17.4E: Cardiac Output
Cardiac output (Q or CO) is the volume of blood pumped by the heart, in particular by the left or right ventricle, in one minute.
- Describe cardiac output and its function in the cardiovascular system
Key Points
- Cardiac output, a measure of how much blood the heart pumps over the course of a minute, is calculated by multiplying heart rate by stroke volume.
- The heart rate is increased by sympathetic nervous stimulation and decreased by parasympathetic nervous stimulation.
- Stroke volume is end diastolic volume (venous return) minus end systolic volume, the amount of blood left over in the heart after systole.
- The ejection fraction is stroke volume divided by end diastolic volume.
- Mean arterial blood pressure is cardiac output multiplied by total peripheral resistance. A twofold change in vascular size will cause a 16-fold change in resistance in the opposite direction.
- Starling’s mechanism states that changes in venous return (preload) to the heart will change cardiac output, which will also change mean arterial blood pressure in the same direction. This means that blood volume and blood pressure are directly related to one another.
Key Terms
- mean arterial blood pressure : A measure of blood pressure based on cardiac output and vascular resistance.
- cardiac output : The volume of blood pumped by the heart, in particular by the left or right ventricle, in the time interval of one minute.
Cardiac output (CO) is a measure of the heart’s performance. While there are many clinical techniques to measure CO, it is best described as a physiological and mathematical relationship between different variables. When one of the variables change, CO as a whole will change as a result. This can also be used to predict other regulated variables, such as blood pressure and blood volume. The mathematical description of CO is that CO=Heart Rate (HR)×Stroke Volume (SV)CO=Heart Rate (HR)×Stroke Volume (SV) . Changes in HR, SV, or their components, will change CO .
Heart Rate
The heart rate is determined by spontaneous action potential generation in the sinoatrial (SA) node and conduction in the atrioventricular (AV) node. It refers to the number of heartbeats over the course of a minute. Sympathetic nervous system activation will stimulate the SA and AV nodes to increase the heart rate, which will increase cardiac output. Parasympathetic nervous system activation will conversely act on the SA and AV nodes to decrease the heart rate, which will decrease cardiac output. For the SA node, the rate of depolarization is altered, while the AV node’s rate of conduction is altered by autonomic nerve stimulation.
Stroke Volume
Stroke volume refers to the amount of blood ejected from the heart during a single beat. It is a measure of the contractility of the heart based on end diastolic volume (EDV), mathematically described as SV=EDV−ESV (end systolic volumeSV=EDV−ESV (end systolic volume. EDV is the volume of blood in the ventricles at the end of diastole, while ESV is the volume of blood left inside the ventricles at the end of systole, making SV the difference between EDV and ESV. Contractility of the heart refers to the variability in how much blood the heart ejects based on changes in stroke volume rather than than changes in heart rate.
Additionally, another indicator known as the ejection fraction (EF) is used to evaluate stroke volume and contractility. It is described as EF=(SVEDV)×100%EF=(SVEDV)×100% and is a measure of the proportion of blood ejected during systole compared to the amount of blood that was present in the heart. A higher EF suggests more efficient heart activity.
Mean Arterial Pressure
Cardiac output is an indicator of mean arterial blood pressure (MAP), the average measure of blood pressure within the body. It is described as MAP=CO×TPR (total peripheral resistance)MAP=CO×TPR (total peripheral resistance). TPR is a measure of resistance in the blood vessels, which acts as the force by which blood must overcome to flow through the arteries determined by the diameter of the blood vessels. The exact relationship is such that a twofold increase in blood vessel diameter (doubling the diameter) would decrease resistance by 16-fold, and the opposite is true as well. When CO increases, MAP will increase, but if CO decreases, MAP will decrease.
Starling’s Law of the Heart
Frank Starling’s Law : This chart indicates stroke volume compared to ventricular preload, with labels for preload dependent zone, responsive patient SVV > 10%, and nonresponsive patient SVV < 10 %.
CO can also predict blood pressure based on blood volume. Starling’s law of the heart states that the SV of the heart increases in response to an increase in EDV when all other factors remain constant. Essentially, this means that higher venous blood return to the heart (also called the preload) will increase SV, which will in turn increase CO. This is because sarcomeres are stretched further when EDV increases, allowing the heart to eject more blood and keep the same ESV if no other factors change.
The main implication of this law is that increases in blood volume or blood return to the heart will increase cardiac output, which will lead to an increase in MAP. The opposite scenario is true as well. For example, a dehydrated person will have a low blood volume and lower venous return to the heart, which will decrease cardiac output and blood pressure. Those that stand up quickly after lying down may feel light-headed because their venous return to the heart is momentarily impaired by gravity, temporarily decreasing blood pressure and supply to the brain. The adjustment for blood pressure is a quick process, while blood volume is slowly altered.
