3.6: The Doppler Effect
Learning Objectives
By the end of this section, you will be able to:
- Explain why the spectral lines of photons we observe from an object will change as a result of the object’s motion toward or away from us
- Describe how we can use the Doppler effect to deduce how astronomical objects are moving through space
The last two sections introduced you to many new concepts, and we hope that through those, you have seen one major idea emerge. Astronomers can learn about the elements in stars and galaxies by decoding the information in their spectral lines. There is a complicating factor in learning how to decode the message of starlight, however. If a star is moving toward or away from us, its lines will be in a slightly different place in the spectrum from where they would be in a star at rest. And most objects in the universe do have some motion relative to the Sun.
Motion Affects Waves
In 1842, Christian Doppler first measured the effect of motion on waves by hiring a group of musicians to play on an open railroad car as it was moving along the track. He then applied what he learned to all waves, including light, and pointed out that if a light source is approaching or receding from the observer, the light waves will be, respectively, crowded more closely together or spread out. The general principle, now known as the Doppler effect, is illustrated in Figure \(\PageIndex{1}\).
In Figure \(\PageIndex{1a}\), the light source (S) is at rest with respect to the observer. The source gives off a series of waves, whose crests we have labeled 1, 2, 3, and 4. The light waves spread out evenly in all directions, like the ripples from a splash in a pond. The crests are separated by a distance, λ, where λ is the wavelength. The observer, who happens to be located in the direction of the bottom of the image, sees the light waves coming nice and evenly, one wavelength apart. Observers located anywhere else would see the same thing.
On the other hand, if the source of light is moving with respect to the observer, as seen in Figure \(\PageIndex{1b}\), the situation is more complicated. Between the time one crest is emitted and the next one is ready to come out, the source has moved a bit, toward the bottom of the page. From the point of view of observer A , this motion of the source has decreased the distance between crests—it’s squeezing the crests together, this observer might say.
In Figure \(\PageIndex{1b}\), we show the situation from the perspective of three observers. The source is seen in four positions, \(S_1\), \(S_2\), \(S_3\), and \(S_4\), each corresponding to the emission of one wave crest. To observer A , the waves seem to follow one another more closely, at a decreased wavelength and thus increased frequency. (Remember, all light waves travel at the speed of light through empty space, no matter what. This means that motion cannot affect the speed, but only the wavelength and the frequency. As the wavelength decreases, the frequency must increase. If the waves are shorter, more will be able to move by during each second.)
The situation is not the same for other observers. Let’s look at the situation from the point of view of observer C , located opposite observer A in Figure \(\PageIndex{1}\). For her, the source is moving away from her location. As a result, the waves are not squeezed together but instead are spread out by the motion of the source. The crests arrive with an increased wavelength and decreased frequency. To observer B , in a direction at right angles to the motion of the source, no effect is observed. The wavelength and frequency remain the same as they were in part (a) of the figure.
We can see from this illustration that the Doppler effect is produced only by a motion toward or away from the observer, a motion called radial velocity . Sideways motion does not produce such an effect. Observers between A and B would observe some shortening of the light waves for that part of the motion of the source that is along their line of sight. Observers between B and C would observe lengthening of the light waves that are along their line of sight.
You may have heard the Doppler effect with sound waves. When a train whistle or police siren approaches you and then moves away, you will notice a decrease in the pitch (which is how human senses interpret sound wave frequency) of the sound waves. Compared to the waves at rest, they have changed from slightly more frequent when coming toward you, to slightly less frequent when moving away from you.
A nice example of this change in the sound of a train whistle can be heard at the end of the classic Beach Boys song “Caroline, No” on their album Pet Sounds . To hear this sound, go to this YouTube version of the song. The sound of the train begins at approximately 2:20.
Color Shifts
When the source of waves moves toward you, the wavelength decreases a bit. If the waves involved are visible light, then the colors of the light change slightly. As wavelength decreases, they shift toward the blue end of the spectrum: astronomers call this a blueshift (since the end of the spectrum is really violet, the term should probably be violetshift , but blue is a more common color). When the source moves away from you and the wavelength gets longer, we call the change in colors a redshift . Because the Doppler effect was first used with visible light in astronomy, the terms “blueshift” and “redshift” became well established. Today, astronomers use these words to describe changes in the wavelengths of radio waves or X-rays as comfortably as they use them to describe changes in visible light.
The greater the motion toward or away from us, the greater the Doppler shift. If the relative motion is entirely along the line of sight, the formula for the Doppler shift of light is
\[\dfrac{\Delta λ}{λ}= \frac{v}{c} \nonumber\]
where \(\lambda\) is the wavelength emitted by the source, \(\Delta \lambda\) is the difference between \(\lambda\) and the wavelength measured by the observer, \(c\) is the speed of light, and \(v\) is the relative speed of the observer and the source in the line of sight. The variable \(v\) is counted as positive if the velocity is one of recession, and negative if it is one of approach. Solving this equation for the velocity, we find
\[ v = c \times \dfrac{\Delta \lambda}{\lambda}. \nonumber\]
If a star approaches or recedes from us, the wavelengths of light in its continuous spectrum appear shortened or lengthened, respectively, as do those of the dark lines. However, unless its speed is tens of thousands of kilometers per second, the star does not appear noticeably bluer or redder than normal. The Doppler shift is thus not easily detected in a continuous spectrum and cannot be measured accurately in such a spectrum. The wavelengths of the absorption lines can be measured accurately, however, and their Doppler shift is relatively simple to detect.
Example \(\PageIndex{1}\): The Doppler Effect
We can use the Doppler effect equation to calculate the radial velocity of an object if we know three things: the speed of light, the original (unshifted) wavelength of the light emitted, and the difference between the wavelength of the emitted light and the wavelength we observe. For particular absorption or emission lines, we usually know exactly what wavelength the line has in our laboratories on Earth, where the source of light is not moving. We can measure the new wavelength with our instruments at the telescope, and so we know the difference in wavelength due to Doppler shifting. Since the speed of light is a universal constant, we can then calculate the radial velocity of the star.
A particular emission line of hydrogen is originally emitted with a wavelength of 656.3 nm from a gas cloud. At our telescope, we observe the wavelength of the emission line to be 656.6 nm. How fast is this gas cloud moving toward or away from Earth?
- Answer
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Because the light is shifted to a longer wavelength (redshifted), we know this gas cloud is moving away from us. The speed can be calculated using the Doppler shift formula:
\[ν=c×\dfrac{\Delta \lambda}{\lambda}= \left(3.0 \times 10^8 ~ \text{m/s} \right) \left( \dfrac{0.3 ~ \text{nm}}{656.3 ~ \text{nm}} \right)= \left(3.0 \times 10^8 ~ \text{m/s} \right) \left(\dfrac{0.3 \times 10^{–9} ~ \text{m}}{656.3 \times 10^{–9} ~ \text{m}} \right)=140,000 ~ \text{m/s} =140 ~ \text{km/s} \nonumber\]
Exercise \(\PageIndex{1}\)
Suppose a spectral line of hydrogen, normally at 500 nm, is observed in the spectrum of a star to be at 500.1 nm. How fast is the star moving toward or away from Earth?
- Answer
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Because the light is shifted to a longer wavelength, the star is moving away from us:
\[ν=c \times \dfrac{\Delta \lambda}{\lambda}= \left(3.0 \times 10^8 ~ \text{m/s} \right) \left(\dfrac{0.1 ~ \text{nm}}{500 ~ \text{nm}} \right)=\left( 3.0 \times 10^8 ~ \text{m/s} \right) \left(\dfrac{0.1 \times 10^{–9} ~ \text{m}}{500 \times 10^{–9} ~ \text{m}} \right)=60,000 ~ \text{m/s}\]
Its speed is 60,000 m/s.
You may now be asking: if all the stars are moving and motion changes the wavelength of each spectral line, won’t this be a disaster for astronomers trying to figure out what elements are present in the stars? After all, it is the precise wavelength (or color) that tells astronomers which lines belong to which element. And we first measure these wavelengths in containers of gas in our laboratories, which are not moving. If every line in a star’s spectrum is now shifted by its motion to a different wavelength (color), how can we be sure which lines and which elements we are looking at in a star whose speed we do not know?
Take heart. This situation sounds worse than it really is. Astronomers rarely judge the presence of an element in an astronomical object by a single line. It is the pattern of lines unique to hydrogen or calcium that enables us to determine that those elements are part of the star or galaxy we are observing. The Doppler effect does not change the pattern of lines from a given element—it only shifts the whole pattern slightly toward redder or bluer wavelengths. The shifted pattern is still quite easy to recognize. Best of all, when we do recognize a familiar element’s pattern, we get a bonus: the amount the pattern is shifted can enable us to determine the speed of the objects in our line of sight.
