the planning process, the actual writing of the informal report should go quickly. In this step, you’ll focus on paragraph structure, wording, and phrasing using the lessons found in Module 2: Writing In Business.
Sometimes, writers hear the term “report” and think their writing style must change. What works well for short messages also works well for informal reports. The primary difference is that a report requires a bit more depth to appropriately communicate its message: there are more words and paragraphs, but the words do not need to be longer or more complex sounding. Write with the same skills taught in Module 2: Writing In Business.
Different companies have different styles for writing reports: you should always match they style of your current institution. Some companies accept a more casual style of writing. This may include the use of personal pronouns such as “I recommend . . . ” or “we completed a survey of 20 people.” Some companies accept use of contractions as in, “The Customer Contact team couldn’t reach a conclusion on types of bags to use,” while others do not.
In all cases, remember that a report may be retained for a long time and may be viewed by many readers. With your current credibility and future credibility possibly at stake, it is generally better to be safe by using slightly more professional tone.
Formatting Your Report
In writing your report, remember that headings guide the reader, but like an email subject line, they are no substitute for clear, descriptive writing that helps the reader stay on track. While writing your report, you should use summary statements as each paragraph or section closes to avoid a jerky, disconnected feel in your writing. Ensure that each new section below a header has a good topic sentence that serves as an introduction to the section.
When writing your report, you can take your preexisting outline (from the planning step) and use your word processor’s pre-formatted heading styles to create the headings for your report. This provides two benefits: it quickly organizes your report in a pleasing way, and it meets ADA (Americans with Disabilities Act) requirements.
When writing a report, writers often tend to add sections simply because they are “supposed to be there,” rather than focusing on the purpose of each section and how it might support the report.
A stronger writing skill is to look to the type of report and the outline prepared for the writing, then select headers that suit the content, rather than content suiting the header. With informal reports, the style is somewhat relaxed, so headers should focus on making information easy for the reader to access.
When writing a report, or in almost any business writing, the writing step takes about 40–50 percent of the total time or effort for the full report. This may surprise many writers who think that this step is all you need to complete for a report. However, if you spend the time to ensure the planning step is well done, writing goes much more quickly, and you’ll produce a better report.
Revising Your Informal Report
As with most documents, the final step in creating a report is the one most frequently skipped or only partially completed by writers; in fact, writers will often intentionally skip this step, likely because it is at the end of a long process, and they are often eager to submit their work to the requester.
Additionally, their familiarity with the content can lead to them seeing what was intended versus what is actually written. For example, the sentence, “In summary, the store should now implement the new plan” can accidentally be typed, “In summary, the store should not implement the new plan” to disastrous results. To combat this, you can use word processing proofreading tools, which will catch some spelling errors. Then, no matter how long it takes, read the report aloud. A team member or peer is an excellent additional reviewing tool.
Another way to fail on this step is to read only for proofreading and grammar mistakes. However, revising should also include going back to the original request for the report and back to the original outline to see if the report is directly focused on the planned purpose. Along the way of data gathering and finding new ideas on a topic, there can be some unintentional shift in the focus of the writing. Look to ensure that just the information needed to address the topic is present. Ensure that the primary purpose comes across clearly in your writing.
In the process of writing a report, or almost any business writing, the revising step takes about 25–30 percent of the total time or effort of the full report.
Contributors and Attributions
- How to Write Informal Report. Authored by : Susan Kendall. Provided by : Lumen Learning. License : CC BY: Attribution


Before going any further, let us spend some time discussing what we mean by culture. When you began reading this chapter what did you think we meant by the word culture? Your answer probably had something to do with people from different countries or of different racial and ethnic backgrounds. You are right—to a certain degree. Culture does include race, nationality, and ethnicity, but goes beyond those identity markers as well. The following are various aspects of our individual identity that we use to create membership with others to form a shared cultural identity: race, ethnicity, nationality, gender, sexual orientation, and social class. In addition to explaining the above identities, we will also discuss ethnocentrism, privilege, advantage, disadvantage, power, whiteness, co-culture, and political correctness as these terms are relevant to understanding the interplay between communication and culture.
When we talk about
culture
we are referring to
belief systems, values, and behaviors that support a particular ideology or social arrangement
. Culture guides language use, appropriate forms of dress, and views of the world. The concept is broad and encompasses many areas of our lives such as the role of the family, individual, educational systems, employment, and gender.
Understanding Race
“When you begin to understand the biology of human variation, you have to ask yourself if race is a good way to describe that.”
–Janis Hutchinson, Biological Anthropologist
Race is often difficult to talk about, not because of the inherent complexity of the term itself, but because of the role that race plays in society. Race is what we call a loaded word because it can bring up strong emotions and connotations. Understandings of race fall into two camps: a biological versus a sociopolitical construction of what it means to belong to a particular racial group. A
biological construction of race
claims that
“pure” races existed and could be distinguished by such physical features as eye color and shape, skin color, and hair.
Moreover, these differences could be traced back to genetic differences. This theory has been debunked by numerous scientists and been replaced with the understanding that there are greater genetic differences within racial groups, not between them. In addition, there is no scientific connection with racial identity and cultural traits or behaviors.
Instead of biology, we draw on a
sociopolitical
understanding of what it means to be of a particular race. This simply means that it is not a person’s DNA that places them into a particular racial grouping, but all of the other factors that create social relations—politics, geography, or migration. We can also examine the reality that the meanings of race have changed across time and space. As dramatized in the 2002 film, “Gangs of New York,” the Irish were once considered a minority with little social or political status. Now, being Irish in America is considered part of the general majority group, white or Caucasian. Noting the change from the biological to the sociopolitical understanding, we refer to race as “a largely social—yet powerful— construction of human difference that has been used to classify human beings into separate value-based categories” (Orbe and Harris 9).
For additional information check out
this website
. The website “look[s] through the eyes of history, science, and lived experience, the RACE Project explains differences among people and reveals the reality – and unreality – of race.”
Related to race are three other distinct concepts: racial prejudice, racial discrimination, and racism.
Racial prejudice
refers to
the practice of holding false or negative beliefs of one racial group for the purpose of making another racial group (usually one’s own) appear superior or normative. Racial discrimination is the outward manifestation of racial prejudice: it is when people act upon their negative beliefs about other races when communicating or setting policy.
Note, it is possible to be prejudiced without acting upon those beliefs and that all races can discriminate against other races. The final concept, racism, combines racial prejudice with social power. Racism is institutional, rather than individual, meaning it
occurs in large institutional contexts such as the representations of particular groups within media or the fact that racial minorities do not have equal access to educational or legal opportunities
(Orbe and Harris 10). Racism often involves the unequal accessibility to resources and power.
Where Do You Come From?
Two other concepts that are often confused with race are ethnicity and nationality.
Ethnicity
refers to
a person’s or people’s heritage and history, and involves shared cultural traditions and beliefs.
A person may identify as Asian-American racially while their ethnicity is Chinese.
Nationality
refers to
a people’s nation-state of residence or where they hold citizenship.
Most often nationality is derived from the country where one was born, but on occasion people give up their citizenship by birth and migrate to a new country where they claim national identity. For example, an individual could have been born and raised in another country but once they migrate to the United States and have American citizenship, their nationality becomes American.
   Gender is discussed in more detail in Chapter 13, but for now think of it as the recognition that one is male, female, or androgynous.
Gender
is part of culture in that every
society has particular gender roles and expectations for males and females.
For example, in the United States, it is considered normal for the female gender to wear makeup, while it is often considered inappropriate for a male to do so. However, in some Native American tribes it was customary for the males to adorn themselves with paint for hunting and ceremonial rituals.
Sexual orientation
refers to
a person’s preference for sexual or romantic relationships; one may prefer a partner of the same sex, the opposite sex, or both.
Sexual orientation influences one’s worldview or politics because while all societies include members who identify as gay or lesbian, these members do not always receive the same social or health benefits as heterosexual couples. However, this is changing. As of 2015, the Supreme Court of the Unites States made gay marriage legal in all 50 states. On top of these specific benefits, those with a nondominant sexual orientation might still have to contend on a daily basis that some people think they are deviant or somehow less than heterosexual people and couples. This may result in strained family relationships or discrimination in the workplace.
