16.3: The Mental Status Exam
By the end of this section, you will be able to:
- Describe the relationship of mental status exam results to cerebral functions
- Explain the categorization of regions of the cortex based on anatomy and physiology
- Differentiate between primary, association, and integration areas of the cerebral cortex
- Provide examples of localization of function related to the cerebral cortex
In the clinical setting, the set of subtests known as the mental status exam helps us understand the relationship of the brain to the body. Ultimately, this is accomplished by assessing behavior. Tremors related to intentional movements, incoordination, or the neglect of one side of the body can be indicative of failures of the connections of the cerebrum either within the hemispheres, or from the cerebrum to other portions of the nervous system. There is no strict test for what the cerebrum does alone, but rather in what it does through its control of the rest of the CNS, the peripheral nervous system (PNS), and the musculature.
Sometimes eliciting a behavior is as simple as asking a question. Asking a patient to state their name is not only to verify that the file folder in a health care provider’s hands is the correct one, but also to be sure that the patient is aware, oriented, and capable of interacting with another person. If the answer to “What is your name?” is “Santa Claus,” the person may have a problem understanding reality. If the person just stares at the examiner with a confused look on their face, the person may have a problem understanding or producing speech.
Functions of the Cerebral Cortex
The cerebrum is the seat of many of the higher mental functions, such as memory and learning, language, and conscious perception, which are the subjects of subtests of the mental status exam. The cerebral cortex is the thin layer of gray matter on the outside of the cerebrum. On average, it is approximately 2.55 mm thick and highly folded to fit within the limited space of the cranial vault. These higher functions are distributed across various regions of the cortex, and specific locations can be said to be responsible for particular functions. There is a limited set of regions, for example, that are involved in language function, and they can be subdivided on the basis of the particular part of language function that each governs.
The basis for parceling out areas of the cortex and attributing them to various functions has its root in pure anatomical underpinnings. The German neurologist and histologist Korbinian Brodmann, who made a careful study of the cytoarchitecture of the cerebrum around the turn of the nineteenth century, described approximately 50 regions of the cortex that differed enough from each other to be considered separate areas (Figure \(\PageIndex{1}\)). Brodmann made preparations of many different regions of the cerebral cortex to view with a microscope. He compared the size, shape, and number of neurons to find anatomical differences in the various parts of the cerebral cortex. Continued investigation into these anatomical areas over the subsequent 100 or more years has demonstrated a strong correlation between the structures and the functions attributed to those structures. For example, the first three areas in Brodmann’s list—which are in the postcentral gyrus—compose the primary somatosensory cortex. Within this area, finer separation can be made on the basis of the concept of the sensory homunculus, as well as the different submodalities of somatosensation such as touch, vibration, pain, temperature, or proprioception. Today, we more frequently refer to these regions by their function (i.e., primary sensory cortex) than by the number Brodmann assigned to them, but in some situations the use of Brodmann numbers persists.
Area 17, as Brodmann described it, is also known as the primary visual cortex. Adjacent to that are areas 18 and 19, which constitute subsequent regions of visual processing. Area 22 is the primary auditory cortex, and it is followed by area 23, which further processes auditory information. Area 4 is the primary motor cortex in the precentral gyrus, whereas area 6 is the premotor cortex. These areas suggest some specialization within the cortex for functional processing, both in sensory and motor regions. The fact that Brodmann’s areas correlate so closely to functional localization in the cerebral cortex demonstrates the strong link between structure and function in these regions.
Areas 1, 2, 3, 4, 17, and 22 are each described as primary cortical areas. The adjoining regions are each referred to as association areas. Primary areas are where sensory information is initially received from the thalamus for conscious perception, or—in the case of the primary motor cortex—where descending commands are sent down to the brain stem or spinal cord to execute movements (Figure \(\PageIndex{2}\)).
A number of other regions, which extend beyond these primary or association areas of the cortex, are referred to as integrative areas. These areas are found in the spaces between the domains for particular sensory or motor functions, and they integrate multisensory information, or process sensory or motor information in more complex ways. Consider, for example, the posterior parietal cortex that lies between the somatosensory cortex and visual cortex regions. This has been ascribed to the coordination of visual and motor functions, such as reaching to pick up a glass. The somatosensory function that would be part of this is the proprioceptive feedback from moving the arm and hand. The weight of the glass, based on what it contains, will influence how those movements are executed.
Cognitive Abilities
Assessment of cerebral functions is directed at cognitive abilities. The abilities assessed through the mental status exam can be separated into four groups: orientation and memory, language and speech, sensorium, and judgment and abstract reasoning.
Orientation and Memory
Orientation is the patient’s awareness of their immediate circumstances. It is awareness of time, not in terms of the clock, but of the date and what is occurring around the patient. It is awareness of place, such that a patient should know where they are and why. It is also awareness of who the patient is—recognizing personal identity and being able to relate that to the examiner. The initial tests of orientation are based on the questions, “Do you know what the date is?” or “Do you know where you are?” or “What is your name?” Further understanding of a patient’s awareness of orientation can come from questions that address remote memory, such as “Who is the President of the United States?”, or asking what happened on a specific date.
There are also specific tasks to address memory. One is the three-word recall test. The patient is given three words to recall, such as book, clock, and shovel. After a short interval, during which other parts of the interview continue, the patient is asked to recall the three words. Other tasks that assess memory—aside from those related to orientation—have the patient recite the months of the year in reverse order to avoid the overlearned sequence and focus on the memory of the months in an order, or to spell common words backwards, or to recite a list of numbers back.
Memory is largely a function of the temporal lobe, along with structures beneath the cerebral cortex such as the hippocampus and the amygdala. The storage of memory requires these structures of the medial temporal lobe. A famous case of a man who had both medial temporal lobes removed to treat intractable epilepsy provided insight into the relationship between the structures of the brain and the function of memory.
Henry Molaison, who was referred to as patient HM when he was alive, had epilepsy localized to both of his medial temporal lobes. In 1953, a bilateral lobectomy was performed that alleviated the epilepsy but resulted in the inability for HM to form new memories—a condition called anterograde amnesia . HM was able to recall most events from before his surgery, although there was a partial loss of earlier memories, which is referred to as retrograde amnesia . HM became the subject of extensive studies into how memory works. What he was unable to do was form new memories of what happened to him, what are now called episodic memory . Episodic memory is autobiographical in nature, such as remembering riding a bicycle as a child around the neighborhood, as opposed to the procedural memory of how to ride a bike. HM also retained his short-term memory , such as what is tested by the three-word task described above. After a brief period, those memories would dissipate or decay and not be stored in the long-term because the medial temporal lobe structures were removed.
The difference in short-term, procedural, and episodic memory, as evidenced by patient HM, suggests that there are different parts of the brain responsible for those functions. The long-term storage of episodic memory requires the hippocampus and related medial temporal structures, and the location of those memories is in the multimodal integration areas of the cerebral cortex. However, short-term memory—also called working or active memory—is localized to the prefrontal lobe. Because patient HM had only lost his medial temporal lobe—and lost very little of his previous memories, and did not lose the ability to form new short-term memories—it was concluded that the function of the hippocampus, and adjacent structures in the medial temporal lobe, is to move (or consolidate) short-term memories (in the pre-frontal lobe) to long-term memory (in the temporal lobe).
The prefrontal cortex can also be tested for the ability to organize information. In one subtest of the mental status exam called set generation, the patient is asked to generate a list of words that all start with the same letter, but not to include proper nouns or names. The expectation is that a person can generate such a list of at least 10 words within 1 minute. Many people can likely do this much more quickly, but the standard separates the accepted normal from those with compromised prefrontal cortices.
Interactive Link
Read this article to learn about a young man who texts his fiancée in a panic as he finds that he is having trouble remembering things. At the hospital, a neurologist administers the mental status exam, which is mostly normal except for the three-word recall test. The young man could not recall them even 30 seconds after hearing them and repeating them back to the doctor. An undiscovered mass in the mediastinum region was found to be Hodgkin’s lymphoma, a type of cancer that affects the immune system and likely caused antibodies to attack the nervous system. The patient eventually regained his ability to remember, though the events in the hospital were always elusive. Considering that the effects on memory were temporary, but resulted in the loss of the specific events of the hospital stay, what regions of the brain were likely to have been affected by the antibodies and what type of memory does that represent?
Language and Speech
Language is, arguably, a very human aspect of neurological function. There are certainly strides being made in understanding communication in other species, but much of what makes the human experience seemingly unique is its basis in language. Any understanding of our species is necessarily reflective, as suggested by the question “What am I?” And the fundamental answer to this question is suggested by the famous quote by René Descartes: “Cogito Ergo Sum” (translated from Latin as “I think, therefore I am”).  It is a confusing topic to delve into, but language is certainly at the core of what it means to be self-aware.
The neurological exam has two specific subtests that address language. One measures the ability of the patient to understand language by asking them to follow a set of instructions to perform an action, such as “touch your right finger to your left elbow and then to your right knee.” Another subtest assesses the fluency and coherency of language by having the patient generate descriptions of objects or scenes depicted in drawings, and by reciting sentences or explaining a written passage. Language, however, is important in so many ways in the neurological exam. The patient needs to know what to do, whether it is as simple as explaining how the knee-jerk reflex is going to be performed, or asking a question such as “What is your name?” Often, language deficits can be determined without specific subtests; if a person cannot reply to a question properly, there may be a problem with the reception of language.