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10.4.12: The Book That Started the Civil War
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- Robin DeRosa, Abby Goode et al.
- Plymouth State University/
First off, let me use this platform to argue my case for Captain Underpants worthy place in the canon of American Literature. JUST KIDDING.
Uncle Tom’s Cabin reads like a beach-read. No, it isn’t 54 pages of vague, homoerotic freestyle (“Song of Myself” ) . If I were to re-write “Self-Reliance ” to be more in the vein of Uncle Tom’s Cabin, it would read something like this: “Do what you want, screw everything else.” Yes, all that self-important pandering could be reduced to seven crude words- sorry Emerson. Uncle Tom’s Cabin isn’t as sophisticated or head-scratch-worthy as the aforementioned works, but it’s arguably more important than them to the American story.
Some literature is canonized not because of its symbolism, elegance or style, but because of its sheer impact on a society. That’s what I’m assuming Jane Tompkins wrote about in her argument for the canonization of Uncle Tom’s Cabin. Yes, the novel may be veering on “sentimental drivel” catered to the popular pallet, but its ease of access and popularity is what gave it such weight in its day. A work shouldn’t be considered worthy of canonization in American Literature because of its ambiguity or complexity, but for the magnitude of impact that it made on America as a whole.
David Reynolds of the New York Daily News wrote an article on Uncle Tom’s Cabin and its impact. He attributes Lincoln’s political victory to Uncle Tom’s Cabin . Its contents polarized a nation. It stirred righteous indignation from the northern abolitionists, while at the same time, says Reynolds, “it stiffened the South’s resolve to defend slavery and demonize the North.” Reynolds writes, “ Uncle Tom’s Cabin . . . created such a stir when it was published in 1852 that Abraham Lincoln reportedly called Stowe ‘the little lady who made this great war.’” Mind that the war Stowe allegedly “made” was the costliest war ever fought on American soil.
One could only imagine the hooplah this book must’ve made, considering that, when it was published, there was no television or radio. Books were the most accessible form of entertainment. One might even say (given a bit of imaginative licence) that Uncle Tom’s Cabin is to 19th-Century America what (insert your favorite widely consumed television show) is to today. But when was the last time Stranger Things started a civil war?
Good Read/Source: http://www.nydailynews.com/opinion/b...ticle-1.112605


11: Metaphor
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The ease with which corpora are accessed via word forms is an advantage as long as it is our aim to investigate words, for example with respect to their relationship to other words, to their internal structure or to their distribution across grammatical structures and across texts and language varieties. As we saw in Chapter 8, the difficulty of accessing corpora at levels of linguistic representation other than the word form is problematic where our aim is to investigate grammar in its own right, but since grammatical structures tend to be associated with particular words and/or morphemes, these difficulties can be overcome to some extent.
When it comes to investigating phenomena that are not lexical in nature, the word-based nature of corpora is clearly a disadvantage and it may seem as though there is no alternative to a careful manual search and/or a sophisticated annotation (manual, semi-manual or based on advanced natural-language technology). However, corpus linguists have actually uncovered a number of relationships between words and linguistic phenomena beyond lexicon and grammar without making use of such annotations. In the final chapter of this book, we will discuss a number of case studies of one such phenomenon: metaphor.


4.7: Graphing Rational Functions
- Identify important behavior of rational functions and use this information to sketch their graphs.
- Find equations of rational functions that model graphical information.
In this section we are going look at sketching graphs of rational functions. Although technology can graph functions for us, it is important that we understand the details of how these functions behave. Window settings for viewing a graph in a graphing calculator can significantly affect what graphical behavior we see. What if you owned a company that produced and sold cookies. Your company would have some costs to produce cookies. In business, it is known that average cost models are rational functions. As a business owner, you’d want to reduce your costs as much as possible. This means you’d want to find the smallest of the local minimums for the functions. If you didn’t know how the function model for your costs behaved, how would you be certain you had reduced your costs as much as possible? This is just one of many examples of why it is important to understand how functions behave.
Graphing Rational Functions
Let’s take a moment to review what we know about rational function behavior. Rational functions have vertical asymptotes or holes for values not in the domain of the function. We know the function values increase or decrease without bound as we near vertical asymptotes. Rational functions have horizontal, slant, or oblique asymptotes which are polynomials that we can use to describe their end behavior. On the ends of the graph, the function will approach one of these asymptotes. To figure out how the function behaves elsewhere, we can use points such as intercepts and also sign charts.
Let \(r(x)\) be a rational function.
- Write \(r\) as a single fraction in factored form.
- Determine the domain of \(r\).
- Find the vertical asymptotes of \(r\), if any.
- Find any holes in the graph, if they exist.
- Find the horizontal, slant, or oblique asymptote for \(r\).
- Find all intercepts for \(r\), if any.
- Find all points where the function intersects the horizontal or slant or oblique asymptote, if any.