The Role of Money
You are probably familiar with the concept of class—what do the labels working class, middle-class, and upper-class bring to mind? Money? Economic standing is only one variable that influences class or socioeconomic standing. As the label suggests, one’s socioeconomic status is influenced by monetary and social factors.  For example, in some middle class families children are expected to go to college just as their parents and grandparents had done. It may also be expected for the children to attend reasonably priced state colleges and universities as opposed to Ivy League Universities, which may be the norm in many upper-class families.
By now you are probably able to think of some other identity markers that shape a person’s culture or worldview. How about spirituality or religion, profession, hobbies, political persuasion, age, abilities? These too are aspects of cultural identity. Spend some time thinking about how these aspects would influence a person’s culture as we have done above.
We may often feel restrained by the constant need to work. We live in a money-centric society where every move we make involves thinking about the monetary gains or losses it will produce. Read Bruce E. Levine’s article on this phenomena.
How America's Obsession With Money Deadens Us
After reading the article, do you believe that we have become more money-centric? Why?
I
ntersectionality
Individuals cannot look simply at race, ethnicity, nationality, gender, sexual orientation, ability, disability, religion, and social class in isolation. Instead, we must look at identity as a combination of these. Cultural identities have many parts that all factor into an individual's identity. Intersectionality is the theory that shows how multiple systems of power and oppression operate on individuals in various degrees that directly corresponds to their identity (Crenshaw, 1989). Intersectionality was a theory articulated by Kimberle Crenshaw, an American civil rights advocate, in the late 1980’s. A United States citizen and a Burmese citizen both growing up in modern times will not have equal opportunities. One might think that the US citizen will have more opportunities based on national privilege. But if the Burmese citizen is from a higher socioeconomic class than the American, the Burmese individual may have more opportunities. Other factors such as race, sexuality and ability also play into how many and which opportunities will be available.
Another example from Kimberle Crenshaw from her 1989 journal article
Demarginalizing the Intersection of Race and Sex: A Black Feminist Critique of Antidiscrimination Doctrine, Feminist Theory, and Antiracist Politics
states:
“Consider an analogy to traffic in an intersection, coming and going in all four directions. Discrimination, like traffic through an intersection, may flow in one direction, and it may flow in another. If an accident happens in an intersection, it can be caused by cars traveling from any number of directions and, sometimes, from all of them. Similarly, if a Black woman is harmed because she is in an intersection, her injury could result from sex discrimination or race discrimination. . . . But it is not always easy to reconstruct an accident: Sometimes the skid marks and the injuries simply indicate that they occurred simultaneously, frustrating efforts to determine which driver caused the harm.”
Intersectionality is important for intercultural communication because individuals must realize not only the duality but the depth of identities. This theory is useful for critical analysis because it illuminates the different ways systems of oppression overlap and how systems of privilege and oppression can manifest simultaneously on an individual (Allen, 2010). It is easy for people to assume because they are one thing that is it their whole identity, just like not every feminist’s agenda is the same and not every Syrian’s views are the same. Intersectionality is a key to intercultural communication because it reminds individuals that systems of power although do not operate uniformly and effect how people will receive and transmit messages.
Contributions and Affiliations
-
Survey of Communication Study.
Authored by
: Scott T Paynton and Linda K Hahn.
Provided by
: Humboldt State University.
Located at
: en.wikibooks.org/wiki/Survey_of_Communication_Study.
License
:
CC BY-SA: Attribution-ShareAlike
-
Image of Thai women.
Authored by
: Morguefile.com.
Located at
: commons.wikimedia.org/wiki/File:Thailandculture.png.
License
:
CC BY: Attribution
-
Image of family on beach.
Authored by
: morguefile.com.
Located at
: commons.wikimedia.org/wiki/File:Gaycouple.png.
License
:
CC BY: Attribution


6.1: Introduction
“I love you when you bow in your mosque, kneel in your temple, pray in your church. For you and I are sons of one religion, and it is the Spirit.”
Khalil Gibran
This chapter is an exploration of the geography of religion. Like language and ethnicity, religion is a cultural characteristic that can be closely bound to individual identity. Religion can provide a sense of community, social cohesion, moral standards, and identifiable architecture. It can also be a source of oppression, social discord, and political instability. Religion is more than metaphysicsmagical explanations for natural phenomena; it is also a governing philosophy of behavior and an organizing cosmology of the universe.
The following pie chart gives us an idea of the relative size (in terms of adherents) of the world’s major religions ( Figure 6.1 ). Bear in mind that these numbers are estimates; there is no world governing body collecting detailed statistics of religion. Christians and Muslims make up over half of the world’s population. The next category, the unaffiliated, are a large but diffuse body containing people who do not identify with any religion. Hindus, who cluster in the Indian subcontinent, are the next largest group. The category of Folk religion is similar to that of Unaffiliated-it is a large group of religions that are bound into one category solely for logical consistency. Folk religions may consist of ancestor worship in China, animism in central Africa, or any other number of indigenous, local religions. The Other category contains newer religions that are just gaining their footholds and other religions that may be fading in the contemporary milieu. Judaism is included in the chart, although it has comparatively few adherents. It is included for two reasons. First, it provided the cultural spoor for both Christianity and Islam, and second, it is the predominant religion of the modern state of Israel.
In some places, religion can be considered a separate element of civil society, but in many others, religion cannot be meaningfully separated from daily life or governance. Charts such as this can be somewhat misleading, as can maps of religion ( Figure 6.2 ). All these methods of tabulating religion rely on estimates with varying degrees of accuracy. One problem is determining the predominance of a particular religion. Does predominance require over fifty percent? What if no religion in a country has a majority? In the case of this map, if no religion has a majority, but there are two large religions (for example Christianity and Islam in Nigeria) then the country is split between the two. If there are numerous fragmented groups, then the group with a plurality is used.
In some parts of the world, some forms of religious expression are discouraged or banned outright. For example, in North Korea, the state ideology is known as Juche, which is not a traditional religion with supernatural elements. The practice of Buddhism or Christianity in North Korea must necessarily be circumspect. In other places, religion has reached the status of being nominal (in name only) in which people identify with a religion, but the practices of that religion have little impact on their daily lives.
State religions are religions that are recognized as the official religion of a country. In some places with a state religion, such as Denmark or the United Kingdom, the official status of one religion has little effect either on the practice of other religions or on the society at large. In places like Saudi Arabia or Pakistan, however, the official religion is closely connected to the power of the state, most obviously in the form of blasphemy laws which allow for state penalties (including death) for violations of religious statutes.
Maps such as this one can be very misleading in the sense that they present homogeneous, religious landscapes by country ( Figure 6.2 ). This is, of course, untrue and it represents one of the problems with mapping anythingthe level of aggregation. As any spatial phenomenon is aggregated into larger and larger groups, the details of those groups are often lost. An example of the importance of scale is seen in the following graphic. Although Mormons make up 88 percent of Utah County, Utah, they are only 61% of the state of Utah, and a mere 1.6% of the United States.
6.1.1 Scale and Predominant Religion
6.1.2 Religions in History
The current religious map of the world is best thought of as a snapshot. The religious landscape has been continuously changing throughout human history and will continue to change in the future. New religions are founded and old ones die out.
New religions are often made using pieces of older religions; Christianity and Islam deriving in part from Judaism and Buddhism deriving from Hinduism are not aberrations, but instead are examples of a common occurrence.
Within the relatively recent past, it has been possible to watch the creation and diffusion of several new religions just within the United States. Mormonism, the Jehovah’s Witnesses, Seventh Day Adventism, and Scientology are all religions that were founded in the U.S. in the relatively recent past.
6.1.3 The Religious Contribution to Culture and Identity
Religions are not isolated social phenomena. They exist within a cultural complex that nurtures and sustains them, or conversely, demeans and undermines them. Religion can be closely bound to other elements of identity -language, or nationality. In many societies, the boundaries between religion and social life, family structure, and law and politics are nonexistent. Religion in those places is the center of all life and everything else revolves around religious concepts. A place that is purely governed by a religious structure is known as a theocracy . There are very few of these in the modern world, although many states have strong religious influence. Many modern societies have built barriers between religious influence and political life. These places are called secular and are much more common in the developed world.