The training of astronomers includes much work on learning to decode light (and other electromagnetic radiation). A skillful “decoder” can learn the temperature of a star, what elements are in it, and even its speed in a direction toward us or away from us. That’s really an impressive amount of information for stars that are light-years away.
Summary
If an atom is moving toward us when an electron changes orbits and produces a spectral line, we see that line shifted slightly toward the blue of its normal wavelength in a spectrum. If the atom is moving away, we see the line shifted toward the red. This shift is known as the Doppler effect and can be used to measure the radial velocities of distant objects.
Glossary
- Doppler effect
- the apparent change in wavelength or frequency of the radiation from a source due to its relative motion away from or toward the observer
- radial velocity
- motion toward or away from the observer; the component of relative velocity that lies in the line of sight
Contributors and Attributions
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Andrew Fraknoi (Foothill College), David Morrison (NASA Ames Research Center), Sidney C. Wolff (National Optical Astronomy Observatory) with many contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/astronomy ).


3.4: Leave Yourself Out of Your Writing
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Author: Joseph Rodríguez, English, The University of Texas at El Paso
“You’re also two people, writer and reader.  This myth both hinders and undermines deeper learning and thinking in the lives of writers, readers, thinkers, and students. It can be interpreted in various ways that range from removing the personal point of view in a document to altogether distancing oneself from the subject and interest that was once held for a topic or concept worthy of writing for an audience. With one’s self removed, the writer is partly absent from the page and conversation. Common lore based around assignments and research papers creates this stance of effacing the writer and essentially one’s being. The opposite is true for writing: The writer needs to be present and breathing on the print or digital page. To make meaning through language, the writer must be present to the audience and mindful of beliefs to produce coherent, meaningful, and engaging writing for the reader. Writing in 1914, Spanish philosopher José Ortega y Gasset asserted, “I am I and my circumstance; and, if I do not save it, I do not save myself.”
The act of writing is influenced by the writing beliefs learned, adopted, and now followed as a habit in practice to record, or save, our selfhood.  Some of these writing beliefs can become pitfalls, or bad ideas, if reflection is missing from our practice. If there is a connection early on in the writing between the subject and writer, then the arguments, observations, and truths presented to the reader can
hold more attention for the audience. By being told to leave yourself out of your writing, the writer is not building a publicly recognized voice with confidence and a level of expertise that builds trust with an audience. The false conception that the writer must be removed or, even worse, nearly dead to one’s audience is anathema. Sometimes, this bad idea is promoted under the assumption that to maintain objectivity the writer must be absent and avoid using the pronoun I . Thus, the first-person point of view is missing. Nonetheless, those who write need to have a presence, and, as Toni Morrison wrote, an I/eye for credibility and connection between writer and reader. The audience is counting on the writer to deliver by being committed and present in the writing.
Why must you leave yourself out of your writing? Why and how? Writers adapt to writing conditions and circumstances, and their adaptation has responsibilities and consequences, too. One of the adaptations includes writing the self in communications instead of gradual disembodiment or complete separation. By being present in one’s writing as one writes, the writer is creating conditions for the reader to better understand a topic as well as the concepts that drive learning and understanding. The writer achieves this by: (1) writing sentences that establish who the writer is through the pronoun I, (2) revealing the purpose and interest for writing about the topic, (3) stating the argument that holds the content together in the first place, and (4) providing evidence and concepts for the argumentative position or exploratory topic. Klinkenborg describes the reader and writer, who are in a kind of union, as a “tremendous asset” that comes to fruition through language and form. The self is significant in the act of writing to make meaning, present an argument, and come into existence on the page. To repress or remove the self from one’s writing is counterproductive for the writer’s purpose.
Sandra Cisneros explains, “As a writer, I continue to analyze and reflect on the power words have over me.” Cisneros’s statement may resonate with emerging writers in classrooms, outside of classrooms, and spaces of their own making to write and create with language. Writers affirm their lives and empower the lives of others who connect with their words and concepts. The first-person point of view is an essential marker in the making of meaning for both the writer and reader and need not be abandoned nor silenced. Even if the self is briefly mentioned and noted in one’s written deliberations, its use will suffice in making oneself known and present in the discussion of ideas, concepts, and perspectives.
Some writers across the disciplines, which include the arts, education, engineering, mathematics, technology, and sciences, attempt in various cloaked forms to remove their sense of self as they write under the assumption that writing must be as universal as possible. Unfortunately, this means less personality and presence. This is done in detrimental forms if the writer fails to acknowledge purpose, actions, and influences to an audience. Furthermore, writers and thinkers are connected to their subjects and arguments, which they deem worthy of explanation or description. As such, writers learn about themselves through their writing and the interconnectedness to thought and argument. In fact, writers give authority and credibility to experience through their expertise and in structure and argument. The reader entrusts the writer’s self not only by the levels of expertise for having done the labor and research, but also by the valuable experiences drawn from analysis. In short, it is a false idea to eliminate one’s first-person point of view in writing, rather than supporting the writer’s personal and communicative voice on the print and digital page.
One learns about oneself as one writes and by asserting the self. Indeed, words and literacies carve the identities of writers who are influenced by the societies they inhabit and the subjects they study. In essence, words empower and define us as Cisneros observes.
Admittedly, writing is an act of free will that calls for the writer to be present with voice while deliberating through language with ideas, concepts, and perspectives for understanding. The new rule to adopt is as follows: Speak up and be present and known in your writing! This can be implemented by valuing your own self-aswriter and by answering the following calls for the writer to be present and alive: (1) As a writer, how will I help the readers know what they will be learning through my writing purpose and human presence? (2) In which ways can I hook and engage the readers? (3) Which experiences and skills do I possess that can deepen understanding through my writing? (4) How can I encourage readers to rethink or reimagine what they may already know through reflection and action? (5) What order of my ideas and concepts will be most optimal for reader understanding? In sum, reliance upon oneself and one’s writing are essential in developing a writing life and an audience. Make your writing and thinking audible on the print and digital page for two: the writer and reader.
Author Bios Rodrigo
Joseph Rodríguez teaches in the department of English at The University of Texas at El Paso, which is located in the borderlands across from Ciudad Juárez, Chihuahua, México. He believes all writing comes to life when the reader and writer enter and maintain a dialogue on the print and digital page. Catch him virtually @escribescribe. Kimberly N. Parker teaches high school and college preparatory courses in the Cambridge Public Schools (Massachusetts). 138 Bad Ideas Her research interests include English language arts with culturally responsive pedagogy and the teaching and study of African American literature. She is an adjunct instructor at Tufts University and president of the New England Association of Teachers of English. Catch her virtually @drkimpossible.


2.2: Introduction to Logic
In the previous section, we jumped in head first and attempted to prove several theorems in the context of number theory without a formal understanding of what it was we were doing. Likely, many issues bubbled to the surface. What is a proof? What sorts of statements require proof? What should a proof entail? How should a proof be structured? Let’s take a step back and do a more careful examination of what it is we are actually doing. In the the next two sections, we will introduce the basics of propositional logic —also referred to as propositional calculus or sometimes zeroth-order logic .
Definition 2.16. A proposition is a sentence that is either true or false but never both. The truth value (or logical value ) of a proposition refers to its attribute of being true or false.
For example, the sentence “All dogs have four legs” is a false proposition. However, the perfectly good sentence “x = 1” is not a proposition all by itself since we do not actually know what x is.
Problem 2.17. Determine whether each of the following is a proposition. Explain your reasoning.
(a) All cars are red.
(b) Every person whose name begins with J has the name Joe.
(c) x 2 = 4.
(d) There exists a real number x such that x 2 = 4.
(e) For all real numbers x, x 2 = 4.
(f) √2 is an irrational number.
(g) p is prime.
(h) Is it raining?
(i) It will rain tomorrow
(j) Led Zeppelin is the best band of all time.
The last two sentences in the previous problem may stir debate. It is not so important that we come to consensus as to whether either of these two sentences is actually a proposition or not. The good news is that in mathematics we do not encounter statements whose truth value is dependent on either the future or opinion.
Given two propositions, we can form more complicated propositions using logical connectives .
Definition 2.18. Let A and B be propositions.
(a) The proposition “ not A” is true if A is false; expressed symbolically as \(¬A\) and called the negation of A.
(b) The proposition “A and B” is true if both A and B are true; expressed symbolically as \(A∧B\) and called the conjunction of A and B.
(c) The proposition “A or B” is true if at least one of A or B is true; expressed symbolically as \(A∨B\) and called the disjunction of A and B.