An important example of multimodal integrative areas is associated with language function (Figure \(\PageIndex{3}\)). Adjacent to the auditory association cortex, at the end of the lateral sulcus just anterior to the visual cortex, is Wernicke’s area . In the lateral aspect of the frontal lobe, just anterior to the region of the motor cortex associated with the head and neck, is Broca’s area. Both regions were originally described on the basis of losses of speech and language, which is called aphasia . The aphasia associated with Broca’s area is known as an expressive aphasia , which means that speech production is compromised. This type of aphasia is often described as non-fluency because the ability to say some words leads to broken or halting speech. Grammar can also appear to be lost. The aphasia associated with Wernicke’s area is known as a receptive aphasia , which is not a loss of speech production, but a loss of understanding of content. Patients, after recovering from acute forms of this aphasia, report not being able to understand what is said to them or what they are saying themselves, but they often cannot keep from talking.
The two regions are connected by white matter tracts that run between the posterior temporal lobe and the lateral aspect of the frontal lobe. Conduction aphasia associated with damage to this connection refers to the problem of connecting the understanding of language to the production of speech. This is a very rare condition, but is likely to present as an inability to faithfully repeat spoken language.
Sensorium
Those parts of the brain involved in the reception and interpretation of sensory stimuli are referred to collectively as the sensorium. The cerebral cortex has several regions that are necessary for sensory perception. From the primary cortical areas of the somatosensory, visual, auditory, and gustatory senses to the association areas that process information in these modalities, the cerebral cortex is the seat of conscious sensory perception. In contrast, sensory information can also be processed by deeper brain regions, which we may vaguely describe as subconscious—for instance, we are not constantly aware of the proprioceptive information that the cerebellum uses to maintain balance. Several of the subtests can reveal activity associated with these sensory modalities, such as being able to hear a question or see a picture. Two subtests assess specific functions of these cortical areas.
The first is praxis , a practical exercise in which the patient performs a task completely on the basis of verbal description without any demonstration from the examiner. For example, the patient can be told to take their left hand and place it palm down on their left thigh, then flip it over so the palm is facing up, and then repeat this four times. The examiner describes the activity without any movements on their part to suggest how the movements are to be performed. The patient needs to understand the instructions, transform them into movements, and use sensory feedback, both visual and proprioceptive, to perform the movements correctly.
The second subtest for sensory perception is gnosis , which involves two tasks. The first task, known as stereognosis , involves the naming of objects strictly on the basis of the somatosensory information that comes from manipulating them. The patient keeps their eyes closed and is given a common object, such as a coin, that they have to identify. The patient should be able to indicate the particular type of coin, such as a dime versus a penny, or a nickel versus a quarter, on the basis of the sensory cues involved. For example, the size, thickness, or weight of the coin may be an indication, or to differentiate the pairs of coins suggested here, the smooth or corrugated edge of the coin will correspond to the particular denomination. The second task, graphesthesia , is to recognize numbers or letters written on the palm of the hand with a dull pointer, such as a pen cap.
Praxis and gnosis are related to the conscious perception and cortical processing of sensory information. Being able to transform verbal commands into a sequence of motor responses, or to manipulate and recognize a common object and associate it with a name for that object. Both subtests have language components because language function is integral to these functions. The relationship between the words that describe actions, or the nouns that represent objects, and the cerebral location of these concepts is suggested to be localized to particular cortical areas. Certain aphasias can be characterized by a deficit of verbs or nouns, known as V impairment or N impairment, or may be classified as V–N dissociation. Patients have difficulty using one type of word over the other. To describe what is happening in a photograph as part of the expressive language subtest, a patient will use active- or image-based language. The lack of one or the other of these components of language can relate to the ability to use verbs or nouns. Damage to the region at which the frontal and temporal lobes meet, including the region known as the insula, is associated with V impairment; damage to the middle and inferior temporal lobe is associated with N impairment.
Judgment and Abstract Reasoning
Planning and producing responses requires an ability to make sense of the world around us. Making judgments and reasoning in the abstract are necessary to produce movements as part of larger responses. For example, when your alarm goes off, do you hit the snooze button or jump out of bed? Is 10 extra minutes in bed worth the extra rush to get ready for your day? Will hitting the snooze button multiple times lead to feeling more rested or result in a panic as you run late? How you mentally process these questions can affect your whole day.
The prefrontal cortex is responsible for the functions responsible for planning and making decisions. In the mental status exam, the subtest that assesses judgment and reasoning is directed at three aspects of frontal lobe function. First, the examiner asks questions about problem solving, such as “If you see a house on fire, what would you do?” The patient is also asked to interpret common proverbs, such as “Don’t look a gift horse in the mouth.” Additionally, pairs of words are compared for similarities, such as apple and orange, or lamp and cabinet.
The prefrontal cortex is composed of the regions of the frontal lobe that are not directly related to specific motor functions. The most posterior region of the frontal lobe, the precentral gyrus, is the primary motor cortex. Anterior to that are the premotor cortex, Broca’s area, and the frontal eye fields, which are all related to planning certain types of movements. Anterior to what could be described as motor association areas are the regions of the prefrontal cortex. They are the regions in which judgment, abstract reasoning, and working memory are localized. The antecedents to planning certain movements are judging whether those movements should be made, as in the example of deciding whether to hit the snooze button.
To an extent, the prefrontal cortex may be related to personality. The neurological exam does not necessarily assess personality, but it can be within the realm of neurology or psychiatry. A clinical situation that suggests this link between the prefrontal cortex and personality comes from the story of Phineas Gage, the railroad worker from the mid-1800s who had a metal spike impale his prefrontal cortex. There are suggestions that the steel rod led to changes in his personality. A man who was a quiet, dependable railroad worker became a raucous, irritable drunkard. Later anecdotal evidence from his life suggests that he was able to support himself, although he had to relocate and take on a different career as a stagecoach driver.
A psychiatric practice to deal with various disorders was the prefrontal lobotomy. This procedure was common in the 1940s and early 1950s, until antipsychotic drugs became available. The connections between the prefrontal cortex and other regions of the brain were severed. The disorders associated with this procedure included some aspects of what are now referred to as personality disorders, but also included mood disorders and psychoses. Depictions of lobotomies in popular media suggest a link between cutting the white matter of the prefrontal cortex and changes in a patient’s mood and personality, though this correlation is not well understood.
Everyday Connection
Left Brain, Right Brain
Popular media often refer to right-brained and left-brained people, as if the brain were two independent halves that work differently for different people. This is a popular misinterpretation of an important neurological phenomenon. As an extreme measure to deal with a debilitating condition, the corpus callosum may be sectioned to overcome intractable epilepsy. When the connections between the two cerebral hemispheres are cut, interesting effects can be observed.
If a person with an intact corpus callosum is asked to put their hands in their pockets and describe what is there on the basis of what their hands feel, they might say that they have keys in their right pocket and loose change in the left. They may even be able to count the coins in their pocket and say if they can afford to buy a candy bar from the vending machine.  They will only put their right hand in their pocket and say they have keys there. They will not even move their left hand, much less report that there is loose change in the left pocket.
The reason for this is that the language functions of the cerebral cortex are localized to the left hemisphere in 95 percent of the population. Additionally, the left hemisphere is connected to the right side of the body through the corticospinal tract and the ascending tracts of the spinal cord. Motor commands from the precentral gyrus control the opposite side of the body, whereas sensory information processed by the postcentral gyrus is received from the opposite side of the body. For a verbal command to initiate movement of the right arm and hand, the left side of the brain needs to be connected by the corpus callosum. Language is processed in the left side of the brain and directly influences the left brain and right arm motor functions, but is sent to influence the right brain and left arm motor functions through the corpus callosum. Likewise, the left-handed sensory perception of what is in the left pocket travels across the corpus callosum from the right brain, so no verbal report on those contents would be possible if the hand happened to be in the pocket.
Interactive Link
Watch the video titled “The Man With Two Brains” to see the neuroscientist Michael Gazzaniga introduce a patient he has worked with for years who has had his corpus callosum cut, separating his two cerebral hemispheres. A few tests are run to demonstrate how this manifests in tests of cerebral function. Unlike normal people, this patient can perform two independent tasks at the same time because the lines of communication between the right and left sides of his brain have been removed. Whereas a person with an intact corpus callosum cannot overcome the dominance of one hemisphere over the other, this patient can. If the left cerebral hemisphere is dominant in the majority of people, why would right-handedness be most common?


11.1: Introduction
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- Jade Miller
- University of California Press
Just as Hollywood production frequently departs the greater Los Angeles area for less expensive shooting locations worldwide, Hollywood studios have also expanded their interests globally, investing in everything from Bollywood studios to telenovela-producing corporations. 1 In this sense, the Los Angeles-based film and television industry is indeed a multilevel “global Hollywood,” as Miller and his colleagues convincingly illustrate in their so-titled book. 2 Accordingly, the individuals who make up global Hollywood’s workforce are both geographically diverse (in “runaway production” locations from New Orleans to Prague) and industrially diverse, working on domestic film and television productions as well as major international projects, all of which increasingly rely on Hollywood capital. Global Hollywood’s workforce, then, may include the labor on a Universal Studios movie shooting in Prague, labor on a Bollywood movie that is partly funded by a subsidiary of Sony Pictures, and labor on a telenovela produced by a company with ownership links to NBC. While the dynamics of employment differ among locations, as other chapters in this book illuminate, interest and investment from Hollywood require significant transparency in distribution and management at the very least. This generally means a corporate structure with a few behemoth companies dominating production and neoliberal governance: local and national policymakers and large private companies working in tandem to establish the sort of regulations, policies, and practices, including reliable copyright and contract enforcement, that are attractive to foreign direct investment (FDI) and formal domestic bank investment.