Religion, along with ethnicity and language, are very often core components of an individual’s identity. It can define the way a person sees the world, what clothing is appropriate, gender roles, employment, and even your position within the greater society. As such, it has tremendous cultural influence, and this influence is visible in the landscape.
6.1.4 Esthetics and religion
Religion has a motivating factor that few other social phenomena can match. When people are doing something for God, they generally have fewer limits than in other spheres of life. One of the ways this lack of limits is manifest in the landscape is through religious architecture.
Sacred spaces can be religious structures, but they can also be historic battlefields, cemeteries, mountains, or rivers. Anything that humans use to generate a sense of the divine can be considered a sacred space. Sacred spaces have expectations of behavior. In some places, it is still possible to claim sanctuary in a sacred place. The small altars that mark roadside fatalities in the United States can be considered sacred spaces, as could a closet that is used as a prayer room.
Elements of culture may be manifest in different types of churches, temples on the landscape, as well as clothing, the food grown, and small home devotionals. Another way that religion manifests in the visible sphere is through codes of acceptable dress as well as acceptable public behaviors.
A less obvious way that religions may influence the landscape is through religious influences on dietary choices. Food production can be influenced by religions. Many religions have doctrine regarding what is acceptable food, and what is not. Halal, Kosher and Ital are all representative of food restrictions. Religions that prohibit the consumption of pork will probably not have swine farms. Cattle wander through the countryside in India, since they are religiously protected from harm. Another effect that religion has on the landscape is the effect of pilgrimage . Many religions have an activity that requires gathering at a particular place. Probably the best-known pilgrimage is the Hajj of Islam, but this isn’t actually the largest in the world. That would be the Hindu Kumbh Mela. The Camino Santiago is a well-known Christian pilgrimage that ends in Santiago de Compostela, Spain. Pilgrimage is not just visible through the pilgrims, but in the entire infrastructure that develops to support the pilgrims.


5.12: Chapter Summary
Blood pressure measurement is important because it provides objective data about the client’s health and illness state. Changes in blood pressure act as a cue for healthcare providers’ diagnostic reasoning. Blood pressure fluctuates with internal and external factors. Therefore, it is important to take more than one measurement before making clinical decisions.
It is always important to ensure correct techniques when taking blood pressure.
In determining the relevance of the blood pressure reading, the healthcare provider considers the client’s baseline blood pressure, previous readings, and health status. The blood pressure reading is always taken in conjunction with a pulse. Diagnostic reasoning takes into account blood pressure, pulse, and subjective and objective client data.


<IP_ADDRESS>: Non-Obvious Relationship
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Digital technologies have given us many new capabilities that simplify and expedite the collection of personal information. Every time we come into contact with digital technologies, information about us is being made available. From our location to our web-surfing habits, our criminal record to our credit report, we are constantly being monitored. This information can then be aggregated to create profiles of each and every one of us. While much of the information collected was available in the past, collecting it and combining it took time and effort. Today, detailed information about us is available for purchase from different companies. Even information not categorized as PII can be aggregated in such a way that an individual can be identified.
This process of collecting large quantities of a variety of information and then combining it to create profiles of individuals is known as non-obvious relationship awareness, or NORA. First commercialized by big casinos looking to find cheaters, NORA is used by both government agencies and private organizations, and it is big business.
In some settings, NORA can bring many benefits, such as in law enforcement. By being able to identify potential criminals more quickly, crimes can be solved more quickly or even prevented before they happen. But these advantages come at a price: our privacy.


3.1: Mathematics and Proofs
The primary activity of research mathematicians is proving mathematical claims. Depending on the depth of the claim, the relationship of the claim to other mathematical claims, and various other factors, a mathematical statement that has been proved is generally called a theorem, proposition, corollary or lemma. A mathematical statement that has not been proved, but that is expected to be true, is commonly called a conjecture. A statement that is accepted as a starting point for arguments without being proved is called an axiom.
Some mathematical results are so fundamental, deep, difficult, surprising or otherwise noteworthy that they are named. Part of your initiation as a member of the community of mathematicians is becoming familiar with some of these named statements - and we shall prove a few of them in this book.
It is likely that most of the mathematics you have studied has been the application of theorems to deriving solutions of relatively concrete problems. Here we begin learning how to prove theorems. Most students find the transition from computational mathematics to mathematical proofs very challenging.
What is a mathematical proof?
The nature of a mathematical proof depends on the context. There is a formal notion of a mathematical proof:
A finite sequence of formal mathematical statements such that each statement either
-
is an axiom or assumption, or
-
follows by formal rules of logical deduction from previous statements in the sequence.
Most mathematicians do not think of mathematical proofs as formal mathematical proofs, and virtually no mathematician writes formal mathematical proofs. This is because a formal proof is a hopelessly cumbersome thing, and is generally outside the scope of human capability, even for the most elementary mathematical statements. Rather, mathematicians write proofs that are sequences of statements in a combination of natural language and formal mathematical symbols (interspersed with diagrams, questions, references and other devices that are intended to assist the reader in understanding the proof) that can be thought of as representing a purely formal argument. A good practical definition of a mathematical proof is:
An argument in favor of a mathematical statement that will convince the preponderance of knowledgeable mathematicians of the truth of the mathematical statement.
This definition is somewhat imprecise, and mathematicians can disagree on whether an argument is a proof, particularly for extremely difficult or deep arguments. However, for virtually all mathematical arguments, after some time for careful consideration, the mathematical community reaches a unanimous consensus on whether it is a proof.
The notion of a mathematical proof for the student is similar to the general idea of a mathematical proof. The differences are due to the type of statement that the student is proving, and the reasons for requesting that the student prove the statement. The statements that you will be proving are known to professional mathematicians or can be proved with relatively little effort by your instructors. Clearly the statements you will be proving require different conditions for a satisfactory proof than those stated above for the professional mathematician. Let’s define a successful argument by the student as follows: An argument for a mathematical statement that
-
the instructor can understand
-
the instructor cannot refute
-
uses only assumptions that the instructor considers admissible.
Note that refuting an argument is not the same as refuting the original claim. The sentence "The square of every real number is nonnegative because all real numbers are non-negative." is a false proof of a true statement. The sentence "The square of every real number is non-negative because all triangles have three sides." fails the first test: while both statements are true, your instructor will not see how the first follows from the second.
In this book, the solutions to the problems will be an exposition in natural language enhanced by mathematical expressions. The student is expected to learn the conventions of mathematical grammar and argument, and use them. Like most conventions, these are often determined by tradition or precedent. It can be quite difficult, initially, to determine whether your mathematical exposition meets the standards of your instructor. Practice, with feedback from a reader experienced in reading mathematics, is the best way to develop good proof-writing skills. Remember, readers of mathematics are quite impatient with trying to decipher what the author means to say - mathematics is sufficiently challenging when the author writes precisely what he or she intends. Most of the burden of communication is on the author of a mathematical proof, not the reader. A proof can be logically correct, but so difficult to follow that it is unacceptable to your instructor.
Why proofs?
Why are proofs the primary medium of mathematics? Mathematicians depend on proofs for certainty and explanation. Once a proof is accepted by the mathematical community, it is virtually unheard of that the result is subsequently refuted. This was not always the case: in the \(19^{\text {th }}\) century there were serious disputes as to whether results had really been proved or not (see Section \(5.3\) for an example, and the book [4] for a very extensive treatment of the development of rigor in mathematical reasoning). This led to our modern notion of a "rigorous" mathematical argument. While one might argue that it is possible that in the \(21^{\text {st }}\) century a new standard of rigor will reject what we currently consider to be proofs, our current ideas have been stable for over a century, and most mathematicians (including the authors of this book) do not expect that there will be a philosophical shift.
For very complicated results, writing a detailed proof helps the author convince himself or herself of the truth of the claim. After a mathematician has hit upon the key idea behind an argument, there is a lot of hard work left developing the details of the argument. Many promising ideas fail as the author attempts to write a detailed argument based on the idea. Finally, proofs often provide a deeper insight into the result and the mathematical objects that are the subject of the proof. Indeed, even very clever proofs which fail to provide mathematical insights are held in lower regard, by some, than arguments that elucidate the topic.
 It is supposed that the existence of an informal mathematical proof is overwhelming evidence for the existence of a formal mathematical proof. If it is not clear that the informal proof could conceivably be interpreted into a formal argument, it is doubtful that the informal argument will be accepted by the mathematical community. Consequently, mathematical arguments have a transparent underlying logical structure.