- Use this information to sketch the graph of \(r(x)\). If needed, construct a sign chart and find other points to help with sketching the graph.
A note is in order about checking to see if \(r(x)\) intersects its horizontal or slant or oblique asymptote. Unlike vertical asymptotes that occur at values not in the domain of \(r(x)\), these asymptotes describe end behavior of the function only. This means that it is possible that \(r(x)\) can have the same function value as the horizontal or slant or oblique asymptote somewhere in between the ends. That is why we must check this.
For the given function, \(r(x)=\dfrac{x^2+2x-3}{x^2+2x-8}\),
- Find the domain and state answer in interval notation.
- Identify all the asymptotes, if any.
- Identify any holes in the graph of \(r\), if any.
- Describe the end behavior of \(r\) using proper notation.
- Find the x and y-intercepts.
- Set up a sign chart and use it to describe where \(r(x)>0\) and \(r(x)<0\).
- Use the information your found above to sketch a graph of \(r\).
- Find the range of \(r(x)\) and state answer in interval notation. Use graphing technology and round to the first decimal place, if applicable.
Solution
- To find domain, we first must factor: \[r(x)=\dfrac{x^2+2x-3}{x^2+2x-8}=\dfrac{(x+3)(x-1)}{(x-2)(x+4)}\nonumber\]From here we can see that \(x=2\) and \(x=-4\) will make \(r\) undefined. Hence, the domain is \((-\infty, -4) \cup (-4,2) \cup (2,\infty)\).
- Since there are no common factors between the numerator and denominator we can conclude that there are two vertical asymptotes at \(x=2\) and \(x=-4\). The degree of the numerator and denominator are the same, degree 2, so there is a horizontal asymptote at \(y=\dfrac{1}{1}=1\).
Remember that we do need to check if the function intersects the horizontal asymptote so let's check this. To do so, we need to find out if there are any points for \(r(x)\) that have a y-coordinate equal to 1. To find out, set \(r(x)=1\)\[\dfrac{x^2+2x-3}{x^2+2x-8}=1\nonumber \]Multiplying both sides by the common denominator, \[(x^2+2x-8)\left(\dfrac{x^2+2x-3}{x^2+2x-8}\right)=(x^2+2x-8)(1)\nonumber\]which simplifies to \[x^2+2x-3=x^2+2x-8\nonumber\]Subtracting \(x^2\) and then \(2x\) from both sides yields \[-3=-8\nonumber\]which is a false statement. We can conclude that \(r\) does not intersect it's horizontal asymptote.
- Since there are no common factors between the numerator and denominator, we can conclude there are no holes in the graph of \(r(x)\).
- As \(x \rightarrow \pm \infty\), \(r(x) \rightarrow 1\).
- To find the x-intercepts set \(y=0\), \[\dfrac{x^2+2x-3}{x^2+2x-8}=0\nonumber\]Multiplying both sides by the common denominator, \[(x^2+2x-8)(\dfrac{x^2+2x-3}{x^2+2x-8})=(x^2+2x-8)(0)\nonumber\]which simplifies to \[x^2+2x-3=0 \nonumber \]Notice that this is just the numerator or \(r(x)\). Now factoring, \[(x+3)(x-1)=0\nonumber\]which gives us \(x=-3\) and \(x=1\). The x-intercepts are \((-3,0)\) and \((1,0)\).
To find the y-intercept, evaluate the function at \(x=0\)\[r(0)=\dfrac{(0)^2+2(0)-3}{(0)^2+2(0)-8}=\dfrac{-3}{-8}=\dfrac{3}{8}\nonumber\]The y-intercept is \(\left(0, \dfrac{3}{8}\right)\).
Note that for the x-intercepts we can just look at the factored numerator of \(r(x)\) without having to actually go through the steps of solving \(r(x)=0\).  Also notice that for the y-intercept, the y-coordinate is just the constant terms of the numerator and denominator of \(r(x)\) in unfactored form.
- The critical values for \(r(x)\) are -4, -3, 1, and 2. Setting up a sign chart:
\(r(x)>0\) on the intervals \((-\infty, -4)\), \((-3, 1)\), and \((2,\infty)\)
\(r(x)<0\) on the intervals \((-4, -3)\) and \((1, 2)\)
- We begin by sketching dashed lines for the vertical asymptotes of \(r(x)\). We use dashed lines since asymptotes describe behavior of the function but they are not actually part of the graph of the function. We know that \(r(x)\) will increase or decrease without bound as the graph nears the vertical asymptotes so we can indicate this behavior on our graph, as shown in Figure \(\PageIndex{2}\).
Next, we sketch a dashed line for the horizontal asymptote. We know that as \(x \rightarrow \pm \infty\), \(r(x) \rightarrow 1\), so we can indicate this behavior on our graph, as shown in Figure \(\PageIndex{3}\).