(d) The proposition “ If A, then B” is true if both A and B are true, or A is false; expressed symbolically as \(A =⇒ B\) and called a conditional proposition (or implication ). In this case, A is called the hypothesis and B is called the conclusion . Note that A = whenever A”. ⇒ B may also be read as “A implies B”, “A only if B”, “B if A”, or “B
(e) The proposition “A if and only if B” (alternatively, “A is necessary and sufficient for B”) is true if both A and B have the same truth value; expressed symbolically as \(A ⇐⇒ B\) and called a biconditional proposition . If A ⇐⇒ B is true, we say that A and B are logically equivalent .
Each of the boxed propositions is called a compound proposition , where A and B are referred to as the components of the compound proposition.
It is worth pointing out that definitions in mathematics are typically written in the form “B if A” (or “B provided that A” or “B whenever A”), where B contains the term or phrase we are defining and A provides the meaning of the concept we are defining. In the case of definitions, we should always interpret “B if A” as describing precisely the collection of “objects” (e.g., numbers, sets, functions, etc.) that should be identified with the term or phrase we defining. That is, if an object does not meet the condition specified in A, then it is never referred to by the term or phrase we are defining. Some authors will write definitions in the form “B if and only if A”. However, a definition is not at all the same kind of statement as a usual biconditional since one of the two sides is undefined until the definition is made. A definition is really a statement that the newly defined term or phrase is synonymous with a previously defined concept.
We can form complicated compound propositions with several components by utilizing logical connectives.
Problem 2.19. Let A represent “6 is an even integer” and B represent “4 divides 6.” Express each of the following compound propositions in an ordinary English sentence and then determine its truth value.
(a) A ∧B
(b) A ∨B
(c) ¬A
(d) ¬B
(e) ¬(A∧B)
(f) ¬(A∨B)
(g) A =⇒ B
Definition 2.20. A truth table for a compound proposition is a table that illustrates all possible combinations of truth values for the components of the compound proposition together with the resulting truth value for each combination.
Example 2.21. If A and B are propositions, then the truth table for the compound proposition A ∧B is given by the following.
| A | B | A∧B |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Notice that we have columns for each of A and B. The rows for these two columns correspond to all possible combinations of truth values for A and B. The third column yields the truth value of A ∧B given the possible truth values for A and B.
Each component of a compound proposition has two possible truth values, namely true or false. Thus, if a compound proposition is built from n component propositions, then the truth table will require 2n rows.
Problem 2.22. Create a truth table for each of the following compound propositions. You should add additional columns to your tables as needed to assist you with intermediate steps. For example, you might need four columns for the third and fourth compound propositions below.
(a) ¬A
(b) A∨B
(c) ¬(A∧B)
(d) ¬A∧¬B
Problem 2.23. A coach promises her players, “If we win tonight, then I will buy you pizza tomorrow.” Determine the cases in which the players can rightly claim to have been lied to. If the team lost the game and the coach decided to buy them pizza anyway, was she lying?
Problem 2.24.  The combination you should pay particular attention to is when the hypothesis is false while the conclusion is true.
In accordance with Definition 2.18(d), a conditional proposition A =⇒ B is only false when the hypothesis is true and the conclusion is false. Perhaps you are bothered by the fact that A =⇒ B is true when A is false no matter what the truth value of B is. The thing to keep in mind is that the truth value of A =⇒ B relies on a very specific definition and may not always agree with the colloquial use of “If. . . , then. . . ” statements that we encounter in everyday language. For example, if someone says, “If you break the rules, then you will be punished”, the speaker likely intends the statement to be interpreted as “You will be punished if and only if you break the rules.” In logic and mathematics, we aim to remove such ambiguity by explicitly saying exactly what we mean. For our purposes, we should view a conditional proposition as a contract or obligation. If the hypothesis is false and the conclusion is true, the contract is not violated. On the other hand, if the hypothesis is true and the conclusion is false, then the contract is broken.
We can often prove facts concerning logical statements using truth tables. Recall that two propositions P and Q (both of which might be complicated compound propositions) are logically equivalent if P ⇐⇒ Q is true (see Definition 2.18(e)). This happens when P and Q have the same truth value. We can verify whether P and Q have the same truth value by constructing a truth table that includes columns for each of the components of P and Q, listing all possible combinations of their truth values, together with columns for P and Q that lists their resulting truth values. If the truth values in the columns for P and Q agree, then P and Q are logically equivalent, and otherwise they are not logically equivalent. When constructing truth tables to verify whether P and Q are logically equivalent, you should add any necessary intermediate columns to aid in your “calculations”. Use truth tables when attempting to justify the next few problems.
Theorem 2.25. If A is a proposition, then ¬(¬A) is logically equivalent to A.
The next theorem, referred to as De Morgan’s Law , provides a method for negating a compound proposition involving a conjunction.
Theorem 2.26 (De Morgan’s Law). If A and B are propositions, then ¬(A ∧ B) is logically equivalent to ¬A ∨ ¬B.
Problem 2.27 (De Morgan’s Law). Let A and B be propositions. Conjecture a statement similar to Theorem 2.26 for the proposition ¬(A∨B) and then prove it. This is also called De Morgan’s Law.
We will make use of both versions De Morgan’s Law on on a regular basis. Sometimes conjunctions and disjunctions are “buried” in a mathematical statement, which makes negating statements tricky business. Keep this in mind when approaching the next problem.
Problem 2.28. Let x be your favorite real number. Negate each of the following statements. Note that the statement in Part (b) involves a conjunction.
(a) x < −1 or x ≥ 3.
(b) 0 ≤ x < 1.
Theorem 2.29. If A and B are propositions, then A ⇐⇒ B is logically equivalent to (A =⇒ B)∧(B =⇒ A).
Theorem 2.30. If A, B, and C are propositions, then (A ∨ B) =⇒ C is logically equivalent to (A =⇒ C)∧(B =⇒ C).
We already introduced the following notion in the discussion following Theorem 2.11
Definition 2.31. If A and B are propositions, then the converse of A =⇒ B is B =⇒ A.
Problem 2.32. Provide an example of a true conditional proposition whose converse is false.
Definition 2.33. If A and B are propositions, then the inverse of A =⇒ B is ¬A =⇒ ¬B.
Problem 2.34. Provide an example of a true conditional proposition whose inverse is false.
Based on Problems 2.32 and 2.34, we can conclude that the converse and inverse of a conditional proposition do not necessarily have the same truth value as the original statement. Moreover, the converse and inverse of a conditional proposition do not necessarily have the same truth value as each other.
Problem 2.35. If possible, provide an example of a conditional proposition whose converse is true but whose inverse is false. If this is not possible, explain why.
What if we swap the roles of the hypothesis and conclusion of a conditional proposition and negate each?
Definition 2.36. If A and B are propositions, then the contrapositive of A =⇒ B is ¬B =⇒ ¬A.
Problem 2.37. Let A and B represent the statements from Problem 2.19. Express each of the following in an ordinary English sentence.
(a) The converse of A =⇒ B.
(b) The contrapositive of A =⇒ B.
Problem 2.38. Find the converse and the contrapositive of the following statement: “If Dana lives in Flagstaff, then Dana lives in Arizona.”
Use a truth table to prove the following theorem.
Theorem 2.39. If A and B are propositions, then A =⇒ B is logically equivalent to its contrapositive.
So far we have discussed how to negate propositions of the form A, A ∧ B, and A ∨ B for propositions A and B. However, we have yet to discuss how to negate propositions of the form A =⇒ B. Prove the following result with a truth table.
Theorem 2.40. If A and B are propositions, then the implication A =⇒ B is logically equivalent to the disjunction ¬A ∨B.
The next result follows quickly from Theorem 2.40 together with De Morgan’s Law. You can also verify this result using a truth table.
Corollary 2.41. If A and B are propositions, then ¬(A =⇒ B) is logically equivalent to A ∧ ¬B.
Problem 2.42. Let A and B be the propositions “√ 2 is an irrational number” and “Every rectangle is a trapezoid,” respectively.
(a) Express A =⇒ B as an English sentence involving the disjunction “or.”
(b) Express ¬(A =⇒ B) as an English sentence involving the conjunction “and.”
Problem 2.43. It turns out that the proposition “If .99··· = \(dfrac{9}{10} + dfrac{9}{100} + dfrac{9}{1000} +···\), then .99 ≠ 1” is false. Write its negation as a conjunction.
Recall that a proposition is exclusively either true or false—it can never be both.
Definition 2.44. A compound proposition that is always false is called a contradiction . A compound proposition that is always true is called a tautology .
Theorem 2.45. If A is a proposition, then the proposition ¬A ∧ A is a contradiction.