Entertainment production worldwide also exists outside Global Hollywood’s networks. In this chapter, I critically examine the relationships among labor, distribution, informality, and power in one such industry: the massively popular southern Nigerian movie industry known as Nollywood. 3 Its productions dominate screens and mediascapes across sub-Saharan Africa and throughout the global African diaspora, though exact numbers about its production output and income are challenging to discern. 4 Counting and demonstrating sales are not just relevant to demonstrating an industry’s importance for academic study or popular journalistic pieces. Rather, opacity (and the accompanying general inability to codify formal sales figures) is a defining part of Nollywood’s structure and strength, 5 shaping nearly every part of the industry’s day-to-day operations and practices. In particular, this opacity reinforces the power of the film distributors known as “marketers,” who leverage their gray-market knowledge to control the Nollywood marketplace. While an industry predicated on personal relationships may appear to risk breeding disorder, Nollywood is in fact quite organized, a result of the marketers’ self-governing practices and the industry’s guild-based infrastructure. Consequently, Nollywood remains largely disconnected from formal global networks of labor organization, financing, and distribution, 6 but is nevertheless a nexus point for its own set of global flows and linkages.
Using a series of onsite observations and interviews with practitioners in the Lagos-based industry, my analysis reveals how global concerns about the precarious nature of local labor are shaped in this context by the particular brand of informality that characterizes Nollywood. If we take precarity and informality to be linked, we can see Nollywood as a particularly informal industry with a particularly precarious workforce, marked by limited recourse for labor grievances. I assess Nollywood’s informality as a phenomenon forged out of a very specific place: Lagos, a rapidly growing, often overflowing megacity and an alternative media capital, a hub for global flows and connections that utilizes few of the formal dominant networks that mark Global Hollywood. 7 In this way, this chapter grounds the reality of local media labor in the specificities of the actual places where that labor works. In short, the structure of Nollywood reflects the specific architecture and shape of Lagos.
I would also like to be specific about what I mean when I discuss industrial informality in Nollywood. Film industries from Hollywood to Bollywood can be said to feature informal elements at many levels of production, especially in relation to labor practices like recruitment. And Nollywood features some formal elements in its production inputs and distribution outlets. 8 The distinction between formality and informality, then, is not a dichotomy. Rather, it is a continuum with no industry falling fully at either extreme. Additionally, the question of what exactly informality is and whether it should be celebrated has been subject to much scholarly debate, particularly in the context of media distribution studies, and Nollywood has been at the core of many of these arguments. 9 There has been concern over the potential to exoticize and “Orientalize” Nollywood via an overabundance of focus on the informality in the industry. 10
Accordingly, to demystify the discussion of Nollywood’s informality, I wish to be very clear about what exactly I mean by informality in Nollywood, and the ways in which it is a conscious choice in a global power play. This study understands the basic intersecting constituents of Nollywood’s industrial informality to be 1) not documenting sales or most other distribution figures in any publicly accessible/scrutinizable fashion, 2) not utilizing legal contracts for employment or other business relationships, 3) not using agents or other formal inputs such as accredited schools for talent recruitment, 4) not pursuing copyright violations via legal frameworks, and 5) privileging undocumented financing and distribution networks and spurning alternatives. It may be noted that four of these five elements begin with the word not . This is because our understanding of informality as an industrial feature worthy of mention exists only because of the existence of formality in these areas in other industries. While there may be some level of informality in other global movie or television industries, the dominance and intersection of these elements in Nollywood’s day-to-day functioning render the industry predominantly informal as opposed to the fragmented informality that characterizes the global media industries that compete for Hollywood’s production, coproductions, partnerships, or investments on the international stage. And point 5 underlines the conscious and active choice by Nollywood’s marketers to utilize informality as a means to maintain power and thwart challengers.


3.3: Integration by Parts
If the integrand consists of two factors, and you know the antiderivative of one of the factors, you can integrate by parts by shifting the derivative onto the other factor: \[\int_a^b dx \; f(x) \, \frac{dg}{dx} \;=\; \Big[\,f(x)\, g(x)\,\Big]_a^b - \int_a^b \frac{df}{dx}\, g(x).\] The first term on the right hand side is a constant denoting \([f(a)g(a) - f(b)g(b)]\) . Hopefully, the integral in the second term is easier to solve than the original integral.
Judicious use of integration by parts is a key step for solving many integrals. For example, consider \[\int_a^b dx\; x \, e^{\gamma x}.\] The integrand consists of two factors, \(x\) and \(e^{\gamma x}\) ; we happen to know the antiderivative of both factors. Integrating by parts lets us replace one of these factors with its antiderivative, while applying an additional derivative on the other factor. The smart thing to do is to apply the derivative on the \(x\) factor, and the antiderivative on the \(e^{\gamma x}\) : \[\begin{align} \int_a^b dx\; x\, e^{\gamma x} \;&=\; \left[\;x\, \frac{e^{\gamma x}}{\gamma}\, \right]_a^b - \int_a^b dx\; \frac{e^{\gamma x}}{\gamma} \\ &=\; \left[\;x\, \frac{e^{\gamma x}}{\gamma} - \frac{e^{\gamma x}}{\gamma^2} \,\right]_a^b.\end{align}\] Whenever we finish doing an integral, it is good practice to double-check the result by making sure the dimensions match up. Note that \(\gamma\) has units of inverse \(x\) , so the integral on the left-hand side has units of \(x^2\) . The solution on the right hand side has two terms, with units \(x/\gamma\) and \(1/\gamma^2\) ; both of these are equivalent to units of \(x^2\) , which is what we need!


4.2: Osmolality
By the end of this section, you should be able to:
- 4.2.1 Describe osmolality.
- 4.2.2 Define tonicity.
Osmolality describes the concentration of solutes in a solution. It is defined as the number of solute particles per 1 kg of solvent, also known as an osmole , or a unit of osmotic concentration. Osmolarity refers to the number of solutes per 1 liter of solvent and provides valuable insight into how solutions will behave under different conditions, enabling the manipulation of their properties to achieve optimal changes within the body.
Osmolality
Osmolality is a measure of the concentration of solutes, such as sodium, glucose, and urea, in a solution; specifically, it refers to the number of particles per kilogram of solvent. It is commonly used to assess the osmotic pressure of body fluids, such as blood or urine, and can provide important information about the body’s fluid and electrolyte balance including:
- Hyperosmolality refers to a condition in which the concentration of solutes in a solution, such as blood or urine, is higher than normal. This can be caused by various factors, such as dehydration, diabetes, and certain drugs. Hyperosmolality may have serious consequences, such as impairment of the cardiovascular and circulatory system, as well as neurologic signs and symptoms such as irritability, seizures, and, in severe cases, coma.
- Hypoosmolality refers to a state in which the concentration of solutes in a solution is lower than normal. This can be caused by a variety of factors, including excessive water intake, certain drugs, disorders affecting the kidneys, and hormonal dysregulation of water balance. Hypoosmolality can have serious consequences, including neurologic signs and symptoms and even coma if not corrected.
- Iso-osmolality occurs when the concentration of solute particles per unit of volume in a solution is equal to a standard reference value.
Tonicity
Although osmolality and tonicity are related, they are not the same. Tonicity refers to the ability of a solution to cause a cell to gain or lose water and is relative to the cytoplasm of a cell (Khan & Farhana, 2023; Maldonado & Mohiuddin, 2022). A change in tonicity can have significant effects on cell function and overall fluid and electrolyte balance within the body, such as:
- Hypertonicity occurs when the tonicity of a solution is greater than that of the cytoplasm of a cell. This state can cause water to move out of the cell, potentially leading to cell shrinkage and changes in cell function. Hypertonicity can be caused by a variety of factors, including dehydration and increased solute concentration in a solution, such as with high glucose levels in the blood.
- Hypotonicity is related to hypoosmolality. It is a state in which the tonicity of a solution is less than that of the cytoplasm of a cell. Water can then move into the cell, potentially leading to swelling and other changes in cell function. Hypotonicity can be caused by a variety of factors, including excessive water intake and certain medical conditions.
- Isotonicity exists when the tonicity of a solution is equal to that of the cytoplasm of a cell. This means the concentration of solutes in the solution is balanced with the concentration of solutes inside the cell, resulting in no net movement of water into or out of the cell.


1.8: Chapter 1- Comments and Solutions
1.
(a) Assuming that the \(2 \times\), \(3 \times\), \(4 \times\), and \(5 \times\) tables are known, and that one has understood that the order of the factors does not matter, all that remains to be learned is \(6 \times 6\), \(6 \times 7\), \(6 \times 8\), \(6 \times 9\); \(7 \times 7\), \(7 \times 8\), \(7 \times 9\); \(8 \times 8\), \(8 \times 9\); and \(9 \times 9\).