For this reason we shall begin our discussion of mathematical proofs with a brief discussion of propositional logic. Despite its abstractness, the topic is straightforward, and most of the claims of this section may be confirmed with some careful, patient thinking.


15.25E: Chikungunya Fever
- Discuss the causes and symptoms associated with chikungunya virus (CHIKV)
Chikungunya Virus (CHIKV)
Chikungunya (in the Makonde language “that which bends up”) virus (CHIKV) is an insect-borne virus of the genus Alphavirus , that is transmitted to humans by virus-carrying Aedes (Ae) mosquitoes. Both Ae. aegypti and Ae. albopictus have been implicated in large outbreaks of CHIKV.
CHIKV infection causes an illness with symptoms similar to dengue fever, with an acute febrile phase of the illness lasting only two to five days, followed by a prolonged arthralgic disease that affects the joints of the extremities. The pain associated with CHIKV infection of the joints persists for weeks or months or, in some cases, years. CHIKV is indigenous to tropical Africa and Asia, where it is transmitted to humans by the bite of infected mosquitoes.
SIGNS AND SYMPTOMS
The incubation period of chikungunya disease ranges from one to 12 days, but usually two to three. Its symptoms include a high fever up to 40 °C (104 °F), a petechial or maculopapular rash of the trunk and occasionally the limbs, and arthralgia or arthritis affecting multiple joints. Other nonspecific symptoms can include headache, conjunctivitis, slight photophobia and partial loss of taste. Typically, the fever lasts for two days and then ends abruptly. However, other symptoms—namely joint pain, intense headache, insomnia and an extreme degree of prostration—last for a variable period; usually for about five to seven days. Patients have complained of joint pains for much longer time periods; some for as long as two years, depending on their age.
DIAGNOSIS
Common laboratory tests for chikungunya include RT-PCR, virus isolation, and serological tests.
Virus isolation provides the most definitive diagnosis, but takes one to two weeks for completion and must be carried out in biosafety level-3 laboratories. The technique involves exposing specific cell lines to samples from whole blood and identifying chikungunya virus-specific responses. RT-PCR using nested-primer pairs is used to amplify several chikungunya-specific genes from whole blood. Results can be determined in one to two days. Serological diagnosis requires a larger amount of blood than the other methods, and uses an ELISA assay to measure chikungunya-specific IgM levels. Results require two to three days.
TREATMENT
There are no specific drugs to cure the disease. Treatment is directed primarily at relieving the symptoms, especially the joint pain. There is no commercial chikungunya vaccine.
CHIKUNGUNYA OUTBREAKS
Chikungunya occurs in Africa, Asia, and the Indian subcontinent. Human infections in Africa have been at relatively low levels for a number of years, but in 1999-2000 there was a large outbreak in the Democratic Republic of the Congo, and in 2007 there was an outbreak in Gabon. Starting in February 2005, a major outbreak occurred in islands of the Indian Ocean. A large number of imported cases in Europe were associated with this outbreak, mostly in 2006 when this epidemic was at its peak.
A large outbreak of chikungunya in India occurred in 2006 and 2007. Several other countries in South-East Asia were also affected. In 2007 transmission was reported for the first time in Europe, in a localized outbreak in north-eastern Italy.
Key Points
- The chikungunya virus is transmitted from human to human by the bites of infected female mosquitoes; the Aedes aegypti and Aedes albopictus mosquitoes.
- The disease occurs in Africa, Asia and the Indian subcontinent. In recent decades mosquito vectors of CHIKV have spread to Europe and the Americas.
- There is no cure for CHIKV. Treatment is focused on relieving the symptoms which include fever and severe joint pain, muscle pain, headache, nausea, fatigue and rash.
Key Terms
- dengue : An acute febrile disease of the tropics caused by a flavivirus, transmitted by mosquitoes, and characterized by high fever, rash, headache, and severe muscle and joint pain.
- Aedes : Aedes is a genus of mosquito originally found in tropical and subtropical zones, but now found on all continents excluding Antarctica.
- chikungunya : A viral fever caused by an alphavirus spread by mosquito bites.


5.2.1: Changing Temperatures
Lesson
Let's add signed numbers.
Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Arrows
Which pair of arrows doesn't belong?
1.
2.
3.
4.
Exercise \(\PageIndex{2}\): Warmer and Colder
1. Complete the table and draw a number line diagram for each situation.
| start (\(^{\circ}\)C) | change (\(^{\circ}\)C) | final (\(^{\circ}\)C) | addition equation | |
|---|---|---|---|---|
| a | \(+40\) | \(10\) degrees warmer | \(+50\) | \(40+10=50\) |
| b | \(+40\) | \(5\) degrees colder | ||
| c | \(+40\) | \(30\) degrees colder | ||
| d | \(+40\) | \(40\) degrees colder | ||
| e | \(+40\) | \(50\) degrees colder |
a.
b.
c.
d.
e.
2. Complete the table and draw a number line diagram for each situation.
| start (\(^{\circ}\)C) | change (\(^{\circ}\)C) | final (\(^{\circ}\)C) | addition equation | |
|---|---|---|---|---|
| a | \(-20\) | \(30\) degrees warmer | ||
| b | \(-20\) | \(35\) degrees warmer | ||
| c | \(-20\) | \(15\) degrees warmer | ||
| d | \(-20\) | \(15\) degrees colder |
a.
b.
c.
d.
Are you ready for more?
For the numbers \(a\) and \(b\) represented in the figure, which expression is equal to \(|a+b|\)?
\(|a|+|b|\qquad |a|-|b| \qquad |b|-|a|\)
Exercise \(\PageIndex{3}\): Winter Temperatures
One winter day, the temperature in Houston is \(8^{\circ}\) Celsius. Find the temperatures in these other cities. Explain or show your reasoning.
- In Orlando, it is \(10^{\circ}\) warmer than it is in Houston.
- In Salt Lake City, it is \(8^{\circ}\) colder than it is in Houston.
- In Minneapolis, it is \(20^{\circ}\) colder than it is in Houston.
- In Fairbanks, it is \(10^{\circ}\) colder than it is in Minneapolis.
- Use the thermometer applet to verify your answers and explore your own scenarios.
Summary
If it is \(42^{\circ}\) outside and the temperature increases by \(7^{\circ}\), then we can add the initial temperature and the change in temperature to find the final temperature.
\(42+7=49\)
If the temperature decreases by \(7^{\circ}\), we can either subtract \(42-7\) to find the final temperature, or we can think of the change as \(-7^{\circ}\). Again, we can add to find the final temperature.
\(42+(-7)=35\)
In general, we can represent a change in temperature with a positive number if it increases and a negative number if it decreases. Then we can find the final temperature by adding the initial temperature and the change. If it is \(3^{\circ}\) and the temperature decreases by \(7^{\circ}\), then we can add to find the final temperature.
\(3+(-7)=-4\)
We can represent signed numbers with arrows on a number line. We can represent positive numbers with arrows that start at 0 and points to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.
We can represent negative numbers with arrows that start at 0 and point to the left. For example, this arrow represents -4 because it is 4 units long and it points to the left.
To represent addition, we put the arrows “tip to tail.” So this diagram represents \(3+5\):
And this represents \(3+(-5)\):
Practice
Exercise \(\PageIndex{4}\)
- The temperature is \(-2^{\circ}\text{C}\). If the temperature rises by \(15^{\circ}\text{C}\), what is the new temperature?
- At midnight the temperature is \(-6^{\circ}\text{C}\). At midday the temperature is \(9^{\circ}\text{C}\). By how much did the temperature rise?
Exercise \(\PageIndex{5}\)
Draw a diagram to represent each of these situations. Then write an addition expression that represents the final temperature.
- The temperature was \(80^{\circ}\text{F}\) and then fell \(20^{\circ}\text{F}\).
- The temperature was \(-13^{\circ}\text{F}\) and then rose \(9^{\circ}\text{F}\).
- The temperature was \(-5^{\circ}\text{F}\) and then fell \(8^{\circ}\text{F}\).
Exercise \(\PageIndex{6}\)
Complete each statement with a number that makes the statement true.