Now add the intercepts to the graph, as shown in Figure \(\PageIndex{4}\)
Since the domain is \((-\infty, -4) \cup (-4,2) \cup (2,\infty)\) there will be three parts to the graph, one part on each interval. On the interval \((-\infty, -4)\), we know that there are no x-intercepts and that the graph does not intersect the horizontal asymptote. This tells us that the graph must be above the horizontal asymptote. We can also see from the sign chart that \(r(x)>0\) on this interval which indicates the graph can not go below the x-axis. On the interval \((-4,2)\), we know the graph does not intersect the horizontal asymptote. This tells us that the graph must come from below the x-axis to be able to pass through the intercepts. On the interval \((2,\infty)\), we know that there are no x-intercepts and that the graph does not intersect the horizontal asymptote. This means that the graph must be above the horizontal asymptote here. We can also see from the sign chart that \(r(x)>0\) on this interval which indicates the graph can not go below the x-axis. The graph of \(r(x)\) is shown in Figure \(\PageIndex{5}\)
- We can see that there is a local maximum within the interval \((-4,2)\) which we will need to approximate using a graphing calculator. The maximum occurs at approximately \((-1,0.4)\). Rounding to the nearest first decimal place, the range is \((-\infty,0.4] \cup (1,\infty)\).
In the previous example, we learned that we can find the x-intercepts quickly by looking at factors of the numerator. Do keep in mind that you must simplify the function first before determining the x-intercepts since there could be common factors between the numerator and denominator which would indicate that a hole exists.
Find the asymptotes, intercepts, domain, range, and end behavior of \(f(x)=\dfrac{3x^2+14x-5}{x+3}\). Round to the first decimal place for the range, if applicable. Then, use this information to sketch \(f(x)\).
Solution
Let's begin with factoring to determine the domain. \(f(x)=\dfrac{3x^2+14x-5}{x+3}=\dfrac{(x+5)(3x-1)}{x+3}\).
From the factored form of \(f(x)\), we find that the domain is \((-\infty,-3) \cup (-3,\infty)\). We can also tell from the factored that there is no hole and that there is a vertical asymptote at \(x=-3\).
Since there are no common factors between the numerator and denominator, we can set the numerator factors equal to 0 to find the x-intercepts, \((-5,0)\) and \((\dfrac{1}{3},0)\).
Evaluating at \(x=0\), \(f(0)=\dfrac{-5}{3}\) so the y-intercept is \(\left(-\dfrac{5}{3},0\right)\).
The degree of the numerator is one degree larger than the degree of the denominator so there is a slant asymptote. Using division, \[f(x)=\dfrac{3x^2+14x-5}{x+3}=3x+5-\dfrac{20}{x+3}\nonumber\]The slant asymptote is \(y=3x+5\).
Checking to see if \(f\) intersects its slant asymptote,
\[\begin{align*}\dfrac{3x^2+14x-5}{x+3} &=3x+5 \\[15pt] 3x^2+14x-5 &= (x+3)(3x+5) \qquad \text {multiply both sides by LCD} \\[15pt] 3x^2+14x-5 &= 3x^2+14x-15 \qquad \text {multiply}\\[15pt] -5 &=-15 \qquad \end{align*}\nonumber\]
This is a false statement so we can conclude that \(f\) does not intersect its slant asymptote.
Since the slant asymptote is a line with a positive slope, the end behavior is: as \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\), and as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\).
Next, sketch the the asymptotes as dashed lines. Fill in the behavior of \(f(x)\) near the asymptotes and plot the intercepts, as shown in Figure \(\PageIndex{6}\).
We know there is an x-intercept on the interval \((-\infty,-3)\) and that the graph does not intercept the slant asymptote. This tells us that the graph must be above the slant asymptote and that as \(x \rightarrow -3\), \(f(x) \rightarrow \infty\). We also know there there is an x and y-intercept on the interval \((-3,\infty)\) and that the graph does not intercept the slant asymptote on this interval. This tells us that the graph must be below the slant asymptote and that as \(x \rightarrow -3\), \(f(x) \rightarrow \infty\). The graph of \(f(x)\) is shown below in Figure \(\PageIndex{7}\).
Vertical asymptote at \(x=-3\), slant asymptote at \(y=3x+5\), intercepts at \((-5,0), \text{ } \left(\dfrac{1}{3},0\right) \text {, and } \left(0,-\dfrac{5}{3}\right)\), domain is \((-\infty,\infty)\), range is \((-\infty,\infty)\), and end behavior is as \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\), and as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\).
Note that a sign chart and other points were not needed to help us determine the general shape of the graph of \(f(x)\) in the previous example. This is not always the case when sketching graphs of rational functions.
Find the asymptotes, intercepts, domain, range, and end behavior of \(B(x)=\dfrac{3x^2-7x-6}{x^2+3x-4}\). Round to the first decimal place for the range, if applicable. Then, use this information to sketch \(B(x)\).