(b) (i) \((4 \times 10^{-3}) \times (2 \times 10^{-2}) = 8 \times 10^{-5} = 0.00008\)
(ii) \((8 \times 10^{-4}) \times (7 \times 10^{-2}) = 56 \times 10^{-6} = 0.000056\)
(iii) \((7 \times 10^{-3}) \times (12 \times 10^{-2}) = 84 \times 10^{-5} = 0.00084\)
(iv) \((7 \times 10^{-3}) \times (12 \times 10^{-2}) = 84 \times 10^{-5} = 0.00084\)
(v) \((8 \times 10^{-2})^{2} = 64 \times 10^{-4} = 0.0064\)
2.
(a) (i) \(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144; 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961\)
(ii) \(1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331\)
(iii) \(1 = 2^{0}, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024\)
(b) (i) \(31 (31^{2} = 96^{1}, 32^{2} = 21^{0} = 1024)\)
(ii) \(99 (100^{2} = 10^{4} = 10000)\)
(iii) \(316 (310^{2} = 96100 \lt 100000 \lt 320^{2}\); so look more carefully between \(310\) and \(320\))
(c) (i) \(9 (9^{3} = 729 \lt 10^{3} = 1000)\)
(ii) \(21 (20^{3} = 8000, 22^{3} = 10648)\)
(iii) \(99 (100^{3} = 10^{6} = 1000000)\)
(d) (i) Those powers \(2^{e}\) of the form \(2^{2n}\) for which the exponent \(e\) is a multiple of \(2\): i.e. \(e \equiv 0\) (mod \(2\)).
(ii) Those powers \(2^{e}\) of the form \(2^{3n}\) for which the exponent \(e\) is a multiple of \(3\): i.e. \(e \equiv 0\) (mod \(3\)).
(e) \(64 = 2^{6} = 4^{3} = 8^{2}. 729 = 3^{6} = 9^{3} = 27^{2}\).
3.
(a) (i) \(7\); (ii) \(12\); (iii) \(21\); (iv) \(13\); (v) \(14\); (vi) \(31\); (vii) \(10^{31} = 31\)
(b) (i) \(2\sqrt{2}\); (ii) \(2\sqrt{3}\); (iii) \(5\sqrt{2}\); (iv) \(7\sqrt{3}\); (v) \(12\sqrt{2}\); (vi) \(21\sqrt{2}\)
(c) (i) \(\angle ABC = 108^{\circ}\). \(\triangle BAC\) is isosceles \((BA = BC)\), so \(\angle BAC = \angle BCA = 36^{\circ}\).
\(\therefore \angle{CAE} = \angle{BAE} - \angle{BAC} = 72^{\circ} = 180^{\circ} - \angle{AED}\).
So \(AC\) is parallel to \(ED\) (since corresponding angles add to \(180^{\circ}\)).
(ii) \(AX\) is parallel to \(ED\); similarly \(DX\) is parallel to \(EA\). Hence \(AXDE\) is a parallelogram, with \(EA = ED\).
(iii) The two triangles are both isosceles and \(\angle AXD = \angle CXB = 108^{\circ}\) (vertically opposite angles). Hence \(\angle XAD = \angle XCB = 36^{\circ}\), and \( \angle XDA = \angle XBC = 36^{\circ}\).
(iv) \(AD : CB = DX : BX\), so \(x : 1 = 1 : (x - 1)\); hence \(x^{2} - x - 1 = 0\) and \(x = \frac{1 + \sqrt{5}}{2} - \text{the} \ Golden \ Ratio\) usually denoted by the Greek letter \(\tau\) (tau) , with approximate value \(0.6180339887...\)
(v) \(BX = x -1 = \frac{\sqrt{5} -1}{2} \ (= \frac{1}{\tau} = \tau -1)\) with approximate value \(0.6180339887...\)
(vi) We may either check that corresponding angles are equal in pairs \((36^{\circ}, 72^{\circ}, 72^{\circ})\), or that corresponding sides are in the same ratio \(x : 1 = 1 : (x -1)\).
(vii) \(\cos{36^{\circ}} = \frac{\sqrt{5} + 1}{4}; \cos{72^{\circ}} = \frac{\sqrt{5} -1}{4}\) (drop perpendiculars from \(D\) to \(AB\) and from \(X\) to \(BC\); or use the Cosine Rule).
(viii) \(\sin^{2}{36^{\circ}} + \cos^{2}{36^{\circ}} = 1: \sin{36^{\circ}} = \frac{\sqrt{10 -2\sqrt{5}}}{4}; \ \sin{72^{\circ}} = \frac{\sqrt{10 + 2\sqrt{5}}}{4}\)
Note : The Golden Ratio crops up in many unexpected places (including the regular pentagon, and the Fibonacci numbers). Unfortunately much that is written about its ubiquity is pure invention. One of the better popular treatments, that highlights the number’s significance, while taking a sober view of spurious claims, is the book The Golden Ratio by Mario Livio.
4.
(a) \(12345 = 5 \times 2469 = 3 \times 5 \times 823\). But is \(823\) a prime number? It is easy to check that \(823\) is not divisible by \(2\), or \(3\), or \(5\), or \(7\), or \(11\). The Square Root Test (displayed below) tells us that it is only necessary to check four more potential prime factors.
Square Root Test: If \(N = a \times b\) with \(a \le b\), then \(a \times a \le a \times b = N\), so the smaller factor \(a \le \sqrt{N}\).
Hence, if \(N (= 823\) say) is not prime, its smallest factor \(\gt 1\) is at most equal to \(\sqrt{N} (= \sqrt{823} \lt 29)\). Checking \(a = 13, 17, 19, 23\) shows that the required prime factorisation is \(12345 = 3 \times 5 \times 823\).
(b) There are \(25 \ (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97)\).
Notes:
(i) For small primes, mental arithmetic should suffice. But one should also be aware of the general Sieve of Eratosthenes (a Greek polymath from the 3rd century BC). Start with the integers \(1–100\) arranged in ten columns, and proceed as follows:
\(\begin{array}{cccccccccc}{1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10} \\ {11} & {12} & {13} & {14} & {15} & {16} & {17} & {18} & {19} & {20} \\ {21} & {22} & {23} & {24} & {25} & {26} & {27} & {28} & {29} & {30} \\ {31} & {32} & {33} & {34} & {35} & {36} & {37} & {38} & {39} & {40} \\ {41} & {42} & {43} & {44} & {45} & {46} & {47} & {48} & {49} & {50} \\ {61} & {62} & {63} & {64} & {65} & {66} & {67} & {68} & {69} & {50} \\ {71} & {72} & {73} & {74} & {75} & {76} & {77} & {78} & {79} & {80} \\ {81} & {82} & {83} & {84} & {85} & {86} & {87} & {88} & {89} & {90} \\ {91} & {92} & {93} & {94} & {95} & {96} & {97} & {98} & {99} & {100}\end{array}\)
Delete \(1\) (which is not a prime: see (ii) below).
Circle the first undeleted integer; remove all other multiples of \(2\).
Circle the first undeleted integer; remove all other multiples of \(3\).
Circle the first undeleted integer; remove all other multiples of \(5\).
Circle the first undeleted integer; remove all other multiples of \(7\).
All remaining undeleted integers \(\lt 100\) must be prime (by the Square Root Test (see part (a)).
(ii) The multiplicative structure of integers is surprisingly subtle. The first thing to notice is that \(1\) has a special role, in that it is the multiplicative identity : for each integer \(n\), we have \(1 \times n = n\). Hence \(1\) is “multiplicatively neutral” – it has no effect.
The “multiplicative building blocks” for integers are the prime numbers : every integer \(\gt 1\) can be broken down, or factorised as a product of prime numbers, in exactly one way. The integer \(1\) has no proper factors, and has no role to play in breaking down larger integers by factorisation . So \(1\) is not a prime. (If we made the mistake of counting \(1\) as a prime number, then we would have to make all sorts of silly exceptions – for example, to allow for the fact that \(2 = 2 \times 1 = 2 \times 1 \times 1 = ...\), so \(2\) could then be factorised in infinitely many ways.)
(iii) Notice that \(91 = 7 \times 13\) is not a prime; so there is exactly one prime in the \(90\)s – namely \(97\).
How many primes are there in the next run of \(10\) (from \(100–109\))?
How many primes are there from \(190–199\)? How many from \(200–210\)?
(c) (i) \(3 = 2^{2} \times 1\).
(ii) Many students struggle with this, and may suggest \(143\), or \(323\), or even \(63\). The problem conceals a (very thinly) disguised message:
One cannot calculate with words.
To make use of mathematics, we must routinely translate words into symbols. As soon as “one less than a square” is translated into symbols, bells should begin to ring. For you know that \(x^{2} - 1 = (x - 1)(x + 1)\), so \(x^{2} - 1\) can only be prime if the smaller factor \((x - 1)\) is equal to \(1\).
5.
(a)(i) If we try to avoid such a “relatively prime pair”, then we must not choose any of \(11, 13, 17, 19\) (since they are prime, and have no multiples in the given range). So we are forced to choose the other six integers: \(10, 12, 14, 15, 16, 18\) – and there are then exactly two pairs which are relatively prime, namely \(14, 15\) and \(15, 16\).