Problem 2.46. Provide an example of a tautology using arbitrary propositions and any of the logical connectives ¬, ∧, and ∨. Prove that your example is in fact a tautology.


9.5: Studying In- Addressing Inequities within Anthropology
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- Jennifer Hasty, David G. Lewis, & Marjorie M. Snipes
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Learning Objectives
By the end of this section, you will be able to:
- Examine the effects of White supremacy in anthropology.
- Give examples of how anthropologists in other subfields are working against White supremacy and colonialism.
- Explain what decolonizing anthropology entails.
This section explores how anthropologists have looked within their own discipline to address ways in which they may be reproducing inequities through their practices and approach. Anthropologist Laura Nader uses the phrase “studying up” (1972) to call for more research on people and institutions with power. Following anthropologist Pamela Runestad (2017), this chapter uses the phrase “studying in” to address how anthropologists have looked at their own practices, training, methodologies, and assumptions and how anthropology as a discipline may in fact be contributing to inequities for students, practitioners, and the communities with which anthropologists engage.
Even though the construct of race has roots in anthropology, anthropologist Leith Mullings (2005) argues that critical studies of race and racism did not originally develop in anthropology. Mullings attributes this to the fact that anthropologists still do not agree about the role of race and racism within the discipline or how the categories of race have emerged and persisted in society. In addition, Mullings argues that many cultural anthropologists have focused on ethnicity, becoming “race avoidant” by not even mentioning race in ethnographies.
In recent years, anthropologists have looked at the ways in which knowledge production and anthropological methods are rooted in White supremacy. Within the subfield of archaeology, anthropologists Maria Franklin and colleagues discuss how “archaeology has been used to justify imperialism, the displacement of Native Americans and Indigenous peoples from their lands, scientific racism, ethnocentrism, and xenophobic nationalism” throughout the world (Franklin et al. 2020, 754). However, archaeology does not exist in a vacuum, and these anthropologists also discuss ways to reimagine archaeology to do anti-racism work, especially in light of the Black Lives Matter movement. These efforts include encouraging growing numbers of members of minority groups as academic colleagues and seeking research sites that represent the lived experiences of minority populations. In 2020, Meredith Poole, a researcher for the Colonial Williamsburg Foundation in Williamsburg, Virginia, began a project identifying previously unrecognized Black archaeologists and excavators who had worked for Colonial Williamsburg in the 1930s. Additionally, Colonial Williamsburg archaeologists are currently working to excavate the First Baptist Church , one of the earliest Black churches in the United States. Projects such as these are critically important to the academic search for truth. Not only does this knowledge correct inaccuracies in the historical record, but it also serves to correct the course of future academic work.
Archaeologists Kylie Quave and colleagues (2020) have written about the ways in which introductory archaeology classes taught in the United States have often been problematic and how those teaching these courses are currently using anti-colonial and decolonial theories to revise curricula to promote equity within the discipline. Quave and her colleagues found that students taking revised curricula developed more complex understandings of the benefits and harms of archeological knowledge production and were better able to articulate the inequities in the discipline.
In April 2021, the Association of Black Anthropologists (ABA), the Society of Black Archaeologists (SBA), and the Black in Bioanthropology Collective (BiBA) released a collective statement regarding the possession and unethical use of the remains of the children of MOVE and the Africa family. In May 1985, the city of Philadelphia dropped two bombs onto the MOVE compound, home of “a revolutionary group of Black people opposed to capitalist growth and committed to environmental justice and interspecies harmony” (ABA, SBA, and BiBA 2021). The bombs killed 11 MOVE members inside the compound, including five children, and destroyed the neighborhood, incinerating at least 61 homes. Two forensic anthropologists, Alan Mann and Janet Monge, were hired by Philadelphia officials to identify the remains. In April 2021, various news outlets revealed that either Mann or Monge kept the remains two child victims, Tree Africa and Delisha Africa, in their personal possession after the investigation, moving them between the University of Pennsylvania and Princeton University. In addition, the family of the deceased were never notified of the remains, nor were the remains returned to the family. In response, the ABA, SBA, and BiBA supported and republished the demands of Mike Africa Jr., who was six years old when the Philadelphia police dropped the bomb on MOVE. The collective statement acknowledged the long history of White supremacy and anti-Blackness within the discipline and called on White anthropologists to actively work to undo the violence committed against non-White communities.
In Decolonizing Anthropology: Moving Further toward an Anthropology for Liberation (2010), edited by anthropologist Faye V. Harrison, the term decolonizing anthropology is used to emphasize the responsibility of anthropologists to work for the enhancement and empowerment of those most alienated and dispossessed. While decolonization refers to different ideas in different disciplines, the principal goal of the Decolonizing Anthropology volume is “to encourage more anthropologists to accept the challenge of working to free the study of humankind from the prevailing forces of global inequality and dehumanization and to locate it firmly in the complex struggle for genuine transformation” (Harrison 2010, 10). This work to decolonize, transform, and liberate anthropology is still happening, and the discipline still has a long way to go in decolonizing each of the subfields of anthropology and decolonizing methods and pedagogy to make classroom spaces more equitable.
Suggested Resources
Documentaries:
Adelman, Larry, prod. 2003. Race: The Power of an Illusion . https://www.racepowerofanillusion.org/ .
Davidson, Kief, and Pedro Kos, dirs. 2017. Bending the Arc . https://bendingthearcfilm.com/ .
Books/Articles:
Cargle, Rachel Elizabeth. 2018. “When Feminism Is White Supremacy in Heels.” Harper’s Bazaar , August 16, 2018. https://www.harpersbazaar.com/cultur...hite-feminism/ .
Crenshaw, Kimberlé, Neil Gotanda, Gary Peller, and Kendall Thomas, eds. 1995. Critical Race Theory: The Key Writings That Formed the Movement . New York: New Press.
Jenkins, Destin, and Justin Leroy, eds. 2021. Histories of Racial Capitalism . New York: Columbia University Press.
Williams, Bianca C., Dian D. Squire, and Frank A. Tuitt, eds. 2021. Plantation Politics and Campus Rebellions: Power, Diversity, and the Emancipatory Struggle in Higher Education . Albany: State University of New York Press.
Websites/Podcasts:
James, Alyssa A. L., and Brendane A. Tynes. 2020–. Zora’s Daughters . Podcast. https://zorasdaughters.com/ .
Smithsonian Institution. 2020. “Talking about Race.” National Museum of African American History and Culture. Last updated June 2, 2020. https://nmaahc.si.edu/learn/talking-about-race .
Mini-Fieldwork Activity
American Census Archive Research
Browse through the US Census Bureau website ( data.census.gov ). Look at the categories from 1790, when the first US census was taken. Compare them to the 2020 census. How are these categories different? Who is being counted and how? Who is excluded?
Go one step further and search through two different zip codes. Look at the demographic makeup of each area. Can you see differences in household income? Education attainment? What if you go to Google Maps? Can you correlate this information with other causes of inequities? Are there grocery stores in these areas? Bus and subway stops? What is the population density of the area? Do you know anything of the history of the zip code?
Construct a visual that best explains the differences between the historical and contemporary censuses. Then, do the same with the two zip codes. Draw conclusions about social inequalities throughout history and in contemporary times.


11.6: Phase Distribution Equilibria
- Phase distribution equilibria play an important role in chemical separation processes on both laboratory and industrial scales. They are also involved in the movement of chemicals between different parts of the environment, and in the bioconcentration of pollutants in the food chain.
It often happens that two immiscible liquid phases are in contact, one of which contains a solute. How will the solute tend to distribute itself between the two phases? One’s first thought might be that some of the solute will migrate from one phase into the other until it is distributed equally between the two phases, since this would correspond to the maximum dispersion (randomness) of the solute. This, however, does not take into the account the differing solubilities the solute might have in the two liquids; if such a difference does exist, the solute will preferentially migrate into the phase in which it is more soluble.
For a solute \(S\) distributed between two phases a and b the process S a = S b is defined by the distribution law
\[K_{a,b} = \dfrac{[S]_a}{[S]_b}\]
in which
- \(K_{a,b}\) is the distribution ratio (also called the distribution coefficient) and
- \([S]_i\) is the solubility of the solute in the phase.
biomagnification
The transport of substances between different phases is of immense importance in such diverse fields as pharmacology and environmental science. For example, if a drug is to pass from the aqueous phase with the stomach into the bloodstream, it must pass through the lipid (oil-like) phase of the epithelial cells that line the digestive tract. Similarly, a pollutant such as a pesticide residue that is more soluble in oil than in water will be preferentially taken up and retained by marine organism, especially fish, whose bodies contain more oil-like substances; this is basically the mechanism whereby such residues as DDT can undergo biomagnification as they become more concentrated at higher levels within the food chain. For this reason, environmental regulations now require that oil-water distribution ratios be established for any new chemical likely to find its way into natural waters. The standard “oil” phase that is almost universally used is octanol, C 8 H 17 OH.