- Answer
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Vertical asymptotes at \(x=-4 \text{ and } x=1\), horizontal asymptote at \(y=3\), graph intersects horizontal asymptote at \(\left(\dfrac{3}{8},3\right)\),
intercepts at \(\left(-\dfrac{2}{3},0\right) \text{, } (3,0) \text{ and } \left(0,\dfrac{3}{2}\right)\),
domain is \((-\infty,-4) \cup (-4,1) \cup (4,\infty)\), range is \((-\infty,\infty)\), as \(x \rightarrow \pm \infty\), \(B(x) \rightarrow 3\).
The graph of \(B(x)\) is shown in Figure \(\PageIndex{8}\).
Find the asymptotes, intercepts, domain, range, and end behavior of \(g(t)=\dfrac{9t+9-t^3-t^2}{t^3+t^2+3t+3}\). Round to the first decimal place for the range, if applicable. Then, use this information to sketch \(g(t)\).
Solution
To begin, we need factor. Write terms in descending order and factoring by grouping,
\[g(t) =\dfrac{9t+9-t^3-t^2}{t^3+t^2+3t+3} = \dfrac{-(t+3)(t-3)(t+1)}{(t^2+3)(t+1)} \nonumber\]
Note that \(t^2+3\) does factor into \((t+\sqrt{3}i)(t-\sqrt{3}i)\) but we are graphing in a real coordinate system so we need not be concerned about these factors.
From the factored form of \(g(t)\), we find that the domain is \((-\infty,-1) \cup (-1,\infty)\). There is a hole at \((-1,2)\) ( remember that the y-coordinate is found by simplifying the function first, then evaluating at -1 ). The t -intercepts are at \((-3,0) \text{ and } (3,0)\).
The y-intercept is \((0,3)\) since \(g(0)=\dfrac{9}{3}=3\).
There are no vertical asymptotes since there are no real numbers that make the denominator zero after simplifying.
There is a horizontal asymptote at \(y=-1\). Verify on your own that \(g\) does not intersect the horizontal asymptote.
Sketch the the asymptote, points, and hole, as shown in Figure \(\PageIndex{9}\).
We are able to connect this information to sketch the curve, as shown in Figure \(\PageIndex{10}\), without further information since the points found are below the asymptote and we know that the graph does not intersect the asymptote.
No vertical asymptotes, horizontal asymptote at \(y=-1\), intercepts at \((-3,0) \text{, } (3,0) \text{, and } (0,3)\), hole at \((-1,2)\),
domain is \((-\infty,\infty)\), range is \((1,3]\), as \(x \rightarrow \pm \infty\), \(g(t) \rightarrow -1\).
Note that we know there is a local maximum from sketching the graph but we are unable to just assume that it is at the y-intercept. The graphing calculator helped us verify that there is a maximum at the y-intercept in this example. In calculus, you will learn ways to find local extrema without aid of graphing technology.
Find the asymptotes, intercepts, domain, and range of \(A(x)=\dfrac{x^2+6x-40}{2x^3-11x^2+10x+8} \). Round to the third decimal place for the range, if applicable. Then, use this information to sketch \(A(x)\).
Solution
First, factor \(A(x)\) ( you will need to use some factoring techniques discussed recently to factor ), \[A(x)=\dfrac{x^2+6x-40}{2x^3-11x^2+10x+8}=\dfrac{(x-4)(x+10)}{(x-4)(x-2)(2x+1)}\nonumber\]
From the factored form of \(A(x)\), we find the domain is \((-\infty,-\dfrac{1}{2}) \cup (-\dfrac{1}{2},2) \cup (2,4) \cup (4, \infty)\). There is a hole at \(\left(4,\dfrac{7}{9}\right)\).
There are vertical asymptotes at \(x=2 \text{ and } x=-\dfrac{1}{2}\).
There is a horizontal asymptote at \(y=0\). We do not need to check if \(A(x)\) intersects the asymptote as we gain this information when finding the x-intercepts since the horizontal asymptote is the x-axis.
The x-intercept is \((-10,0)\). The y-intercept is \((0,-5)\).
Sketch the the asymptote, points, and hole as shown in Figure \(\PageIndex{11}\).
On the interval \(\left(-\infty,-\dfrac{1}{2}\right)\), we know that as \(x \rightarrow -\infty \text{, } A \rightarrow 0\). Also, as \(x \rightarrow -\dfrac{1}{2}^-\), \(A \rightarrow -\infty\) or \(A \rightarrow \infty\). There is also an x-intercept at \((-10,0)\) within this interval. Applying what we have learned about polynomials, the graph must pass through the x-intercept since it comes from the factor \((x+10)\) which is a single root of multiplicity.