(ii) If we try to avoid such a pair, then we can choose at most one even integer. So we are then forced to choose all five available odd integers, and our list will be: “unknown even, \(11, 13, 15, 17, 19\)”. If the even integer is chosen to be \(14\), or \(16\), then every pair in my list has \(\text{LCM} = 1\). So the answer is “No”.
(b)(i) If we try to avoid such a pair, then we must not choose \(97\) (the only prime number in the nineties). And we must not choose \(95 = 5 \times 19\) (which is relatively prime to all other integers in the given range – except for \(90\)); and we must not choose \(91 = 7 \times 13\) (which is relatively prime to all other integers in the given range – except for \(98\)). So we are forced to choose six integers from \(90, 92, 93, 94, 96, 98, 99\). Whichever integer we then omit leaves a pair which is relatively prime.
(ii) If we try to avoid such a pair, then we can choose at most one even integer. So we are then forced to choose all five available odd integers, and our list will be: “unknown even, \(91, 93, 95, 97, 99\)”, and so must include the pair \(93, 99\) – with common factor \(3\). So the answer is “Yes”.
(c) In parts (a) and (b), the possible integers are limited (in (a) to the “teens”, and in (b) to the “nineties”); so it is natural to reach for ad hoc arguments as we did above. But in part (c) you know nothing about the numbers chosen.
Note: The question says that “ I choose”, and asks whether “ you ” can be sure. So you have to find either a general argument that works for any \(n\), or a counterexample. And the theme of this chapter indicates that it should not require any extended calculation.
The relevant “general idea” is the Pigeon Hole Principle which we may meet in the second part of this collection. So this problem may be viewed as a gentle introduction.
(i) Group the \(2n\) consecutive integers
\(a + 1, a + 2, …, a +2n\)
into \(n\) pairs of consecutive integers
\(\{a+1, a+2\},\{a+3, a+4\}, \ldots,\{a+(2 n-1), a+2 n\}\)
- If we choose at most one integer from each pair, then we never get more than \(n\) integers.
- So as soon as we choose \(n + 1\) integers from \(2n\) consecutive integers, we are forced to choose both integers in some pair \(k\), \(k + 1\), and this pair of consecutive integers is always relatively prime (see Problem 6 (b)(i)).
(ii) We saw in part (a)(ii) that, if \(n = 5\) and the \(2n\) integers start at \(10\), then we can choose six integers (either \(11, 13, 14, 15, 17, 19\), or \(11, 13, 15, 16, 17, 19\)), and in each case every pair has \(\text{LCM} = 1\). So for \(n = 5\) the answer is “No” (because there is at least one case where one cannot be sure).
However, as soon as \(n\) is at least \(6\), we show that the argument in part (a)(ii) breaks down. As before, if we try to choose a subset in which no pair has a common factor, then we can choose at most one even integer. So we are forced to choose all the odd integers. But any run of at least six consecutive odd integers includes two multiples of \(3\). So for \(n \ge 6\), the answer is “Yes”.
6.
(a)(i) Suppose \(k\) is a factor \(of\) \(m\) and \(n\). Then we can write \(m = kp\) and \(n = kq\) for some integers \(p, q\). Hence \(m-n = k(p-q)\), so \(k\) is a factor of \(m-n\). Also \(m +n = k(p + q)\), so \(k\) is a factor of \(m + n\).
(ii) Any factor of \(m\) and \(n\) is also a factor of their difference \(m + n\); so the set of common factors of \(m\) and \(n\) is a subset of the set of common factors of \(m-n\) and \(n\).
And any factor of \(m-n\) and \(n\) is also a factor of their sum \(m\); so the set of common factors of \(m - n\) and \(n\) is a subset of the set of common factors of \(m\) and \(n\).
Hence the two sets of common factors are identical. In particular, the two “highest common factors” are equal.
(iii) Subtract \(91\) from \(1001\) ten times to see that
\(HCF(1001, 91) = HCF(1001 - 910, 91) = 91\).
(b)(i) Subtract \(m\) from \(m + 1\) once to see that
\(HCF( m + 1, m) = HCF(1,m) = 1\)
(ii) Subtract \(m\) from \(2m + 1\) twice to see that
\(HCF(2m +1, m) = HCF(m+1, m) = HCF(1,m) = 1\)
(iii) Subtract \(m -1\) from \(m^{2} +1\) “\(m +1\) times” to see that
\(HCF(m^{2}+1, m-1) = HCF((m^2+1) - (m^{2} -1), m -1) = HCF(2, m-1)\)
Hence, if \(m\) is odd, the \(\text{HCF} = 2\); if \(m\) is even, the \(\text{HCF} = 1\).
7. They are equal. (The first is
\(\frac{17}{100} \times 19000000\)
the second is
\(\frac{19}{100} \times 17000000\)
which are equal since multiplication is commutative and associative.)
8.
(a)
\(\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \frac{6}{5} = \frac{6}{2} = 3\)
(b)
\(\sqrt{\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \frac{6}{5} \times \frac{7}{6} \times \frac{8}{7}} = \sqrt{\frac{8}{2}} = 2\)
(c)
\(10 \times 9 \times8 \times7 \times6 \times5 \times4 \times3 \times2 \times1 \ \text{seconds}\)
\(\begin {array} {align} = \frac{10 \times 9 \times8 \times7 \times6 \times5 \times4 \times3 \times2 \times1 }{60} \ \text{minutes} \\ = \frac{10 \times 9 \times8 \times7 \times6 \times5 \times4 \times3 \times2 \times1 }{60 \times 60} \ \text{hours} \\ = \frac{10 \times 9 \times8 \times7 \times6 \times5 \times4 \times3 \times2 \times1 }{60 \times 60 \times 24} \ \text{days} \\ = \frac{10 \times 9 \times8 \times7 \times6 \times5 \times4 \times3 \times2 \times1 }{60 \times 60 \times 24 \times 7} \ \text{weeks} \\ = 6 \ \text{weeks (after cancelling).} \end {array}\)
Note: These three questions underline what we mean by structural arithmetic . Fractions should never be handled by evaluating numerators and denominators. Instead one should always be on the lookout for structural features which simplify calculation – such as cancellation .
9.
(a) Suppose a rectangle in the “DIN A” series has dimensions \(a\) by \(b\), with \(a \lt b\). Folding in half produces a rectangle of size \(\frac{b}{2}\) by \(a\). Hence \(b : a = a : \frac{b}{2}\), so \(b^{2} = 2a^{2}\), and \(b : a = \sqrt{2}:1\).
(b) (i) \(\frac{1}{r}\)
(ii) \(r\)
10.
(a) “\(15\%\) discount” means the price actually charged is “\(85\%\) of the marked price”. Hence each marked price needs to be multiplied by \(0.85\) .
The distributive law says we may add the marked prices first and then multiply the total (exactly \(£80\)) by \(0.85\) to get
\(£\left(\frac{85}{100} \times 80\right) = £(17 \times 4) = £68\)
Note: The context (shopping, sales tax, and discount) is mathematically uninteresting. What matters here is the underlying multiplicative structure of the solution, which arises in many different contexts.
(b) “Add \(20\%\) VAT” means multiplying the discounted pre-VAT total (\(£68\)) by \(1.2\), or \(\frac{6}{5}\). Hence the final price, with VAT added, is \(£(1.2 \times 0.85 \times 80)\).
If the VAT were added first, the price before discount would be \(£(1.2 \times 80)\), and the final price after allowing for the discount would be \(£(0.85 \times 1.2 \times 80)\). Since multiplication is commutative, the two calculations have the same result, so the order does not matter (just as the final result in Problem 9 is the same whether one first enlarges A4 to A3 and then reduces A3 to A4, or first reduces A4 to A5 and then enlarges A5 to A4).
Note: Notice that we did not evaluate the two answers to see that they gave the same output \(£81.60\). If we had, then the equality might have been a fluke due to the particular numbers chosen. Instead we left the answer unevaluated, in structured form , which showed that the equality would hold for any input.
(c) To cancel out multiplying by \(\frac{6}{5}\) we need to multiply by \(\frac{5}{6}\) – a discount of \(\frac{1}{6}\), or \(16\frac{2}{3}\%\).
Note: This question has nothing to do with financial applications. It is included to underline the fact that although percentage change questions use the language of addition and subtraction (“increase”, or “decrease”), the mathematics suggests they should be handled multiplicatively .
11.
(a)(i) \(2 \times 5 \lt 3 \times 4\), so
\(\frac{7}{2 \times 5} \gt \frac{7}{3 \times 4}\)
Hence
\(\frac{1}{2} + \frac{1}{5} \gt \frac{1}{3} + \frac{1}{4}\)
(ii) At first sight, “\(10 \lt 12\)” may not seem related to “\(\frac{1}{2} + \frac{1}{5} \gt \frac{1}{3} + \frac{1}{4}\)”. Yet the crucial fact we started from in part (i) was “\(2 \times 5 = 10 \lt 12 = 3 \times 4\)”.