- _____ < \(7^{\circ}\text{C}\)
- _____ < \(-3^{\circ}\text{C}\)
- \(-0.8^{\circ}\text{C}\) < _____ < \(-0.1^{\circ}\text{C}\)
- _____ > \(-2^{\circ}\text{C}\)
(From Unit 5.1.1)
Exercise \(\PageIndex{7}\)
Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would be the constant of proportionality?
-
The number of wheels on a group of buses.
number of buses number of wheels wheels per bus \(5\) \(30\) \(8\) \(48\) \(10\) \(60\) \(15\) \(90\) Table \(\PageIndex{3}\) -
The number of wheels on a train.
number of train cars number of wheels wheels per train car \(20\) \(184\) \(30\) \(264\) \(40\) \(344\) \(50\) \(424\) Table \(\PageIndex{4}\)
(From Unit 2.3.1)
Exercise \(\PageIndex{8}\)
Noah was assigned to make 64 cookies for the bake sale. He made 125% of that number. 90% of the cookies he made were sold. How many of Noah's cookies were left after the bake sale?
(From Unit 4.2.2)


11.1: Voting Methods
A preference ballot is a ballot in which voters rank their choices in order.
Here are four methods to determine which candidate is the winner of an election.
In this method, the candidate with the most first place votes is declared the winner.
This method is sometimes mistakenly called the majority method, or “majority rules”, but it is not necessary for a choice to have gained a majority of votes to win. A majority is over 50%; it is possible for a winner to have a plurality without having a majority.
In our election from previous pages, we had the preference table:
\(\begin{array}{|l|l|l|l|l|}
\hline & 1 & 3 & 3 & 3 \\
\hline 1^{\text {st }} \text { choice } & \text { A } & \text { A } & \text { O } & \text { H } \\
\hline 2^{\text {nd }} \text { choice } & \text { O } & \text { H } & \text { H } & \text { A } \\
\hline 3^{\text {rd }} \text { choice } & \text { H } & \text { O } & \text { A } & \text { O } \\
\hline
\end{array}\)
Solution
For the plurality method, we only care about the first choice options. Totaling them up:
Anaheim: 1+3 = 4 first-choice votes
Orlando: 3 first-choice votes
Hawaii: 3 first-choice votes
Anaheim is the winner using the plurality voting method.
Notice that Anaheim won with 4 out of 10 votes, 40% of the votes, which is a plurality of the votes, but not a majority.
Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)[1]. The voting schedule is shown below. Which candidate wins under the Plurality Method?
\(\begin{array}{|l|l|l|l|l|}
\hline & 44 & 14 & 20 & 70 & 22 & 80 & 39 \\
\hline 1^{\text {st }} \text { choice } & \text { G } & \text { G } & \text { G } & \text { M } & \text { M } & \text { B } & \text { B } \\
\hline 2^{\text {nd }} \text { choice } & \text { M } & \text { B } & \text { } & \text { G } & \text { B } & \text { M } & \text { } \\
\hline 3^{\text {rd }} \text { choice } & \text { B } & \text { M } & \text { } & \text { B } & \text { G } & \text { G } & \text { } \\
\hline
\end{array}\)
Note: In the third column and last column, those voters only recorded a first-place vote, so we don’t know who their second and third choices would have been.
- Answer
-
Using plurality method:
G gets \(44+14+20 = 78\) first-choice votes
M gets \(70+22 = 92\) first-choice votes
B gets \(80+39 = 119\) first-choice votes
Bunney (B) wins under Plurality Method.
Plurality with Elimination is a modification of the plurality method.
In this method, the candidate with the least first-place votes is then eliminated from the election, and any votes for that candidate are redistributed to the voters’ next choice. This continues until a candidate has a majority (over 50%).
This is similar to the idea of holding runoff elections, but since every voter’s order of preference is recorded on the ballot, the runoff can be computed without requiring a second costly election.
This voting method is used in several political elections around the world, including election of members of the Australian House of Representatives, and was used for county positions in Pierce County, Washington until it was eliminated by voters in 2009. A version of IRV is used by the International Olympic Committee to select host nations.
Consider the preference schedule below, in which a company’s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity.
Initial votes
\(\begin{array}{|l|l|l|l|l|l|l|}
\hline & 3 & 4 & 4 & 6 & 2 & 1 \\
\hline 1^{\text {st choice }} & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{B} & \mathrm{E} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{C} & \mathrm{A} & \mathrm{D} & \mathrm{C} & \mathrm{E} & \mathrm{A} \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{A} & \mathrm{D} & \mathrm{C} & \mathrm{A} & \mathrm{A} & \mathrm{D} \\
\hline 4^{\text {th }} \text { choice } & \mathrm{D} & \mathrm{B} & \mathrm{A} & \mathrm{E} & \mathrm{C} & \mathrm{B} \\
\hline 5^{\text {th }} \text { choice } & \mathrm{E} & \mathrm{E} & \mathrm{E} & \mathrm{B} & \mathrm{D} & \mathrm{C} \\
\hline
\end{array}\)
If this was a plurality election, note that B would be the winner with 9 first-choice votes, compared to 6 for D, 4 for C, and 1 for E.
There are total of 3+4+4+6+2+1 = 20 votes. A majority would be 11 votes. No one yet has a majority, so we proceed to elimination rounds.
Solution
Round 1 : We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice
\(\begin{array}{|l|l|l|l|l|l|l|}
\hline & 3 & 4 & 4 & 6 & 2 & 1 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{B} & \mathrm{E} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{C} & & \mathrm{D} & \mathrm{C} & \mathrm{E} & \\
\hline 3^{\text {rd }} \text { choice } & & \mathrm{D} & \mathrm{C} & & & \mathrm{D} \\
\hline 4^{\text {th }} \text { choice } & \mathrm{D} & \mathrm{B} & & \mathrm{E} & \mathrm{C} & \mathrm{B} \\
\hline 5^{\text {th }} \text { choice } & \mathrm{E} & \mathrm{E} & \mathrm{E} & \mathrm{B} & \mathrm{D} & \mathrm{C} \\
\hline
\end{array}\)
We then shift everyone’s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.
\(\begin{array}{|l|l|l|l|l|l|l|}
\hline & 3 & 4 & 4 & 6 & 2 & 1 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{B} & \mathrm{E} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{C} & \mathrm{D} & \mathrm{D} & \mathrm{C} & \mathrm{E} & \mathrm{D} \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{D} & \mathrm{B} & \mathrm{C} & \mathrm{E} & \mathrm{C} & \mathrm{B} \\
\hline 4^{\text {th }} \text { choice } & \mathrm{E} & \mathrm{E} & \mathrm{E} & \mathrm{B} & \mathrm{D} & \mathrm{C} \\
\hline
\end{array}\)
Round 2 : We make our second elimination. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone’s options to fill the gaps.
\(\begin{array}{|l|l|l|l|l|l|l|}
\hline & 3 & 4 & 4 & 6 & 2 & 1 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{B} & \mathrm{D} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{C} & \mathrm{D} & \mathrm{D} & \mathrm{C} & \mathrm{C} & \mathrm{B} \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{D} & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{C} \\
\hline
\end{array}\)
Notice that the first and fifth columns have the same preferences now, we can condense those down to one column.
\(\begin{array}{|l|l|l|l|l|l|}
\hline & 5 & 4 & 4 & 6 & 1 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{D} & \mathrm{D} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{C} & \mathrm{D} & \mathrm{D} & \mathrm{C} & \mathrm{B} \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{D} & \mathrm{B} & \mathrm{C} & \mathrm{B} & \mathrm{C} \\
\hline
\end{array}\)
Now B has 9 first-choice votes, C has 4 votes, and D has 7 votes. Still no majority, so we eliminate again.
Round 3 : We make our third elimination. C has the fewest votes.
\(\begin{array}{|l|l|l|l|l|l|}
\hline & 5 & 4 & 4 & 6 & 1 \\
\hline 1^{\text {st }} \text { choice } & \text { B } & \text { D } & \text { B } & \text { D } & \text { D } \\
\hline 2^{\text {nd }} \text { choice } & \text { D } & \text { B } & \text { D } & \text { B } & \text { B } \\
\hline
\end{array}\)
Condensing this down:
\(\begin{array}{|l|l|l|}
\hline & 9 & 11 \\
\hline 1^{\text {st }} \text { choice } & \text { B } & \text { D } \\
\hline 2^{\text {nd }} \text { choice } & \text { D } & \text { B } \\
\hline
\end{array}\)
D has now gained a majority, and is declared the winner under Plurality with Elimination.