Currently, this is not enough information to sketch the graph on this interval since we do not know if the graph will be above or below the x-axis on the interval \((-\infty,-10)\). This is also the case for the interval \(\left(-10,-\dfrac{1}{2}\right)\). We can either set up a sign chart of find a point within one of these intervals to find out.
Using the latter, let's evaluate at \(x=-6\). \[A(-6)=\dfrac{1}{22}\nonumber\]so we can conclude that the graph is above the x-axis on \(\left(-10,-\dfrac{1}{2}\right)\). The graph must be below the x-axis on \((-\infty,-10)\) because we know the graph passed through the x-intercept at \((-10,0)\).
On the interval \(\left(-\dfrac{1}{2},2\right)\), we know there is a y-intercept at \((0,-5)\) and no x-intercept. This indicates that the graph will be below the x-axis. Since we know that \(A \rightarrow -\infty\) as we approach the vertical asymptotes, we also know that there will be a local maximum within this interval.
Currently, we do not have enough information to sketch the graph on the interval \((2,\infty)\). We know that \(A\) has to approach the asymptotes, but we do not know if the graph is above or below the x-axis. Again, we may use a sign chart or find a point within this interval.
Let's evaluate at \(x=5\) which is within this interval. \[A(5)=\dfrac{5}{11}\nonumber\]so we can conclude that the graph is above the x-axis.
The graph of \(A\) is shown in Figure \(\PageIndex{12}\).
We still need to find the range. Upon viewing the graph, we can see the range is not all real numbers so we will need to approximate the range using graphing technology. We can see there is a maximum within the interval \(\left(-\dfrac{1}{2},2\right)\). Using our graphing calculator, we find the maximum is \(\approx -3.428\).
Remember that there is is an x-intercept at \((-10,0)\). Although not easily viewable in the window used to sketch the graph in Figure \(\PageIndex{12}\), we know that they must be a local minimum within the interval \(\left(-\dfrac{1}{2},2\right)\). To approximate it using graphing technology you will need to find an appropriate window to view the local minimum so that you can then find it. The local minimum is \(\approx -0.012\). We can conclude from this that the range is \((-\infty,-3.428] \cup [-0.012,\infty)\).
One important thing to note about the graph of \(A(x)\). Most graphing viewing windows will not show the local minimum. It is important for hand drawn sketches to include this information. Also, this example also illustrates the importance of understanding how functions behave as we would have not known there was a local minimum within the interval \(\left(-\infty,-\dfrac{1}{2}\right)\) had we not investigated the behavior of the function.
Find the asymptotes, intercepts, domain, and range of \(w(t)=\dfrac{3}{t}-\dfrac{3t+2}{t^2+1}\). Round to the third decimal place for the range, if applicable. Then, use this information to sketch \(w(t)\).
- Answer
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Note that the function must be simplified before finding the graphical information we are asked to find. \[w(t)=\dfrac{3}{t}-\dfrac{3t+2}{t^2+1}=\dfrac{3-2t}{t(t^2+1)}\nonumber \]
The vertical asymptote is \(x=0\). The horizontal asymptote is \(y=0\).
There is a t- intercept at \((\dfrac{3}{2},0)\). This is the only intercept.
The domain is \((-\infty,0) \cup (0,\infty)\). The range is (\(-\infty,\infty)\). ( note that there is a local minimum at \(\approx (2.382,-0.111)\).
Graph is shown in Figure \(\PageIndex{13}\).
Finding Equations of Rational Functions that Model Graphical Information
Now that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can use information given by a graph to write the function. A rational function written in factored form will have an x -intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable discontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set of x -intercepts by introducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by introducing a corresponding set of factors.
Write an equation for the rational function shown in Figure \(\PageIndex{14}\).
Solution
The graph appears to have x-intercepts at \(x=–2\) and \(x=3\). At both, the graph passes through the intercept, suggesting linear factors for the numerator. The graph has two vertical asymptotes at \(x=–1\) and \(x=2\) indicating denominator factors. Notice that at \(x=2\) that the graph approaches the asymptote from the same side. This indicates the factor \(x-2\) would be squared ( think about what we know about sign charts and polynomial behavior ). See Figure \(\PageIndex{15}\).
We can use this information to write a function of the form \[f(x)=a\dfrac{(x+2)(x−3)}{(x+1){(x−2)}^2}\nonumber\]
To find the stretch factor, \(a\), we can evaluate at a clear point on the graph, such as the y-intercept \((0,–2)\).
\[\begin{align*}−2&=a\dfrac{(0+2)(0−3)}{(0+1){(0−2)}^2} \\[6pt] -2&=a\left(\dfrac{−6}{4} \right) \\[6pt] a&=\dfrac{−8}{−6}=\dfrac{4}{3} \end{align*}\nonumber\]
This gives us a final function of \(f(x)=\dfrac{4(x+2)(x−3)}{3(x+1){(x−2)}^2}\).