(b) \(10 \lt 12\) so
\((x+ 2)(x + 5) = x^{2} + 7x + 10 \lt x^{2} + 7x + 12 = (x +3)(x+4)\)
(i) If all four brackets are positive (i.e. if \(x \gt -2\)), then we also have \(2x + 7 \gt 0\), and it follows that
\(\begin{array} {align} \frac{1}{x + 2} + \frac{1}{x +5} &= \frac{2x + 7}{(x+2)(x+5)} \\ &\gt \frac{2x + 7}{(x+3)(x+4)} \\ &= \frac{1}{x + 3} + \frac{1}{x+4} \end{array}\)
(ii) When calculating with the given algebraic expression, the values
\(x = -2, -3, -4, -5\)
are "forbidden values."
If \(x \gt -2\), then (as in part (i)) we have
\(\begin{array} {align} \frac{1}{x + 2} + \frac{1}{x +5} &= \frac{2x + 7}{(x+2)(x+5)} \\ &\gt \frac{2x + 7}{(x+3)(x+4)} \\ &= \frac{1}{x + 3} + \frac{1}{x+4} \end{array}\)
For permitted values of \(x \lt -2\), one or more of the brackets \((x + 2), (x + 5), (x + 3), (x + 4)\) will be negative. However, one can still carry out the algebra to simplify
\(\frac{1}{x + 2} + \frac{1}{x +5} = \frac{2x + 7}{(x+2)(x+5)}\) and \(\frac{1}{x + 3} + \frac{1}{x +4} = \frac{2x + 7}{(x+3)(x+4)}\)
When \(x = -\frac{7}{2}\) both expressions are equal , and equal to \(0\). The simplified numerators are both positive if \(x \gt -\frac{7}{2}\), and both negative if \(x \lt -\frac{7}{2}\); and the sign of the denominators changes as one moves through the four intervals \(-3 \lt x \lt -2, -4 \lt x \lt -3, -5 \lt x \lt -4, x \lt -5\), with the inequality switching from
\(\begin{array}{align} “\gt” \ (\text{for} \ x \gt -2) &\text{to} \ “\lt” \ (\text{for} -3 \lt x\lt -2), \\ &\text{to} \ “\gt” \ (\text{for} -3.5 \lt x\lt -3), \\ &\text{to} \ “\lt” \ (\text{for} -4 \lt x\lt -3.5), \\ &\text{to} \ “\gt” \ (\text{for} -5 \lt x\lt -4), \\ &\text{to} \ “\lt” \ (\text{for}\ x\lt -5)
\end{array}\)
12.
(a)(i) \(3 + 2\sqrt{2}\);
(ii) \(3- 2\sqrt{2}\);
(iii) \(7+5\sqrt{2}\)
Note: Notice that you can write down the answer to (ii) as soon as you have finished (i), without doing any further calculation.
(b)(i) \( 2+ \sqrt{6}\);
(ii) \(\sqrt{2} + \sqrt{3}\);
(iii) \(\frac{1+\sqrt{5}}{2}\) (the Golden Ratio \(\frac{1+\sqrt{5}}{2} = \tau \) is the larger root of the quadratic equation \(x^{2} - x -1 = 0\). Hence \(\frac{3+\sqrt{5}}{2} = \tau + 1 = \tau^{2}\);
(iv) \(\sqrt{10-2\sqrt{5}}\): this does not simplify further.
Note: Some readers may think an apology is in order for part (iv). The lesson here is that, while one should always try to simplify, there is no way of knowing in advance whether a simplification is possible. And there is no way out of this dilemma. So one is reduced to thinking: any simplification would involve \(\sqrt{5}\), and if one tries to solve (a + b\sqrt{5})^{2} = 10 -2\sqrt{5}, then the solutions for \(a\) and \(b\) do not lead to anything “simpler”. (This repeated surd should perhaps have rung bells, as it was equal to the exact expression for \(4\sin{36^{\circ}}\) in Problem 3 (c). It was included here partly because the question of its simplification should already have arisen when it featured in that context.)
13.
In reconstructing the missing digits the number of possible solutions is determined by the highest common factor of the multiplier and 10. At the first step (in the units column):
because \(HCF(6, 10) = 2, \square \times 6 = 8 \ (\text{mod} \ 10)\) has two solutions which differ by \(5\) – namely \(3\) and \(8\).
The first possibility then requires us to solve \((\square \times 3) + 1 = 2 \ (\text{mod} \ 10)\): because \(HCF(3, 10) = 1 \), this has just one solution – namely \(7\). This gives rise to the solution \(76 \times 3 = 228\) .
The second possibility requires us to solve \((\square \times 8) + 4 = 2 \ (\text{mod} \ 10)\): because \(HCF(8, 10) = 2 \), this has two solutions which differ by \(5\) – namely \(1\) and \(6\). This gives rise to two further solutions: \(16 \times 8 = 128\) , and \(66 \times 8 = 528\) .
14.
(a) The solutions are entirely elementary, with no trickery. But they can be surprisingly elusive. And since this elusiveness is the only reason for including the problem, we hesitate to relieve any frustration by giving the solution. The whole thrust of the “\(24\) game” is to underline the scope for “getting to know” the many faces of a number like \(24\): for example, as \(24 = 12 + 12 \ (= 3 \times 4 + 3 \times 4\) for \(3, 3, 4, 4)\); as \(24 = 25 - 1 \ (= 5 \times 5 -3 \div 3\) for \(3, 3, 5, 5)\); and as \(24 = 27 -3 \ (= 3 \times 3 \times 3 - 3\) for \(3, 3, 3, 3)\). So one should be looking for ways of exploiting other important arithmetical aspects of \(24\) – in particular, as \(4 \times 6\) and as \(3 \times 8\).
(b)(i)
\(\begin{array} {align} 0 = (4-4) + (4-4); 1 = (4 \div 4) \times (4 \div 4); 2 = (4 \div 4) + (4 \div 4); 3 = (4 + 4 + 4) \div 4; \\ 4 = ((4-4) \times 4) + 4; 5 = (( 4 \times 4) + 4) \div 4); 6 = 4 + ((4 + 4) \div 4); 7 = 4 + 4 - (4 \div 4); \\ 8 = (4 + 4) \times (4 \div 4); 9 = (4+ 4) + (4 \div 4). \end{array}\)
The output \(10\) seems to be impossible with the given restrictions.
(ii) With squaring and \(\sqrt{\ \ \ \ }\) allowed we can manage \(10 = 4 + 4 + 4 - \sqrt{4}\). Indeed, one can make everything up to \(40\) except (perhaps) \(39\).
15.
(a)(i) \(a^{2} + 2ab + b^{2}; a^{3} + 3a^{2}b + 3ab^{2} + b^{3}\)
(ii) Replace \(b\) by \((b)\): \(a^{2} - 2ab + b^{2}; a^{3} - 3a^{2}b + 3ab^{2} - b^{3}\)
(b)(i) \((x +1)^{2}\)
(ii) \((x^{2} - 1)^{2}\)
(iii) \((x^{2} - 1)^{3}\)
(c)(i) \(a^{2} - b^{2}\)
(ii) Replace "\(b\)" by "\(b+ c\)": \(a^{2}-(b+c)^{2} = a^{2} - b^{2} - c^{2}- 2bc\)
Replace "\(b\)" by "\(b- c\)": \(a^{2}-(b-c)^{2} = a^{2} - b^{2} - c^{2} + 2bc\)
(d) One way is to rewrite this expression as a difference of two squares:
\(\begin{array} {align} (2x)^{2} - (x^{2} - 2x +1) &= (2x)^{2} - (x-1)^{2} \\ &= (2x -(x-1))(2x +(x -1)) \\ &= (x +1)(3x -1)
\end{array}\)
Note: As so often, the messages here are largely implicit. In part (a)(ii) we explicitly highlight the intention to use what you already know (by simply substituting "\(-b\)” in place of “\(b\)”. In part (b), you are expected to recognise (i), and then to see (ii) and (iii) as mild variations on the expansions of \((a - b)^{2}\) and \((a- b)^{3}\) in part (a). Part (c) repeats (in silence) the message of (a)(ii): think – don’t slog it out. And part (d) encourages you to keep an eye out for thinly disguised instances of “a difference of two squares”.
16.
(a) Final digits : "block" \(4\), \(6\) of length \(2\);
Leading digits : “block” \(4\), \(1\), \(6\), \(2\), \(1\) of length \(5\).
(b) Claim The sequence of “units digits” really does recur.
Proof Given a power of \(4\) that has units digit \(4\), the usual multiplication algorithm for multiplying by \(4\) produces a number with units digit \(6\).
Given this new power of \(4\) with units digit \(6\), the usual multiplication algorithm for multiplying by \(4\) produces a number with units digit \(4\).
At this stage the sequence of units digits begins a new cycle.
[Alternatively: The units digit is simply equal to the relevant power of \(4 \ (\text{mod} \ 10)\). Multiplying by \(4\) changes \(4\) to \(6 \ (\text{mod} \ 10)\); multiplying by \(4\) changes \(6\) to \(4 \ (\text{mod} \ 10)\); – and the cycle repeats.]
(c) The sequence of leading digits seems to recur every \(5\) terms, because \(4^{5} = 2^{10} = 1024\) is almost exactly equal to \(1000\). Each time we move on \(5\) steps in the sequence, we multiply by \(4^{5} = 1024\). As far as the leading digit is concerned, this has the same effect as multiplying the initial term \((4)\) by slightly more than \(1.024\) (then adding any ‘carries’), which is very like multiplying by \(1\) – and so does not change the leading digit (yet).