In preparative chemistry it is frequently necessary to recover a desired product present in a reaction mixture by extracting it into another liquid in which it is more soluble than the unwanted substances. On the laboratory scale this operation is carried out in a separatory funnel as shown below. The two immiscible liquids are poured into the funnel through the opening at the top. The funnel is then shaken to bring the two phases into intimate contact, and then set aside to allow the two liquids to separate into layers, which are then separated by allowing the more dense liquid to exit through the stopcock at the bottom.
If the distribution ratio is too low to achieve efficient separation in a single step, it can be repeated; there are automated devices that can carry out hundreds of successive extractions, each yielding a product of higher purity. In these applications our goal is to exploit the Le Chatelier principle by repeatedly upsetting the phase distribution equilibrium that would result if two phases were to remain in permanent contact.
Video \(\PageIndex{1}\) : How to perform a liquid-liquid extraction using a separating funnel.
The distribution ratio for iodine between water and carbon disulfide is 650. Calculate the concentration of I 2 remaining in the aqueous phase after 50.0 mL of 0.10M I 2 in water is shaken with 10.0 mL of CS 2 .
Solution
The equilibrium constant is
\[K_d = \dfrac{C_{CS_2}}{C_{H_2O}} = 650 \nonumber\]
\[((5.00 – m_1) mmol / 10 mL) ÷ (m_1 mmol / 50 mL) = 650 \nonumber\]
Simplifying and solving for m 1 yields
\[ \dfrac{(0.50 – 0.1)m_1}{(0.02 m_1} = 650 \nonumber\]
with m 1 = 0.0382 mmol.
The concentration of solute in the water layer is (0.0382 mmol) / (50 mL) = 0.000763 M , showing that almost all of the iodine has moved into the CS 2 layer.


7: Pharmacologic Issues Across the Lifespan
- Name three physiologic pharmacokinetic issues that can affect drug response in neonates and infants.
- Discuss the pharmacologic variables associated with each pediatric age group (neonate, infants, children).
- Name the four considerations for increasing medication compliance for the elderly.
- Describe the physiologic changes in pregnancy that may affect absorption, distribution, metabolism, and excretion of medications.
- Summarize the Pregnancy and Lactation Rule endorsed by the FDA.
7.0 Overview
Pharmakinetic and pharmacodynamic processes varies with age. These differences are due to the normal physiologic changes that occur over a normal lifespan, especially evident are changes in infants and the elderly. One important difference across the lifespan is the amount of total body water, which changes with age and condition.
7.1 Neonates, Infants, Children
Drug treatment for neonates (less than one month of age) and infants (less than one year of age) is complicated by the lack of evidence in this age group. Rather, data derived from adult clinical trials is applied to determine dosing and efficacy. The dose-response relationship may change in the first few weeks after birth, making it a challenge to establish guidelines.
7.1.1 Absorption
Some of the physiologic changes associated with the first few months of life that affect absorption include a higher-than-normal gastric pH. In neonates and pediatric patients, the acid-producing cells of the stomach are immature until around the age of one to two years. They also experience prolonged and irregular gastric emptying until 6-8 months of age.
7.1.2 Distribution
Changes that affect distribution include lower adipose tissue, decreased plasmin albumin, and higher total body water. The developing blood–brain barrier allows more drugs to enter the brain. 1
7.1.3 Metabolism
The developing liver in neonates and infants produces decreased levels of microsomal enzymes. This may decrease the ability to metabolize medications, necessitating lower drug doses. As an infant’s liver matures and liver metabolism increases, the infant may require higher doses of medications. Age (in weeks) is an important consideration when prescribing for neonates and infants as liver function may change weekly. Hepatic function reaches maturity at about two years old.
7.1.4 Elimination
Neonates, infants, and children have immature kidneys with decreased glomerular filtration, reabsorption, and tubular secretion. As a result, medications are not cleared as efficiently from the body. Dosing for most medications used to treat infants and pediatric patients is commonly based on weight in kilograms, and a smaller dose is usually prescribed. In addition, pediatric patients may have higher levels of free circulating medication than anticipated and may become toxic quickly.
7.2 Overview of Older Adults
The rationale use of drugs in persons over 65 years of age is complicated by physiologic changes, increased incidence of chronic disease and co-morbidities, and polypharmacy. Financial issues, limited resources, and poor nutritional status may all influence an elderly person’s ability to comply.
Cardiac output starts declining 1% at about age 30, and in the elderly, there is a redistribution of blood flow favoring the brain, heart, and kidney. Lean body mass declines with age, and total body water decreases. Plasma albumin levels may decrease, especially in the poorly nourished. Glomerular filtration rate and plasma renal blood flow steadily decline with aging.
7.2.1 Absorption
As a natural result of aging, older adults will experience decreased blood flow to tissues within the GI tract. In addition, there may be changes in the gastric pH that may alter the absorption of certain medications. Older adult patients may also experience variations in available plasma proteins, which can impact drug levels of medications that are highly protein-bound. Consideration must also be given to the use of subcutaneous and intramuscular injections in older patients experiencing decreased cardiac output. Decreased drug absorption of medications can occur when peripheral circulation is reduced. Finally, as adults age, they often have less body fat, resulting in reduced absorption of medication from transdermal patches that require adequate subcutaneous fat stores for proper absorption.
7.2.2 Distribution
The aging adult patient will experience a decrease in total body water, muscle mass, and serum albumin while total body fat increases. This results in lipid-soluble drugs such as diazepam and lidocaine having a larger Vd, whereas water-soluble drugs such as acetaminophen and ethanol have a smaller Vd. Serum albumin often also decreases, resulting in more active free drug within the body. This is one reason why many older adult patients require lower levels of medication. Be aware that a person who has taken medication for a chronic condition for many years may need dosage adjustment as they age.
7.2.3 Metabolism
Hepatic metabolism may significantly decline in older adults. As a result, dosages should be adjusted according to the patient’s liver function and anticipated metabolic rate. First-pass metabolism is also decreased with aging; therefore, older adults may have higher “free” circulating drug concentrations and be at higher risk for side effects and toxicities.
7.2.4 Elimination
Kidney and liver function often decreases with age, leading to decreased medication excretion. Subsequently, medication may have a prolonged half-life with a greater potential for toxicity from elevated circulating drug levels. Two important principles need to be applied to dosing for the elderly. First, most elderly patients do not have “normal” renal function despite having a serum creatinine level that appears normal. Second, most elderly persons require a dose adjustment for a medication or drug metabolite eliminated through the kidneys. Smaller doses of medications are often recommended for older patients because of these factors, commonly referred to as “start low and go slow.”
7.3 Overview of Pregnancy and Breastfeeding
Drug use during pregnancy is common: many people take at least one medication. Indications may include pregnancy-related issues such as nausea, constipation, and pre-eclampsia or chronic conditions such as epilepsy and asthma. When drugs are used during pregnancy, there is a risk to the fetus as well. However, most drugs are not tested during pregnancy, and the risk to the fetus remains unknown. Therefore, the provider should initiate a shared decision approach and weigh the risks and benefits.
For some chronic conditions such as asthma, the risk to the fetus is clearly greater when the pregnant person stops the asthma medications. The incidence of stillbirth is high among pregnant persons who do not take medications for asthma. Note due to the lack of data from clinical trials, evidence for the medication safety and risk is derived from histories of women who have delivered children with and without birth defects.
During pregnancy, changes in the kidney, liver, and GI tract affect drug disposition. These changes may be associated with a specific trimester. During the third trimester, renal blood flow doubles, causing an increase in glomerular filtration and an accelerated clearance of drugs, such as lithium, that are eliminated by this process.
In December 2014, the FDA passed the Pregnancy and Lactation Labeling Rule (PLLR). This rule supersedes the Pregnancy Risk Categories and was enacted on June 30, 2015. By 2020, all drugs ceased using lettered categories. The PLLR requires labeling for (1) pregnancy, (2) lactation, and (3) females and males of reproductive potential. More information is available at https://www.fda.gov/drugs/labeling-information-drug-products/pregnancy-and-lactation-labeling-drugs-final-rule
From a pharmacological perspective, the greatest concern during pregnancy is exposing the fetus to a teratogen . A teratogen is any substance, organism, or physical agent that causes a permanent malformation of an embryo. The risk is dependent upon the dose of the drug and time during gestation the exposure occurred. In general, clinicians should assume all drugs cross the placenta and may reach the fetus.