Consider again the election from Try it Now 1. Find the winner using Plurality with Elimination.
\(\begin{array}{|l|l|l|l|l|l|l|l|}
\hline & 44 & 14 & 20 & 70 & 22 & 80 & 39 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{G} & \mathrm{G} & \mathrm{G} & \mathrm{M} & \mathrm{M} & \mathrm{B} & \mathrm{B} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{M} & \mathrm{B} & & \mathrm{G} & \mathrm{B} & \mathrm{M} & \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{B} & \mathrm{M} & & \mathrm{B} & \mathrm{G} & \mathrm{G} & \\
\hline
\end{array}\)
- Answer
-
G has the fewest first-choice votes, so is eliminated first. The 20 voters who did not list a second choice do not get transferred - they simply get eliminated
\(\begin{array}{|l|l|l|}
\hline & 136 & 133 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{M} & \mathrm{B} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{B} & \mathrm{M} \\
\hline
\end{array}\)McCarthy (M) now has a majority, and is declared the winner.
Borda Count is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.
In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner.
A group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia. Their approximate locations on a map are shown to the right.
The votes for where to hold the conference were:
\(\begin{array}{|l|l|l|l|l|}
\hline & 51 & 25 & 10 & 14 \\
\hline 1^{\text {st }} \text { choice } & \text { Seattle } & \text { Tacoma } & \text { Puyallup } & \text { Olympia } \\
\hline 2^{\text {nd }} \text { choice } & \text { Tacoma } & \text { Puyallup } & \text { Tacoma } & \text { Tacoma } \\
\hline 3^{\text {rd }} \text { choice } & \text { Olympia } & \text { Olympia } & \text { Olympia } & \text { Puyallup } \\
\hline 4^{\text {th }} \text { choice } & \text { Puyallup } & \text { Seattle } & \text { Seattle } & \text { Seattle } \\
\hline
\end{array}\)
Solution
In each of the 51 ballots ranking Seattle first, Puyallup will be given 1 point, Olympia 2 points, Tacoma 3 points, and Seattle 4 points. Multiplying the points per vote times the number of votes allows us to calculate points awarded:
\(\begin{array}{|l|l|l|l|l|}
\hline & 51 & 25 & 10 & 14 \\
\hline 1^{\text {st choice }} & \text { Seattle } & \text { Tacoma } & \text { Puyallup } & \text { Olympia } \\
4 \text { points } & 4 \cdot 51=204 & 4 \cdot 25= 100 & 4 \cdot 10=40 & 4 \cdot 14=56 \\
\hline 2^{\text {nd choice }} & \text { Tacoma } & \text { Puyallup } & \text { Tacoma } & \text { Tacoma } \\
3 \text { points } & 3 \cdot 51=153 & 3 \cdot 25=75 & 3 \cdot 10=30 & 3 \cdot 14=42 \\
\hline 3^{\text {rd }} \text { choice } & \text { Olympia } & \text { Olympia } & \text { Olympia } & \text { Puyallup } \\
2 \text { points } & 2 \cdot 51=102 & 2 \cdot 25=50 & 2 \cdot 10=20 & 2 \cdot 14=28 \\
\hline 4^{\text {th }} \text { choice } & \text { Puyallup } & \text { Seattle } & \text { Seattle } & \text { Seattle } \\
1 \text { point } & 1 \cdot 51=51 & 1 \cdot 25=25 & 1 \cdot 10=10 & 1 \cdot 14=14 \\
\hline
\end{array}\)
Adding up the points:
- Seattle: \(204 + 25 + 10 + 14 = 253\) points
- Tacoma: \(153 + 100 + 30 + 42 = 325\) points
- Puyallup: \(51 + 75 + 40 + 28 = 194\) points
- Olympia: \(102 + 50 + 20 + 56 = 228\) points
Under the Borda Count method, Tacoma is the winner of this vote.
Consider again the election from Try it Now 1. Find the winner using Borda Count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate 1 point, the points they would normally get for last place.
\(\begin{array}{|l|l|l|l|l|l|l|l|}
\hline & 44 & 14 & 20 & 70 & 22 & 80 & 39 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{G} & \mathrm{G} & \mathrm{G} & \mathrm{M} & \mathrm{M} & \mathrm{B} & \mathrm{B} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{M} & \mathrm{B} & & \mathrm{G} & \mathrm{B} & \mathrm{M} & \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{B} & \mathrm{M} & & \mathrm{B} & \mathrm{G} & \mathrm{G} & \\
\hline
\end{array}\)
- Answer
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Using Borda Count:
We give 1 point for 3 rd place, 2 points for 2 nd place, and 3 points for 1 st place.
\(\begin{array}{|l|l|l|l|l|l|l|l|}
\hline & 44 & 14 & 20 & 70 & 22 & 80 & 39 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{G} & \mathrm{G} & \mathrm{G} & \mathrm{M} & \mathrm{M} & \mathrm{B} & \mathrm{B} \\
& 132 \mathrm{pt} & 42 \mathrm{pt} & 60 \mathrm{pt} & 210 \mathrm{pt} & 66 \mathrm{pt} & 240 \mathrm{pt} & 117 \mathrm{pt} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{M} & \mathrm{B} & & \mathrm{G} & \mathrm{B} & \mathrm{M} & \\
& 88 \mathrm{pt} & 28 \mathrm{pt} & & 140 \mathrm{pt} & 44 \mathrm{pt} & 160 \mathrm{pt} & \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{B} & \mathrm{M} & \mathrm{M} 20 \mathrm{pt} & \mathrm{B} & \mathrm{G} & \mathrm{G} & \mathrm{M} 39 \mathrm{pt} \\
& 44 \mathrm{pt} & 14 \mathrm{pt} & \mathrm{B} 20 \mathrm{pt} & 70 \mathrm{pt} & 22 \mathrm{pt} & 80 \mathrm{pt} & \mathrm{G} 39 \mathrm{pt} \\
\hline
\end{array}\)G: \(132+42+60+140+22+80+39 = 515\) pts
M: \(88+14+20+210+66+160+39 = 597\) pts
B: \(44+28+20+70+44+240+117 = 563\) pts
McCarthy (M) would be the winner using Borda Count.
The fourth method is the Pairwise Comparison Method.
In this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded \(\frac{1}{2}\) point. After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.
Consider our vacation group example from the beginning of the chapter. Determine the winner using Pairwise Comparison Method.
\(\begin{array}{|l|l|l|l|l|}
\hline & 1 & 3 & 3 & 3 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{A} & \mathrm{A} & \mathrm{O} & \mathrm{H} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{O} & \mathrm{H} & \mathrm{H} & \mathrm{A} \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{H} & \mathrm{O} & \mathrm{A} & \mathrm{O} \\
\hline
\end{array}\)
Solution
We need to look at each pair of choices, and see which choice would win in a one-to-one comparison. For example, comparing Hawaii vs Orlando, we see that 6 voters, those shaded below in the first table below, would prefer Hawaii to Orlando. Note that Hawaii doesn’t have to be the voter’s first choice – we’re imagining that Anaheim wasn’t an option. If it helps, you can imagine removing Anaheim, as in the second table below.
Based on this, in the comparison of Hawaii vs Orlando, Hawaii wins, and receives 1 point.
Comparing Anaheim to Orlando, the 1 voter in the first column clearly prefers Anaheim, as do the 3 voters in the second column. The 3 voters in the third column clearly prefer Orlando. The 3 voters in the last column prefer Hawaii as their first choice, but if they had to choose between Anaheim and Orlando, they'd choose Anaheim, their second choice overall. So, altogether 1+3+3=7 voters prefer Anaheim over Orlando, and 3 prefer Orlando over Anaheim. So, comparing Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point.
All together,
\(\begin{array} {ll} {\text{Hawaii vs Orlando: }} & {6\text{ votes to }4\text{ votes: Hawaii gets }1\text{ point}} \\ {\text{Anaheim vs Orlando:}} & {7\text{ votes to }3\text{ votes: Anaheim gets }1\text{ point}} \\ {\text{Hawaii vs Anaheim: }} & {6\text{ votes to }4\text{ votes: Hawaii gets }1\text{ point}} \end{array}\)
Hawaii is the winner under Pairwise Comparison Method, having earned the most points.