5.2: Arithmetic Sequences
In this section, you will:
- Find the common difference for an arithmetic sequence.
- Write terms of an arithmetic sequence.
- Use a recursive formula for an arithmetic sequence.
- Use an explicit formula for an arithmetic sequence.
Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation . One method of calculating depreciation is straight-line depreciation , in which the value of the asset decreases by the same amount each year.
As an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.
Finding Common Differences
The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.
The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.
An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference . If \( a_1 \) is the first term of an arithmetic sequence and \(d\) is the common difference, the sequence will be:\[ \{a_n\} = \{ a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \ldots \}. \nonumber \]
Is each sequence arithmetic? If so, find the common difference.
- \( \{a_n\} = \{1,2,4,8,16,\ldots\}\)
- \( \{ b_n \} = \{−3,1,5,9,13,\ldots\}\)
- Solution
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Subtract each term from the subsequent term to determine whether a common difference exists.
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The sequence is not arithmetic because there is no common difference.
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The sequence is arithmetic because there is a common difference. The common difference is 4.
The graph of each of these sequences is shown in Figure \( \PageIndex{ 1 } \). We can see from the graphs that, although both sequences show growth, \(\{a_n\}\) is not linear whereas \(\{b_n\}\) is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.
Figure \( \PageIndex{ 1 } \) -
The sequence is not arithmetic because there is no common difference.
If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?
No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.
Is the given sequence arithmetic? If so, find the common difference.
- \( \{c_n\} = \{18,16,14,12,10, \ldots \}\)
- \( \{ d_n \} = \{1,3,6,10,15, \ldots \}\)
Writing Terms of Arithmetic Sequences
Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of \(n\) and \(d\) into formula below.\[ a_n = a_1 +(n−1)d. \nonumber \]
Write the first five terms of the arithmetic sequence with \( a_1 = 17\) and \(d = −3\).
- Solution
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Adding \(−3\) is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.
The first five terms are \(\{17,14,11,8,5\}\)
As expected, the graph of the sequence consists of points on a line as shown in Figure \( \PageIndex{ 2 } \).
Figure \( \PageIndex{ 2 } \)
List the first five terms of the arithmetic sequence with \( a_1 =1\) and \(d=5\).
Given \( a_1 =8\) and \( a_4 =14\), find \( a_5 \).
- Solution
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The sequence can be written in terms of the initial term, 8, and the common difference \(d\).\[ \{8, 8+d, 8+2d, 8+3d \}. \nonumber \]We know the fourth term equals 14; we know the fourth term has the form \( a_1 + 3d = 8+3d\).
We can find the common difference \(d\).\[ \begin{array}{rrclcl} & a_n & = & a_1 + (n - 1)d & & \\[6pt] \implies & 14 & = & 8 + 3d & \quad & \left( \text{substituting }n = 4, a_1 = 8, \text{ and }a_4 = 14 \right) \\[6pt] \implies & 6 & = & 3d & \quad & \left( \text{subtracting }8 \text{ from both sides} \right) \\[6pt] \implies & 2 & = & d & \quad & \left( \text{dividing both sides by }3 \right) \\[6pt] \end{array} \nonumber \]Find the fifth term by adding the common difference to the fourth term.\[ a_5 = a_4 + 2 = 16. \nonumber \]
Notice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation \( a_n = a_1 + ( n−1 )d\).
Given \( a_3 =7\) and \( a_5 =17\), find \( a_2 \).
Using Recursive Formulas for Arithmetic Sequences
Arithmetic sequences can also be defined in terms of the previous term using a recursive formula. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.\[ a_n = a_{n−1} + d, \quad n \geq 2 \nonumber \]
The recursive formula for an arithmetic sequence with common difference \(d\) is:\[ a_n = a_{n−1} +d, \quad n \geq 2. \nonumber \]
Write a recursive formula for the arithmetic sequence.\[ \{−18, −7, 4, 15, 26, \ldots \} \nonumber \]
- Solution
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The first term is given as \(−18\) . The common difference can be found by subtracting the first term from the second term.\[ d = −7 − (−18) = 11 \nonumber \]Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.\[ \begin{array}{rcl} a_1 & = & 18 \\[6pt] a_n & = & a_{n - 1} + 11, \text{ for }n \geq 2 \\[6pt] \end{array} \nonumber \]
We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure \( \PageIndex{ 3 } \). The growth pattern of the sequence shows the constant difference of 11 units.
Figure \( \PageIndex{ 3 } \)
Do we have to subtract the first term from the second term to find the common difference?
No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.
Write a recursive formula for the arithmetic sequence.\[\{ 25, 37, 49, 61, \ldots \} \nonumber \]
Using Explicit Formulas for Arithmetic Sequences
We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.\[ a_n = a_1 + d ( n − 1 ) \nonumber \]To find the \( y \)-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.