7.3.1 Absorption
Hormonal changes and abdominal pressure as the uterus expands may affect absorption. Gastric tone and motility slow during pregnancy, requiring a longer time to absorb drugs. This may require a reduction in drug dose.
7.3.2 Distribution
The increase in blood volume and decrease in plasma proteins in a pregnant person causes dilution of drugs levels. In addition, increased cardiac output and plasma volume alter regional blood flow. Blood flow to the uterus, kidneys, and skin is increased. Changes in lipid levels may alter distribution, especially in the third trimester. Fat-soluble drugs readily distribute into breast milk.
7.3.3. Metabolism
Drug metabolism may increase during pregnancy, requiring a higher dose for certain drugs, such as anticonvulsants. Carbamazepine, phenytoin, and valproic acid may require higher doses during pregnancy.
7.3.4 Elimination
Blood flow through the kidneys is increased by the third trimester by over 50%. Because drug excretion rates are increased, medication doses may need to be adjusted.
7.4 Lactation
A large number of drugs are secreted into breast milk, but few are absolutely contraindicated. The potential effects of drugs on the infant are influenced by the amount of drug that reaches the infant’s circulation. Some drugs may not be absorbed or may be destroyed in the infant’s GI system. General recommendations for pharmacotherapy during lactation are: (1) postponing therapy until the infant is weaned, (2) administering the medication immediately after breastfeeding to maximize time in between feedings, (3) prescribing medications with a short half-life to reduce the drug exposure to the infant, (4) avoiding breastfeeding when the medication is at its peak, (5) selecting medications with high protein-binding properties as they are less likely to be secreted into breast milk, and (6) avoiding herbals and supplements that have not been reviewed with a provider.
References:
Fernandez, E., Perez, R., Hernandez, A., Tejada, P., Arteta, M., & Ramos, J. T. (2011). Factors and mechanisms for pharmacokinetic differences between pediatric population and adults. Pharmaceutics, 3(1), 53–72. https://doi.org/10.3390/pharmaceutics3010053
This chapter, titled Pharmacologic Issues Across the Lifespan , is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Karen Vuckovic from Introduction to Pharmacology by Carl Rosow, David Standaert, & Gary Strichartz ( MIT OpenCourseWare ) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Figures by Riley Cutler.


7.3: Additional Resources
This section moves us from the Medieval period of literature to the Renaissance. The Renaissance in England started later than the Renaissance in Italy (which is the place most people think about when they hear the term “Renaissance”). The resource reading from the Encylopedia Brittanica provides a good overview of the European Renaissance.
This module’s play, The Tragical History of Dr. Faustus, shares some roots with morality plays (such as Everyman , which we read a few weeks ago), but as you’ll see from the from the resource readings, the view of religion and its central role in society has changed. The resource reading from Dartmouth addresses this.
In terms of literary style, the play is written in blank verse, a verse style also used by Shakespeare. For many, Shakespeare is assumed to be the only English Renaissance playwright, but as you can see, others also existed.
According to one definition: “Blank verse is a literary device defined as un-rhyming verse written in iambic pentameter . In poetry and prose , it has a consistent meter with 10 syllables in each line ( pentameter ); where, unstressed syllables are followed by stressed ones and five of which are stressed but do not rhyme . It is also known as un-rhymed iambic pentameter.” (See http://literarydevices.net/blank-verse/ for further elaboration of the definition and to access the hyperlinks in the definition.)
You can test the meter by counting the syllables in a line. Normally there will be 10 in each line with every other syllable stressed (meaning accented, or pronounced with more emphasis). Occasional exceptions can happen in the line length and stress patterns, but if it varies significantly or makes a new pattern (such as suddenly using much shorter lines or switching to rhymed lines), you can assume this is done for some literary reason, such as highlighting something about a character or the plot.
Background and Interpretation
Encyclopedia Brittanica’s article on the “Renaissance: European History”
Read this overview of the Renaissance from the Encyclopedia Brittanica, keeping in mind that our class’s focus is on the Renaissance in Britain. It did not happen in a vacuum, however, so knowing more about the overall Renaissance in Europe puts the English Renaissance into perspective.
Dartmouth College’s article on “Christopher Marlowe’s Doctor Faustus”
This is a good introductory note from Dartmouth about Marlowe’s The Tragical History of Dr. Faustus (which sometimes may be referred to by the shortened title of Dr. Faustus ).
Luminarium’s article on “Christopher Marlowe (1564-1593)”
This is a detailed and useful biography of Marlowe and an overview of his works from Luminarium.org. Marlowe had a colorful life, as you will see, and some of the information in the biography may shed some light on some of the ideas in the play.
Audio/Video Files
Full length video of a stage production of Dr. Faustus from the Oxford Theatre Guild. Just as with movies, stage productions represent the director’s and other artists’ interpretation of the written play. Keep this in mind as you watch any stage production of any play. Our focus is on the content of the text; the staging, sets, casting, music, or other aspects of the performance in a production are not intrinsic to the text we are studying. Suggestion: In order to better follow along, have a copy of the play with you while you watch the video.
https://youtu.be/ILwZmZdk28Y?list=PL...O2r-QIL6pFxROi
Clip of a stage production from Shakespeare’s Globe Theater (you’ll notice the difference in costumes, sets, casting, and staging compared with the Oxford video).
Full length audio recording of the play by LibraVox/GreenAudioBooks (follow along with the print text).
- Survey of English Literature I. Authored by : Wendy Howard Gray. Provided by : Reynolds Community College. Located at : http://www.reynolds.edu/ . License : CC BY: Attribution
- Doctor Faustus - Marlowe - Shakespeare's Globe Theatre. Authored by : Kultur Films. Located at : https://youtu.be/ILwZmZdk28Y?list=PLfL2Ni1dDONNv_a_SvrO2r-QIL6pFxROi . License : All Rights Reserved . License Terms : Standard YouTube License
- Christopher Marlowe's Doctor Faustus, performed by Oxford Theatre Guild, December 2013. Authored by : Mike Taylor. Located at : https://youtu.be/V97pMacSp-0 . License : All Rights Reserved . License Terms : Standard YouTube License
- Doctor Faustus - FULL Audio Play - by Christopher Marlowe. Authored by : GreenAudioBooks. Located at : https://youtu.be/ZF3o-svLH5o . License : All Rights Reserved . License Terms : Standard YouTube License


29: Introduction to Quantum Physics
-
- 29.1: Quantization of Energy
- Energy is quantized in some systems, meaning that the system can have only certain energies and not a continuum of energies, unlike the classical case. This would be like having only certain speeds at which a car can travel because its kinetic energy can have only certain values. We also find that some forms of energy transfer take place with discrete lumps of energy.
-
- 29.2: The Photoelectric Effect
- When light strikes materials, it can eject electrons from them. This is called the photoelectric effect, meaning that light (photo) produces electricity. One common use of the photoelectric effect is in light meters, such as those that adjust the automatic iris on various types of cameras. In a similar way, another use is in solar cells, as you probably have in your calculator or have seen on a roof top or a roadside sign.
-
- 29.5: The Particle-Wave Duality of Light
- We have long known that EM radiation is a wave, capable of interference and diffraction. We now see that light can be modeled as photons, which are massless particles. This may seem contradictory, since we ordinarily deal with large objects that never act like both wave and particle. An ocean wave, for example, looks nothing like a rock. To understand small-scale phenomena, we make analogies with the large-scale phenomena we observe directly.
-
- 29.7: Probability and The Heisenberg Uncertainty Principle
- Experiments show that you will find the electron at some definite location, unlike a wave. But if you set up exactly the same situation and measure it again, you will find the electron in a different location, often far outside any experimental uncertainty in your measurement. Repeated measurements will display a statistical distribution of locations that appears wavelike.
Thumbnail: Sometimes matter behaves as a particle and sometimes a wave. Quantum physics is the study of this phenomena. Imag used with permission (Public domain; Maschen).


5.4: Multiply Polynomials
By the end of this section, you will be able to:
- Multiply monomials
- Multiply a polynomial by a monomial
- Multiply a binomial by a binomial
- Multiply a polynomial by a polynomial
- Multiply special products
- Multiply polynomial functions
Before you get started, take this readiness quiz.
Multiply Monomials
We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.