Consider the advertising group’s vote we explored earlier. Determine the winner using Pairwise Comparison Method.
\(\begin{array}{|l|l|l|l|l|l|l|}
\hline & 3 & 4 & 4 & 6 & 2 & 1 \\
\hline 1^{\text {st }} \text { choice } & \text { B } & \text { C } & \text { B } & \text { D } & \text { B } & \text { E } \\
\hline 2^{\text {nd }} \text { choice } & \text { C } & \text { A } & \text { D } & \text { C } & \text { E } & \text { A } \\
\hline 3^{\text {rd }} \text { choice } & \text { A } & \text { D } & \text { C } & \text { A } & \text { A } & \text { D } \\
\hline 4^{\text {th }} \text { choice } & \text { D } & \text { B } & \text { A } & \text { E } & \text { C } & \text { B } \\
\hline 5^{\text {th }} \text { choice } & \text { E } & \text { E } & \text { E } & \text { B } & \text { D } & \text { C } \\
\hline
\end{array}\)
Solution
With 5 candidates, there are 10 comparisons to make:
\( \begin{array} {ll} {\text{A vs B: }11\text{ votes to }9\text{ votes}} & {\text{A gets }1\text{ point}} \\ {\text{A vs C: }3\text{ votes to }17\text{ votes}} & {\text{C gets }1\text{ point}} \\ {\text{A vs D: }10\text{ votes to }10\text{ votes }} & {\text{A gets }\frac{1}{2}\text{ point, D gets }\frac{1}{2}\text{ point}} \\ {\text{A vs E: }17\text{ votes to }3\text{ votes}} & {\text{A gets }1\text{ point}} \\ {\text{B vs C: }10\text{ votes to }10\text{ votes}} & {\text{B gets }\frac{1}{2}\text{ point, C gets }\frac{1}{2}\text{ point}} \\ {\text{B vs D: }9\text{ votes to }11\text{ votes}} & {\text{D gets }1\text{ point}} \\ {\text{B vs E: }13\text{ votes to }7\text{ votes}} & {\text{B gets }1\text{ point}} \\ {\text{C vs D: }9\text{ votes to }11\text{ votes}} & {\text{D gets }1\text{ point}} \\ {\text{C vs E: }17\text{ votes to }3\text{ votes}} & {\text{C gets }1\text{ point}} \\ {\text{D vs E: }17\text{ votes to }3\text{ votes}} & {\text{D gets }1\text{ point}} \end{array}\)
Totaling these up:
A gets \(2\frac{1}{2}\) points
B gets \(1\frac{1}{2}\) points
C gets \(2\frac{1}{2}\) points
D gets \(3\frac{1}{2}\) points
E gets \(0\) points
Using Pairwise Comparison Method, we declare D as the winner.
Consider again the election from Try it Now 1. Find the winner using Pairwise Comparison method. Since we have some incomplete preference ballots, we’ll have to adjust. For example, when comparing M to B, we’ll ignore the 20 votes in the third column which do not rank either candidate.
\(\begin{array}{|l|l|l|l|l|l|l|l|}
\hline & 44 & 14 & 20 & 70 & 22 & 80 & 39 \\
\hline 1^{\text {st }} \text { choice } & \mathrm{G} & \mathrm{G} & \mathrm{G} & \mathrm{M} & \mathrm{M} & \mathrm{B} & \mathrm{B} \\
\hline 2^{\text {nd }} \text { choice } & \mathrm{M} & \mathrm{B} & & \mathrm{G} & \mathrm{B} & \mathrm{M} & \\
\hline 3^{\text {rd }} \text { choice } & \mathrm{B} & \mathrm{M} & & \mathrm{B} & \mathrm{G} & \mathrm{G} & \\
\hline
\end{array}\)
- Answer
-
Using Pairwise Comparison Method:
Looking back at our work from Try it Now #2, we see
G vs M: \(44+14+20 = 78\) prefer G, \(70+22+80=172\) prefer M: M preferred – 1 point
G vs B: \(44+14+20+70=148\) prefer G, \(22+80+39 = 141\) prefer B: G preferred – 1 point
M vs B: \(44+70+22=136\) prefer M, \(14+80+39=133\) prefer B: M preferred – 1 point
M earns 2 points; G earns 1 point. M wins under Pairwise Comparison Method.
[1] This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See www.co.pierce.wa.us/xml/abtus...ec/summary.pdf


24.7: Relativistic momentum and energy
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In this section, we show how to define momentum and energy in a way that is consistent with the postulates of Special Relativity. We expect that, since time and space depend on the frame of reference of the observer, so too will the momentum and the energy of an object. Consider an object of mass \(m_0\) , moving in a frame of reference, \(S\) , with velocity, \(\vec u\) (we reserve \(\vec v\) to represent the speed between two inertial frames of reference), in the \(x\) direction. At some time, \(t\) , the object will be at position, \(x\) , along the \(x\) axis. We define the relativistic momentum as:
\[\begin{aligned} p = m_0\frac{dx}{dt'}\end{aligned}\]
where \(t'\) is the time as measured in the rest frame of the object. By defining momentum in terms of the proper time of the object, all observers will agree on the value of \(t'\) . In the frame of reference, \(S\) , (with time \(t\) ) this corresponds to:
\[\begin{aligned} p = m_0\frac{dx}{dt'}=m_0\frac{dt}{dt'}\frac{dx}{dt}=m_0\frac{dt}{dt'}u\end{aligned}\]
where \(u\) is the speed of the particle in frame, \(S\) . We can use time dilation to re-express the derivative:
\[\begin{aligned} \Delta t &= \gamma \Delta t'\\[4pt] \frac{\Delta t}{\Delta t'}&=\gamma\\[4pt] \therefore \frac{dt}{dt'}&=\gamma\end{aligned}\]
where in the last line, we simply took the limit of an infinitesimally short time interval. Therefore, the relativistic momentum of the particle, in frame, \(S\) , can be defined:
\[\vec p=m_{0}\gamma\vec u=\frac{m_{0}\vec u}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\]
where \(\gamma\) is calculated with the same speed, \(u\) , since that is the speed of the reference frame of the object relative to \(S\) . Note that as the speed, \(u\) , of the particle approaches the speed of light, the factor of \(\gamma\) approaches infinity. This means that an object with a mass can never reach the speed of light, as it would have an infinite momentum. In order to define momentum in a way that resembles the classic definition, one can think of the mass of the object as depending on the speed of the object. We define the rest-mass, \(m_0\) , of the object as the mass that is measured when the object is at rest. We can then model the mass of the object as increasing with its speed:
\[\begin{aligned} m(u) = \gamma m_0 = \frac{m_0}{\sqrt{1-\frac{u^2}{c^2}}}\end{aligned}\]
so that the relativistic momentum would be defined as:
\[\begin{aligned} \vec p = m(u)\vec u\end{aligned}\]
In this case, we can think of the mass of the object as increasing with its speed. The object would acquire infinite mass if it were to reach the speed of light.
With the relativistic definition of momentum, Newton’s Second Law can be written as:
\[\begin{aligned} \vec F = \frac{d\vec p}{dt}=\frac{d}{dt}m_0\gamma\vec u \end{aligned}\]
Example \(\PageIndex{1}\)
A constant force of \(1\times 10^{-22}\text{N}\) is applied to an electron (with mass \(m_e=9.11\times 10^{-31}\text{kg}\) ) in order to accelerate it from rest to a speed of \(u=0.99c\) . Compare the length of time over which the force must be applied using classical and relativistic dynamics.