However, each time we move on \(10\) steps in the sequence, we multiply by \(4^{10} = 1024^{2} = 1048576\). As far as the leading digit is concerned, this has the same effect as multiplying by slightly more than \(1.048576\).
When we move on \(25\) steps, we multiply by \(4^{25} =<PHONE_NUMBER>842624\). And as far as the leading digit is concerned, this has the same effect as multiplying by slightly more than \(1.12599906842624\). And so on.
Eventually, the multiplier becomes large enough to change one of the leading digits.
17. The total is \(100\).
Having found this by direct calculation, we should think indirectly and notice that \(100 = 10^{2}\).
And we should then ask: “Why \(10\)? What has \(10\) got to do with the \(4 \times\) multiplication table?”

(a) \(\sin{45^{\circ}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} = \cos{45^{\circ}}; \tan{45^{\circ}} = 1\)
(b) \(AM = \sqrt{3}; \ \sin{30^{\circ}} = \frac{1}{2}, \cos{30^{\circ}} = \frac{\sqrt{3}}{2}, \tan{30^{\circ}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}; \ \sin{60^{\circ}} = \frac{\sqrt{3}}{2}, \cos{60^{\circ}} = \frac{1}{2}, \tan{60^{\circ}} = \sqrt{3}\)
(c)(iii) \(\cos{315^{\circ}} = \cos{45^{\circ}} = \frac{\sqrt{2}}{2}; \sin{225^{\circ}} = -\sin{45^{\circ}} = -\frac{\sqrt{2}}{2}; \tan{210^{\circ}} = \tan{30^{\circ}} = \frac{\sqrt{3}}{3}; \cos{120^{\circ}} = -\cos{60^{\circ}} = -\frac{1}{2}; \sin{960^{\circ}} = \sin{240^{\circ}} = -\sin{60^{\circ}} = -\frac{\sqrt{3}}{2}; \tan{(-135^{\circ})} = \tan{45^{\circ}} = 1\)
(d) Cut the \(n\)-gon into \(n\) “cake slices”, and use the formula “\(\frac{1}{2}ab\sin{C}\)” for each slice.
(i) \(\frac{3\sqrt{3}}{4}\)
(ii) \(2\)
(iii) \(\frac{3\sqrt{3}}{2}\)
(iv) \(2\sqrt{2}\)
(v) \(3\)
(e) Work out the side length of the \(n\)-gon, then cut the \(n\)-gon into \(n\) "slices," and use the formula "\frac{1}{2}(\text{base} \times \text{height})" for each slice.
(i) \(3\sqrt{3}\)
(ii) \(4\)
(iii) \(2\sqrt{3}\)
(iv) \(8(\sqrt{2} - 1)\)
(v) \(12(2 -\sqrt{3})\)
Note: There is no hidden trig here: all you need is Pythagoras’ Theorem. For example, in part (e)(iv) we can extend alternate sides of the regular octagon to form the circumscribed \(2\) by \(2\) square. The four corner triangles are isosceles right angled triangles with hypotenuse of length \(s\) (the side of the octagon). Hence each side of the square is equal to \(s + 2 \cdot \frac{s}{\sqrt{2}} = 2\) whence \(s = 2(\sqrt{2} -1)\).
20.
(a) \(\sqrt{8}\)
(b) \(\sqrt{2}\), \(\sqrt{2}\)
(c) \(\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\)
21. Construct the perpendicular from \(A\) to \(BC\) (possibly extended); let this meet the line \(BC\) at \(X\). There are four possibilities:
(i) Either \(X = C\), in which case \(\angle{BCA}\) is a right angle as required; or \(X = B\), in which case \(b^{2} = a^{2} + c^{2}\), contradicting \(a^{2} + b^{2} = c^{2}\);
(ii) \(X \ne B,C\), and \(C\) lies between \(B\) and \(X\);
(iii) \(X \ne B,C\), and \(X\) lies between \(B\) and \(C\);
(iv) \(X \ne B,C\), and \(B\) lies between \(X\) and \(C\).
We analyse case (ii) and leave cases (iii) and (iv) to the reader.
(ii) \(\triangle{AXC}\) and \(\triangle{AXB}\) are both right angled triangles; so by Pythagoras’ Theorem we know that
\(\begin{array}{align} AC^{2} &= AX^{2} + XC^{2}, \text{and} \\ AB^{2} &= AX^{2} + XB^{2} \\ &= AX^{2} + (XC + CB)^{2} \\ &=AX^{2} + XC^{2} + CB^{2} + 2XC \cdot CB \\ &= AC^{2} + CB^{2} + 2XC \cdot CB
\end{array}\)
Since we are told that \(AC^{2} + CB^{2} = AB^{2}\), it follows that \(2XC \cdot CB = 0\), contrary to \(X \ne C\).
Note : Notice that the proof of the converse of Pythagoras’ Theorem makes use of Pythagoras’ Theorem itself.
22.
(a) \( c = b+1\), so \(a^{2} = c^{2} - b^{2} = 2b + 1\). Hence \(a\) is odd, and we can write \(a = 2m + 1\).
(b) Suppose \(b = 2n-1\) is also odd. Then \(c^{2} = 4n^{2}\) is divisible by \(4\) – which contradicts the fact that \(b^{2} = 4(n^{2} -n) +1\), and \(a^{2} = 4(m^{2} +m) +1\), so \(a^{2} + b^{2}\) leaves remainder \(2\) on division by \(4\).
Hence \(b = 2n\) is even and \(c = 2n+1\) is odd. But then
\((2m + 1)^{2} + (2n)^{2} = a^{2} + b^{2} = c^{2} = (2n +1)^{2}\)
so \(4(m^{2} + m) = 4n\), and \(n = m(m+1)\).
23.
(a) If \(a\) and \(b\) are both even, then \(HCF(a,b) \ne 1\), so the triple would not be primitive.
If \(a\) and \(b\) are both odd, we use the idea from Problem 22 (b). Suppose \(a = 2m +1, \ b = 2n +1\); then \(a^{2} = 4(m^{2} + m) +1\), and \(b^{2} = 4(n^{2} + n) +1\), so \(a^{2} + b^{2} = 2 \times (2(m^{2} + m + n^{2} +n) +1)\). But this is “twice an odd number”, so cannot be equal to \(c^{2}\) (since \(c\) would have to be even, and any even square must be a multiple of \(4\)).
Hence we may assume that \(a\) is odd and \(b\) is even: so \(c\) is odd.
(b) Then \(a^{2} + b^{2} = c^{2}\) yields \(b^{2} = c^{2} - a^{2} = (c - a)(c +a)\), so
\(\left(\frac{b}{2}\right)^{2} = \left( \frac{c -a}{2} \right) \left( \frac{c +a}{2} \right)\)
Any common factor of \(\frac{c+a}{2}\) and \(\frac{c-a}{2}\) divides their sum \(c\) and their difference \(a\), so \(HCF\left( \frac{c-a}{2}, \frac{c+a}{2}\right) = 1\). Since the difference of these two factors is \(a\), which is odd, they have opposite parity.
(c) If two integers are relatively prime, and their product is a square, then each of the factors has to be a square (consider their prime factorisations). Hence \(\frac{c+a}{2} = p^2\) and \(\frac{c-a}{2} = q^2\), where \(HCF(p,q) = 1\) and \(p\) and \(q\) have opposite parity.
Therefore
\(c = p^{2} + q^{2}, \ \ \ a = p^{2} - q^{2} , \ \ \ b = 2pq\)
(d) It is easy to check that any triple of the given form is (i) primitive, and (ii) satisfies \(a^{2} + b^{2} = c^{2}\).


29.5: Modern Architecture
European Expressionist Architecture
Expressionist architecture was a European movement of the twentieth century that came about in parallel with the Expressionist visual and performing arts.
Describe some common stylistic elements found in European Expressionist architecture in the early 20th century
Key Points
- The term Expressionist architecture initially described the activity of the German, Dutch, Austrian, Czech, and Danish avant-garde from 1910 until 1930.
- The style was characterized by an early modernist adoption of novel materials, formal innovation, and very unusual massing—sometimes inspired by natural biomorphic forms and sometimes by the new technical possibilities offered by the mass production of brick, steel, and glass.
- Expressionist architecture was individualistic and in many ways eschewed aesthetic dogma, but some aims common to its proponents include: the distortion of form, a striving for the new, the notion of architecture as a work of art, and the aim to express an inner experience
Key Terms
- Expressionist architecture : An architectural movement that developed in Europe during the first decades of the 20th century in parallel with the expressionist visual and performing arts.
Modern European Architecture
Expressionist architecture was an architectural movement that developed in Europe during the first decades of the twentieth century in parallel with the expressionist visual and performing arts. The term Expressionist architecture initially described the activity of the German, Dutch, Austrian, Czech, and Danish avant-garde from 1910 until 1930.
Subsequent redefinitions extended the term backwards to 1905 and also widened it to encompass the rest of Europe. Today the meaning has broadened even further to refer to architecture of any date or location that exhibits some of the qualities of the original movement such as: distortion, fragmentation, or the communication of violent or overstressed emotion.
Expressionism
The style was characterized by an early modernist adoption of novel materials, formal innovation, and very unusual massing—sometimes inspired by natural biomorphic forms and sometimes by the new technical possibilities offered by the mass production of brick, steel, and glass.