Multiply:
- \((3x^2)(−4x^3)\)
- \(\left(\frac{5}{6}x^3y\right)(12xy^2).\)
- Answer a
-
\(\begin{array} {ll} {} &{(3x^2)(−4x^3)} \\ {\text{Use the Commutative Property to rearrange the terms.}} &{3·(−4)·x^2·x^3} \\ {\text{}} &{−12x^5} \\ \end{array} \)
- Answer b
-
\(\begin{array} {ll} {} &{\left(\frac{5}{6}x^3y\right)(12xy^2)} \\ {\text{Use the Commutative Property to rearrange the terms.}} &{\frac{5}{6}·12·x^3·x·y·y^2} \\ {\text{Multiply.}} &{10x^4y^3} \\ \end{array} \)
Multiply:
- \((5y^7)(−7y^4)\)
- \((25a^4b^3)(15ab^3)\)
- Answer a
-
\(−35y^{11}\)
- Answer b
-
\(375 a^5b^6\)
Multiply:
- \((−6b^4)(−9b^5)\)
- \((23r^5s)(12r^6s^7).\)
- Answer a
-
\(54b^9\)
- Answer b
-
\(276 r^{11}s^8\)
Multiply a Polynomial by a Monomial
Multiplying a polynomial by a monomial is really just applying the Distributive Property.
Multiply:
- \(−2y(4y^2+3y−5)\)
- \(3x^3y(x^2−8xy+y^2)\).
- Answer a
-
Distribute. Multiply. - Answer b
-
\(\begin{array} {ll} {} &{3x^3y(x^2−8xy+y^2)} \\ {\text{Distribute.}} &{3x^3y⋅x^2+(3x^3y)⋅(−8xy)+(3x^3y)⋅y^2} \\ {\text{Multiply.}} &{3x^5y−24x^4y^2+3x^3y^3} \\ \end{array} \)
Multiply:
- \(-3y(5y^2+8y^{7})\)
- \(4x^2y^2(3x^2−5xy+3y^2)\)
- Answer a
-
\(−15y^3−24y^8\)
- Answer b
-
\(12x^4y^2−20x^3y^3+12x^2y^4\)
Multiply:
- \(4x^2(2x^2−3x+5)\)
- \(−6a^3b(3a^2−2ab+6b^2)\)
- Answer a
-
\(8x^4−12x^3+20x^2\)
- Answer b
-
\(−18a^5b+12a^4b^2−36a^3b^3\)
Multiply a Binomial by a Binomial
Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.
Multiply:
- \((y+5)(y+8)\)
- \((4y+3)(2y−5)\).
- Answer
-
ⓐ
Distribute \((y+8)\). Distribute again. Combine like terms. ⓑ
Distribute. Distribute again. Combine like terms.
Multiply:
- \((x+8)(x+9)\)
- \((3c+4)(5c−2)\).
- Answer a
-
\(x^2+17x+72\)
- Answer b
-
\(15c^2+14c−8\)
Multiply:
- \((5x+9)(4x+3)\)
- \((5y+2)(6y−3)\).
- Answer a
-
\(20x^2+51x+27\)
- Answer b
-
\(30y^2−3y−6\)
If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms,
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.
Let’s multiply \((x+3)(x+7)\) using both methods.
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
Multiply:
- \((y−7)(y+4)\)
- \((4x+3)(2x−5)\).
- Answer
-
a.
b.
Multiply:
- \((x−7)(x+5)\)
- \((3x+7)(5x−2)\).
- Answer
-
a. \(x^2−2x−35\)
b. \(15x^2+29x−14\)
Multiply:
- \((b−3)(b+6)\)
- \((4y+5)(4y−10)\).
- Answer
-
a. \(b^2+3b−18\)
b. \(16y^2−20y−50\)
The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.
Multiply:
- \((n^2+4)(n−1)\)
- \((3pq+5)(6pq−11)\).
- Answer
-
a.
Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms. Step 5. Combine like terms—there are none. b.
Step 1. Multiply the First terms. Step 2. Multiply the Outer terms. Step 3. Multiply the Inner terms. Step 4. Multiply the Last terms. Step 5. Combine like terms.
Multiply:
- \((x^2+6)(x−8)\)
- \((2ab+5)(4ab−4)\).
- Answer
-
a. \(x^3−8x^2+6x−48\)
b. \(8a^2b^2+12ab−20\)
Multiply:
- \((y^2+7)(y−9)\)
- \((2xy+3)(4xy−5)\).
- Answer
-
a. \(y^3−9y^2+7y−63\)
b. \(8x^2y^2+2xy−15\)
The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
Now we’ll apply this same method to multiply two binomials.
Multiply using the Vertical Method: \((3y−1)(2y−6)\).
- Answer
-
It does not matter which binomial goes on the top.
\(\begin{align*} & & &\quad\; \;\;3y - 1\\[4pt]
& & &\underline{\quad \times \;2y-6}\\[4pt]
&\text{Multiply }3y-1\text{ by }-6. & &\quad -18y + 6 & & \text{partial product}\\[4pt]
&\text{Multiply }3y-1\text{ by }2y. & & \underline{6y^2 - 2y} & & \text{partial product}\\[4pt]
&\text{Add like terms.} & & 6y^2 - 20y + 6 \end{align*} \)Notice the partial products are the same as the terms in the FOIL method.
Multiply using the Vertical Method: \((5m−7)(3m−6)\).
- Answer
-
\(15m^2−51m+42\)
Multiply using the Vertical Method: \((6b−5)(7b−3)\).
- Answer
-
\(42b^2−53b+15\)
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.
To multiply binomials, use the:
- Distributive Property
- FOIL Method
- Vertical Method
Multiply a Polynomial by a Polynomial
We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.
Multiply \((b+3)(2b^2−5b+8)\) using ⓐ the Distributive Property and ⓑ the Vertical Method.
- Answer
-
a.
Distribute. Multiply. Combine like terms. b. It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.
Multiply \((2b^2−5b+8)\) by 3.
Multiply \((2b^2−5b+8)\) by \(b\).Add like terms.
Multiply \((y−3)(y^2−5y+2)\) using ⓐ the Distributive Property and ⓑ the Vertical Method.
- Answer
-
a. \(y^3−8y^2+17y−6\)
b. \(y^3−8y^2+17y−6\)
Multiply \((x+4)(2x^2−3x+5)\) using a) the Distributive Property and b) The Vertical Method.
- Answer
-
a. and b. \(2x^3+5x^2−7x+20\)
We have now seen two methods you can use to multiply a polynomial by a polynomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.
To multiply a trinomial by a binomial, use the:
- Distributive Property
- Vertical Method
Multiply Special Products
Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and multiplying them, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for a pattern.
Look at these results. Do you see any patterns?
What about the number of terms? In each example we squared a binomial and the result was a trinomial.
\[(a+b)^2=\text{___}+\text{___}+\text{___} \nonumber\]
Now look at the first term in each result. Where did it come from?
The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!
\[(a+b)^2=a^2+\text{___}+\text{___} \nonumber\]
To get the first term of the product, square the first term.
Where did the last term come from? Look at the examples and find the pattern.
The last term is the product of the last terms, which is the square of the last term.
\[(a+b)^2=\text{___}+\text{___}+b^2 \nonumber\]
To get the last term of the product, square the last term.
Finally, look at the middle term . Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.
\[(a+b)^2=\text{___}+2ab+\text{___} \nonumber\]
\[(a−b)^2=\text{___}−2ab+\text{___} \nonumber\]
To get the middle term of the product, multiply the terms and double their product.
Putting it all together:
If a and b are real numbers,
To square a binomial, square the first term, square the last term, double their product.
Multiply: a. \((x+5)^2\) b. \((2x−3y)^2\).
- Answer
-
a.
Square the first term. Square the last term. Double their product. Simplify. b.
Use the pattern. Simplify.
Multiply: a.\((x+9)^2\) b. \((2c−d)^2\).
- Answer
-
a. \(x^2+18x+81\)
b. \(4c^2−4cd+d^2\)
Multiply: a. \((y+11)^2\) b. \((4x−5y)^2\).
- Answer
-
a. \(y^2+22y+121\)
b. \(16x^2−40xy+25y^2\)
We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.
A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form \((a−b)\), \((a+b)\).
A conjugate pair is two binomials of the form
\[(a−b), (a+b). \nonumber\]
The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.
There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier. Let’s look for the pattern by using FOIL to multiply some conjugate pairs.
What do you observe about the products?
The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.
Each first term is the product of the first terms of the binomials, and since they are identical it is the square of the first term.
\[(a+b)(a−b)=a^2−\text{___} \nonumber\]
To get the first term, square the first term.
The last term came from multiplying the last terms, the square of the last term.
\[(a+b)(a−b)=a^2−b^2 \nonumber\]
To get the last term, square the last term .
Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.
The product of conjugates is always of the form \(a^2−b^2\). This is called a difference of squares .
This leads to the pattern:
If a and b are real numbers,
The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, write it as a difference of squares.
Multiply using the product of conjugates pattern: a. \((2x+5)(2x−5)\) b. \((5m−9n)(5m+9n)\).
- Answer
-
a.