Solution
In both cases, we can start with Newton’s Second Law:
\[\begin{aligned} \vec F &= \frac{d\vec p}{dt}\\[4pt] \therefore \int \vec F dt &= \Delta \vec p = \vec p\end{aligned}\]
where \(\vec p\) is the final momentum of the electron (which is different depending on whether we use the classical or the relativistic definition of momentum). Since the force is constant:
\[\begin{aligned} \int \vec F dt&=\vec F \Delta t=\vec p\\[4pt] \therefore \Delta t &=\frac{p}{F}\end{aligned}\]
where \(\Delta t\) is the length of time over which the force is applied. With the classical definition of momentum, the time is given by:
\[\begin{aligned} \Delta t &=\frac{p}{F} = \frac{mu}{F}=\frac{(9.11\times 10^{-31}\text{kg})(0.99)(3\times 10^{8}\text{m/s})}{(1\times 10^{-22}\text{N})}=2.71\text{s}\end{aligned}\]
With the relativistic definition of momentum, we first need the gamma factor:
\[\begin{aligned} \gamma = \frac{1}{\sqrt{1-\frac{u^2}{c^2}}}=\frac{1}{\sqrt{1-(0.99)^2}}=7.1\end{aligned}\]
We can then calculate the time over which the force needs to be applied:
\[\begin{aligned} \Delta t &=\frac{p}{F} = \frac{\gamma m_0u}{F}=\gamma \frac{m_0u}{F}=(7.1)(2.71\text{s})=19.2\text{s}\end{aligned}\]
Discussion
When using the relativistic definition of momentum, we find that the time over which the force must be applied to reach a given speed is longer. This makes sense, since it will take infinitely long to reach the speed of light. Also, note that the time that is required using relativistic dynamics is just the time-dilated time that is required in classical dynamics.
Recall how we defined kinetic energy, in Section 7.2, by defining the change in kinetic energy of an object as the net work done on that object. We use the same formalism here to redefine kinetic energy using relativistic dynamics.
The work done by the net force, \(\vec F\) , on an object that goes from a position \(A\) to a position \(B\) , is given by
\[\begin{aligned} W = \int_A^B \vec F\cdot d\vec l=\int_0^t \left(\frac{d}{dt}m_0\gamma\vec u \right)\cdot(\vec u dt)\end{aligned}\]
where we recognized that a infinitesimal segment \(d\vec l\) along the path of the object is given by \(d\vec l=\vec u dt\) . The time infinitesimals, \(dt\) , cancel, and we are left with:
\[\begin{aligned} W&=\int_0^t \left(\frac{d}{dt}m_0\gamma\vec u \right)\cdot(\vec u dt)\\[4pt] &=\int d(m_0\gamma \vec u)\cdot \vec u\end{aligned}\]
which we can integrate by parts. We can integrate this over the speed, \(u\) , and we assume that the object started with a speed of \(u=0\) at the beginning of the path and has a speed, \(u=U\) , at the end of the path:
\[\begin{aligned} W&=\int_0^U d(m_0\gamma \vec u)\cdot \vec u = \Big[\gamma m_0 \vec u\cdot\vec u\Big]_0^U-\int_0^U m_0\gamma u du\quad\text{(int. by parts)}\\[4pt] &=\gamma m_0 U^2-m_0\int_0^U\frac{udu}{\sqrt{1-\frac{u^2}{c^2}}}\\[4pt] &=\gamma m_0 U^2-m_0\Big[ c^2\sqrt{1-\frac{u^2}{c^2}} \Big]_0^U\\[4pt] &=\gamma m_0 U^2-m_0c^2+m_0c^2\sqrt{1-\frac{U^2}{c^2}}\\[4pt] &=\gamma \left(m_0 U^2+m_0c^2\left(1-\frac{U^2}{c^2}\right)\right)-m_0c^2\\[4pt] &=m_0c^2(\gamma -1) \end{aligned}\]
Since the object started at rest (with a speed \(u=0\) ) the above integral corresponds to what we would call the kinetic energy of the object, with a speed, \(u\) :
\[\begin{aligned} K=m_0c^2(\gamma -1)=m_0c^2\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)\end{aligned}\]
This form for the relativistic kinetic energy of the object is not at all similar to the form that we obtained in classical physics. As the speed of the object approaches the speed of light, the \(\gamma\) factor approaches infinity, as does the kinetic energy. Thus, it would take an infinite amount of work to accelerate an object to the speed of light, and again, we see that it is impossible for anything with mass to ever reach the speed of light. The formula above, however, should always be correct, even in the non-relativistic limit, when \(v<<c\) . We can approximate the gamma factor using the binomial expansion for the case where \(x<<1\) :
\[\begin{aligned} (1+x)^n\sim 1 + nx +\dots \end{aligned}\]
So that, when \(v<<c\) (and \(v^2/c^2<<1\) ), the gamma factor is approximated by:
\[\begin{aligned} \gamma=\left(1-\frac{u^2}{c^2}\right)^{-\frac{1}{2}}\sim 1+\frac{1}{2}\frac{u^2}{c^2}\end{aligned}\]
In this limit, the relativistic kinetic energy reduces to:
\[\begin{aligned} \lim_{v<<c}K=\lim_{v<<c}m_0c^2(\gamma -1)\sim m_0c^2\left(1+\frac{1}{2}\frac{u^2}{c^2} - 1 \right)=\frac{1}{2}mu^2\end{aligned}\]
which is the classical definition of kinetic energy. The kinetic energy is also zero when the speed is zero.
The kinetic energy has two terms in it:
\[\begin{aligned} K=m_0c^2\gamma -m_0c^2\end{aligned}\]
The first term increases with speed and behaves as we would expect.  We can think of this in slightly different terms. Let us define the total energy, \(E\) , of the object as:
\[\begin{aligned} E=m_{0}c^{2}\gamma \end{aligned}\]
\[\therefore E=K+m_{0}c^{2}\]
so that the total energy is just the rest mass energy plus the kinetic energy. This highlights a key aspect of Special Relativity. An object will have energy, \(E\) , even when it is at rest. That energy, at rest, is called the rest mass energy, and corresponds to energy that an object has by virtue of having mass. This is, of course, Einstein’s famous equation:
\[E=m_{0}c^{2}\quad\text{(rest mass energy)}\]
This equation implies that mass can be thought of as a form of energy. Nuclear reactors function by converting a small amount of mass of uranium atoms into energy (in the form of heat), that is then used to produce high pressure steam to rotate a turbine.
Einstein’s relation is often used to express the mass of subatomic particles in terms of energy. For example, an electron has a mass of \(511\times 10^{3}\text{eV/c}^{2}\) in these units.
Example \(\PageIndex{2}\)
What is the mass of a proton, \(m_p=1.67\times 10^{-27}\text{kg}\) , in units of \(\text{MeV/c}^{2}\) (where the \(\text{M}\) stands for “Mega”, and corresponds to \(1\text{MeV}=1\times 10^{6}\text{eV}\) )?
Solution
We can first calculate the rest mass energy of the proton in Joules:
\[\begin{aligned} E=m_pc^2=(1.67\times 10^{-27}\text{kg})(3\times 10^{8}\text{m/s})^2=1.503\times 10^{-10}\text{J}\end{aligned}\]
We can then convert from Joules to electron-volts:
\[\begin{aligned} \frac{(1.503\times 10^{-10}\text{J})}{(1.6\times 10^{-19}\text{J/eV})}=939.4\times 10^{6}\text{eV}=939.4\text{MeV}\end{aligned}\]
The mass of the proton can then be expressed as \(m_p=939.4\text{MeV/c}^{2}\) .
Finally, it is interesting to examine the relationship between the momentum and the energy of a relativistic object. Consider the quantity \(c^2p^2\) :
\[\begin{aligned} c^2p^2 &= c^2(\gamma m_0 u)^2=c^2\gamma^2m_0^2u^2=c^4\gamma^2m_0^2\frac{u^2}{c^2}=c^4\gamma^2m_0^2\left(1- \frac{1}{\gamma^2}\right)\\[4pt] &=c^4\gamma^2m_0^2 - c^4m_0^2\\[4pt] &=E^2-c^4m_0^2\end{aligned}\]
where we recognized that \(c^4\gamma^2m_0^2\) is simply the energy, \(E\) , squared. This is generally called the “energy-momentum” relation and written:
\[E^{2}=p^{2}c^{2}+m_{0}^{2}c^{4}\]
An interesting consequence of this relationship is that particles with no mass will still have a momentum. For example, the photon, which is a particle of light and must thus have a mass of zero (or it could not move at the speed of light), will have a momentum given by:
\[\begin{aligned} p=\frac{E}{c}\end{aligned}\]
Thus, one can use light to impart momentum to something. This is how a solar sail, a proposed propulsion mechanism for space travel, operates.