Many expressionist architects fought in World War I and their experiences, combined with the political turmoil and social upheaval that followed the German Revolution of 1919, resulted in a Utopian outlook and a romantic socialist agenda. Hence, ephemeral exhibition buildings were numerous and highly significant during this period.
Likewise, scenography for theater and films provided another outlet for the expressionist imagination, and provided supplemental incomes for designers attempting to challenge conventions in a harsh economic climate.
Features of Expressionist Architecture
Expressionist architecture was individualistic and in many ways eschewed aesthetic dogma. While the movement was very broad, some points can be found as recurring in works of Expressionist architecture, and are evident in some degree in each of its works.
- A distortion of form for an emotional effect.
- The subordination of realism to symbolic or stylistic expression of inner experience.
- An underlying effort at achieving the new, original, and visionary.
- A profusion of works on paper, and models, with discovery and representations of concepts being more important than pragmatic finished products.
- Often hybrid solutions, irreducible to a single concept.
- Themes of natural romantic phenomena, such as caves, mountains, lightning, crystal and rock formations.
- Utilizes the creative potential of artisan craftsmanship.
- A tendency towards the gothic than the classical.
- Draws as much from Moorish, Islamic, Egyptian, and Indian art and architecture as from Roman or Greek.
- Conceives architecture as a work of art.
Form also played a defining role in setting apart expressionist architecture from its immediate predecessor, art nouveau, or Jugendstil. While art nouveau had an organic freedom with ornament, expressionist architecture strove to free the form of the whole building instead of just its parts.
An example of a built expressionist project that is formally inventive is Erich Mendelsohn’s Einstein Tower. This sculpted building shows a relativistic and shifting view of geometry: devoid of applied ornament, form and space are shaped in fluid concrete to express concepts of the architect and the building’s namesake.
Expressionist architecture utilized curved geometries and a recurring form in the movement is the dome. Another expressionist motif was the emphasis on either horizontality or verticality for dramatic effect, which was influenced by new technologies like cruise liners and skyscrapers.
American Art Deco Architecture
Art Deco and Streamline Moderne were two key styles of early 20th-century American architecture.
Distinguish between Art Deco and Streamline Moderne in American architecture in the first half of the 20th century
Key Points
- Art Deco emerged from the Interwar period and combines traditional craft motifs with Machine-Age imagery and materials.
- Art Deco can be recognized by its heavy ornament, bold geometric forms, and rich colors.
- Streamline Moderne was an aesthetic first created by industrial designers who stripped Art Deco design of its ornament in favor of the aerodynamic pure- line concept of motion and speed developed from scientific thinking: it is embodied in cylindrical forms and long, horizontal windowing.
- The Streamline Moderne was both a reaction to Art Deco and a reflection of austere economic times. Gone was unnecessary ornament: sharp angles were replaced with simple, aerodynamic curves.
Key Terms
- Streamline Moderne : Sometimes referred to as Art Moderne, a late type of the Art Deco design style that emerged during the 1930s. Its architectural style emphasized curving forms, long, horizontal lines, and sometimes nautical elements.
- Art Deco : A style of decorative art and architecture originating in the 1920s and 1930s, characterized by bold geometric forms and simple composition.
Overview
Modern American architecture is usually divided into the two styles of Art Deco and Streamline Moderne. Art Deco, which emerged in the 1920s and flourished in the 1930s to 1940s, is an eclectic style that combines traditional craft motifs with Machine Age imagery and materials.
Streamline Moderne, also known as Art Moderne, was a late type of the Art Deco design style that emerged during the 1930s. Its architectural style emphasized curving forms, long, horizontal lines, and sometimes nautical elements.
Art Deco
The Art Deco style is often characterized by its use of rich colors, symmetry, bold geometric shapes, simple composition, rectilinear rather than curvilinear shapes, and lavish ornamentation. Emerging during the Interwar period when rapid industrialization was transforming culture, one of the major attributes of Art Deco was its embrace of technology.
During its heyday, Art Deco represented luxury, glamor, exuberance, and faith in social and technological progress. The urban United States has many examples of Art Deco architecture, especially in New York, Chicago, and Detroit. The famous skyscrapers in these cities are the best known, but notable Art Deco buildings can be found in other neighborhoods.
Streamline Moderne
As the Great Depression decade of the 1930s progressed, Americans saw a new decorative element of the Art Deco style emerge in the marketplace: streamlining. Streamline Moderne was a concept first created by industrial designers, who stripped Art Deco design of its ornament in favor of the aerodynamic pure-line concept of motion and speed developed from scientific thinking.
This aesthetic was embodied through the use of cylindrical forms and long, horizontal windowing. An array of designers quickly ultra-modernized and streamlined the designs of everyday objects, such as toasters.
Streamline Moderne was both a reaction to Art Deco and a reflection of austere economic times. Gone was unnecessary ornament. Sharp angles were replaced with simple, aerodynamic curves. Exotic woods and stone were replaced with cement and glass.
Some common characteristics of Streamline Moderne include horizontal orientation, rounded edges, corner windows, glass blocks, porthole windows, chrome hardware, smooth exterior wall surfaces (usually stucco), horizontal wall grooves, and subdued colors.
However, Art Deco and Streamline Moderne were not necessarily opposites. Streamline Moderne buildings with a few Deco elements were not uncommon, and sometimes there is so much crossover that it can be difficult to differentiate between the two styles.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
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- Chrysler Building. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Chrysler_Building#Design_beginnings. License : CC BY-SA: Attribution-ShareAlike
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- Friedrichstrasse Mies. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Friedrichstrasse_Mies.jpg. License : Public Domain: No Known Copyright
- Babelsberg Einsteinturm. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Babelsberg_Einsteinturm.jpg. License : Public Domain: No Known Copyright
- Hecht warehouse washington dc. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Hecht_warehouse_washington_dc.jpg. License : CC BY-SA: Attribution-ShareAlike
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- Hollywood Palladium. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Hollywood_Palladium.jpg. License : CC BY-SA: Attribution-ShareAlike
- Chrysler Building spire, Manhattan, by Carol Highsmith (LOC highsm.04444). Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Chrysler_Building_spire,_Manhattan,_by_Carol_Highsmith_(LOC_highsm.04444).png. License : Public Domain: No Known Copyright


8.5B: Classification of Prokaryotes
- Discuss the origins of prokaryotic organisms in terms of the geologic timeline
Evolution of Prokaryotes
In the recent past, scientists grouped living things into five kingdoms (animals, plants, fungi, protists, and prokaryotes) based on several criteria such as: the absence or presence of a nucleus and other membrane-bound organelles, the absence or presence of cell walls, multicellularity, etc. In the late 20 th century, the pioneering work of Carl Woese and others compared sequences of small-subunit ribosomal RNA (SSU rRNA) which resulted in a more fundamental way to group organisms on earth. Based on differences in the structure of cell membranes and in rRNA, Woese and his colleagues proposed that all life on earth evolved along three lineages, called domains. The domain Bacteria comprises all organisms in the kingdom Bacteria, the domain Archaea comprises the rest of the prokaryotes, and the domain Eukarya comprises all eukaryotes, including organisms in the kingdoms Animalia, Plantae, Fungi, and Protista.
The current model of the evolution of the first, living organisms is that these were some form of prokaryotes, which may have evolved out of protobionts. In general, the eukaryotes are thought to have evolved later in the history of life. However, some authors have questioned this conclusion, arguing that the current set of prokaryotic species may have evolved from more complex eukaryotic ancestors through a process of simplification. Others have argued that the three domains of life arose simultaneously, from a set of varied cells that formed a single gene pool.
Two of the three domains, Bacteria and Archaea, are prokaryotic. Based on fossil evidence, prokaryotes were the first inhabitants on Earth, appearing 3.5 to 3.8 billion years ago during the Precambrian Period. These organisms are abundant and ubiquitous; that is, they are present everywhere. In addition to inhabiting moderate environments, they are found in extreme conditions: from boiling springs to permanently frozen environments in Antarctica; from salty environments like the Dead Sea to environments under tremendous pressure, such as the depths of the ocean; and from areas without oxygen, such as a waste management plant, to radioactively-contaminated regions, such as Chernobyl. Prokaryotes reside in the human digestive system and on the skin, are responsible for certain illnesses, and serve an important role in the preparation of many foods.
Key Points
- All living things can be classified into three main groups called domains; these include the Archaea, the Bacteria, and the Eukarya.
- Prokaryotes arose during the Precambrian Period 3.5 to 3.8 billion years ago.
- Prokaryotic organisms can live in every type of environment on Earth, from very hot, to very cold, to super haline, to very acidic.
- The domains Bacteria and Archaea are the ones containing prokaryotic organisms.
- The Archaea are prokaryotes that inhabit extreme environments, such as inside of volcanoes, while Bacteria are more common organisms, such as E. coli .
Key Terms
- prokaryote : an organism whose cell (or cells) are characterized by the absence of a nucleus or any other membrane-bound organelles
- domain : in the three-domain system, the highest rank in the classification of organisms, above kingdom: Bacteria, Archaea, and Eukarya
- archaea : a taxonomic domain of single-celled organisms lacking nuclei, formerly called archaebacteria, but now known to differ fundamentally from bacteria