Are the binomials conjugates? It is the product of conjugates. Square the first term, 2x.2x. Square the last term, 5.5. Simplify. The product is a difference of squares. b.
This fits the pattern. Use the pattern. Simplify.
Multiply: a. \((6x+5)(6x−5)\) b. \((4p−7q)(4p+7q)\).
- Answer
-
a. \(36x^2−25\)
b. \(16p^2−49q^2\)
Multiply: a. \((2x+7)(2x−7)\) b.\((3x−y)(3x+y)\).
- Answer
-
a. \(4x^2−49\) b. \(9x^2−y^2\)
We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.
| Binomial Squares | Product of Conjugates |
|---|---|
| \((a+b)^2=a^2+2ab+b^2\) | \((a−b)(a+b)=a^2−b^2\) |
| \((a−b)^2=a^2−2ab+b^2\) | |
| • Squaring a binomial | • Multiplying conjugates |
| • Product is a trinomial | • Product is a binomial. |
| • Inner and outer terms with FOIL are the same. | • Inner and outer terms with FOIL are opposites. |
| • Middle term is double the product of the terms | • There is no middle term. |
Choose the appropriate pattern and use it to find the product:
a. \((2x−3)(2x+3)\) b. \((8x-5)^2\) c. \((6m+7)^2\) d. \((5x−6)(6x+5)\).
- Answer
-
a. \((2x−3)(2x+3)\)
These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.
Use the pattern. Simplify. b. \((8x−5)^2\)
We are asked to square a binomial. It fits the binomial squares pattern.
Use the pattern. Simplify. c. \((6m+7)^2\)
Again, we will square a binomial so we use the binomial squares pattern.
Use the pattern. Simplify. d. \((5x−6)(6x+5)\)
This product does not fit the patterns, so we will use FOIL.
\(\begin{array} {ll} {} &{(5x−6)(6x+5)} \\ {\text{Use FOIL.}} & {30x^2+25x−36x−30} \\ {\text{Simplify.}} & {30x^2−11x−30} \\ \end{array}\)
Choose the appropriate pattern and use it to find the product:
a. \((9b−2)(2b+9)\) b. \((9p−4)^2\) c. \((7y+1)^2\) d. \((4r−3)(4r+3)\).
- Answer
-
a. FOIL; \(18b^2+77b−18\)
b. Binomial Squares; \(81p^2−72p+16\)
c. Binomial Squares; \(49y^2+14y+1\)
d. Product of Conjugates; \(16r^2−9\)
Choose the appropriate pattern and use it to find the product:
a. \((6x+7)^2\) b. \((3x−4)(3x+4)\) c. \((2x−5)(5x−2)\) d. \((6n−1)^2\).
- Answer
-
a. Binomial Squares; \(36x^2+84x+49\) b. Product of Conjugates; \(9x^2−16\) c. FOIL; \(10x^2−29x+10\) d. Binomial Squares; \(36n^2−12n+1\)
Multiply Polynomial Functions
Just as polynomials can be multiplied, polynomial functions can also be multiplied.
For functions \(f(x)\) and \(g(x)\),
\[(f·g)(x)=f(x)·g(x)\]
For functions \(f(x)=x+2\) and \(g(x)=x^2−3x−4\), find:
- \((f·g)(x)\)
- \((f·g)(2)\).
- Answer
-
a.
\(\begin{array} {ll} {} &{(f·g)(x)=f(x)·g(x)} \\ {\text{Substitute for } f(x) \text{ and } g(x)} &{(f·g)(x)=(x+2)(x^2−3x−4)} \\ {\text{Multiply the polynomials.}} &{(f·g)(x)=x(x^2−3x−4)+2(x^2−3x−4)} \\ {\text{Distribute.}} &{(f·g)(x)=x3−3x^2−4x+2x^2−6x−8} \\ {\text{Combine like terms.}} &{(f·g)(x)=x3−x^2−10x−8} \\ \end{array}\)
b. In part a. we found \((f·g)(x)\) and now are asked to find \((f·g)(2)\).
\(\begin{array} {ll} {} &{(f·g)(x)=x^3−x^2−10x−8} \\ {\text{To find }(f·g)(2), \text{ substitute } x=2.} &{(f·g)(2)=2^3−2^2−10·2−8} \\ {} &{(f·g)(2)=8−4−20−8} \\ {} &{(f·g)(2)=−24} \\ \end{array}\)
For functions \(f(x)=x−5\) and \(g(x)=x^2−2x+3\), find
- \((f·g)(x)\)
- \((f·g)(2)\).
- Answer a
-
\((f·g)(x)=x^3−7x^2+13x−15\)
- Answer b
-
\((f·g)(2)=−9\)
For functions \(f(x)=x−7\) and \(g(x)=x^2+8x+4\), find
- \((f·g)(x)\)
- \((f·g)(2)\).
- Answer a
-
\((f·g)(x)=x^3+x^2−52x−28\)
- Answer a
-
\((f·g)(2)=−120\)
Access this online resource for additional instruction and practice with multiplying polynomials.
- Introduction to special products of binomials
Key Concepts
-
How to use the FOIL method to multiply two binomials.
-
Multiplying Two Binomials:
To multiply binomials, use the:
- Distributive Property
- FOIL Method
-
Multiplying a Polynomial by a Polynomial:
To multiply a trinomial by a binomial, use the:
- Distributive Property
- Vertical Method
-
Binomial Squares Pattern
If a and b are real numbers, -
Product of Conjugates Pattern
If a, b are real numbers
The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, write it as a difference of squares. -
Comparing the Special Product Patterns
Binomial Squares Product of Conjugates \((a+b)^2=a^2+2ab+b^2\) \((a−b)^2=a^2−2ab+b^2\) \((a−b)(a+b)=a^2−b^2\) • Squaring a binomial • Multiplying conjugates • Product is a trinomial • Product is a binomial. • Inner and outer terms with FOIL are the same. • Inner and outer terms with FOIL are opposites. • Middle term is double the product of the terms • There is no middle term. -
Multiplication of Polynomial Functions:
-
For functions \(f(x)\) and \(g(x)\),
\[(f⋅g)(x)=f(x)⋅g(x) \nonumber\]
-
For functions \(f(x)\) and \(g(x)\),
Glossary
- conjugate pair
- A conjugate pair is two binomials of the form \((a−b)\) and \((a+b)\). The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.


9.0: William Shakespeare- Biography
William Shakespeare (26 April 1564 (baptised) – 23 April 1616) was an English poet, playwright, and actor, widely regarded as the greatest writer in the English language and the world’s pre-eminent dramatist. He is often called England’s national poet and the “Bard of Avon”. His extant works, including some collaborations, consist of around38 plays, 154 sonnets, two long narrative poems, and a few other verses, of which the authorship of some is uncertain. His plays have been translated into every major living language and are performed more often than those of any other playwright.
Shakespeare was born and brought up in Stratford-upon-Avon. At the age of 18, he married Anne Hathaway, with whom he had three children: Susanna, and twins Hamnet and Judith. Between 1585 and 1592, he began a successful career in London as an actor, writer, and part-owner of a playing company called the Lord Chamberlain’s Men, later known as the King’s Men. He appears to have retired to Stratford around 1613 at age 49, where he died three years later. Few records of Shakespeare’s private life survive, and there has been considerable speculation about such matters as his physical appearance, sexuality, religious beliefs, and whether the works attributed to him were written by others.
Shakespeare produced most of his known work between 1589 and 1613. His early plays were mainly comedies and histories and these works remain regarded as some of the best work produced in these genres. He then wrote mainly tragedies until about 1608, including Hamlet , Othello , King Lear , and Macbeth , considered some of the finest works in the English language. In his last phase, he wrote tragicomedies, also known as romances, and collaborated with other playwrights.
Many of his plays were published in editions of varying quality and accuracy during his lifetime. In 1623, John Heminges andHenry Condell, two friends and fellow actors of Shakespeare, published the First Folio, a collected edition of his dramatic works that included all but two of the plays now recognised as Shakespeare’s. It was prefaced with a poem by Ben Jonson, in which Shakespeare is hailed, presciently, as “not of an age, but for all time”. In the 20th and 21st centuries, his work has been repeatedly adopted and rediscovered by new movements in scholarship and performance. His plays remain highly popular today and are constantly studied, performed, and reinterpreted in diverse cultural and political contexts throughout the world.
Additional information on Shakespeare’s life, work, and influence can be found here.
- William Shakespeare. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/William_Shakespeare . License : CC BY-SA: Attribution-ShareAlike
- Image of Shakespeare. Authored by : John Taylor. Located at : https://commons.wikimedia.org/wiki/File:Shakespeare.jpg . License : Public Domain: No Known Copyright