The common difference is \(−50\), so the sequence represents a linear function with a slope of \(−50\). To find the \(y\)-intercept, we subtract \(−50\) from 200:\[ 200 − (−50) = 200 + 50 = 250. \nonumber \]The graph is shown in Figure \( \PageIndex{ 4 } \).
Figure \( \PageIndex{ 4 } \)
Recall the slope-intercept form of a line is \(y = mx+b\). When dealing with sequences, we use \( a_n \) in place of \(y\) and \(n\) in place of \(x\). If we know the slope and vertical intercept of the function, we can substitute them for \(m\) and \(b\) in the slope-intercept form of a line. Substituting \(−50\) for the slope and \(250\) for the vertical intercept, we get the following equation:\[ a_n = −50n + 250. \nonumber \]We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is \( a_n = 200 − 50(n−1)\), which simplifies to \( a_n = −50n + 250\).
Write an explicit formula for the arithmetic sequence.\[ \{2, 12, 22, 32, 42, \ldots \} \nonumber \]
- Solution
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The common difference can be found by subtracting the first term from the second term.\[ \begin{array}{rcl} d & = & a_2 - a_1 \\[6pt] & = & 12 - 2 \\[6pt] & = & 10 \\[6pt] \end{array} \nonumber \]The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.\[ \begin{array}{rrcl} & a_n & = & 2 + 10(n - 1) \\[6pt] \implies & a_n & = & 10n - 8 \\[6pt] \end{array} \nonumber \]The graph of this sequence, represented in Figure \( \PageIndex{ 5 } \), shows a slope of 10 and a vertical intercept of \(−8\) .
Figure \( \PageIndex{ 5 } \)
Write an explicit formula for the following arithmetic sequence.\[\{50,47,44,41, \ldots \} \nonumber \]
Finding the Number of Terms in a Finite Arithmetic Sequence
Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.
Find the number of terms in the finite arithmetic sequence.\[\{ 8, 1, –6, \ldots, –41 \} \nonumber \]
- Solution
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The common difference can be found by subtracting the first term from the second term.\[1−8=−7 \nonumber \]The common difference is \(−7\) . Substitute the common difference and the initial term of the sequence into the \(n^{\text{th}}\) term formula and simplify.\[ \begin{array}{rrclcl} & a_n & = & a_1 + d(n - 1) & & \\[6pt] \implies & a_n & = & 8 + (-7)(n - 1) & \quad & \left( \text{substituting} \right) \\[6pt] \implies & a_n & = & 15 - 7n & \quad & \left( \text{simplifying} \right) \\[6pt] \end{array} \nonumber \]Substitute \(−41\) for \( a_n \) and solve for \(n\)\[ \begin{array}{rrclcl} & -41 & = & 15 - 7n & & \\[6pt] \implies & -56 & = & -7n & \quad & \left( \text{subtracting }15\text{ from both sides} \right) \\[6pt] \implies & 8 & = & n & \quad & \left( \text{dividing both sides by }-7 \right) \\[6pt] \end{array} \nonumber \]There are eight terms in the sequence.
Find the number of terms in the finite arithmetic sequence.\[\{6, 11, 16, \ldots, 56 \} \nonumber \]
Solving Application Problems with Arithmetic Sequences
In many application problems, it often makes sense to use an initial term of \( a_0 \) instead of \( a_1 \). In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:\[ a_n = a_0 + d n. \nonumber \]
A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.
- Write a formula for the child’s weekly allowance in a given year.
- What will the child’s allowance be when he is 16 years old?
- Solution
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The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2. Let \(A\) be the amount of the allowance and \(n\) be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:\[ A_n = 1 + 2 n. \nonumber \]
- We can find the number of years since age 5 by subtracting.\[16 − 5 = 11 \nonumber \]We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.\[ A_{11} = 1 + 2 ( 11 ) = 23 \nonumber \]The child’s allowance at age 16 will be $23 per week.
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A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?
Access this online resource for additional instruction and practice with arithmetic sequences.


18.2: Problems of Work Adjustment
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- How do you recognize the symptoms of stress in yourself and in others?
Failure to adjust to work represents a major problem in industry today. It has been estimated that between 80 and 90 percent of industrial accidents are caused by personal factors. 1 Turnover, absenteeism, drug abuse, alcoholism, and sabotage remain relatively permanent fixtures of most contemporary work organizations. To the extent that individuals are unable to adjust to work, we would expect them to persist in counterproductive behavior.
W. S. Neff has identified five types of people who have problems adjusting to work. He suggests that each of the five types represents a “clinical picture of different varieties of work psychopathology”:
- Type I: People who lack motivation to work. These individuals have a negative conception of the work role and choose to avoid it.
- Type II: People whose predominating response to the demand to be productive is fear or anxiety.
- Type III: People who are characterized predominantly by open hostility and aggression.
- Type IV: People who are characterized by marked dependency. These people often exhibit the characteristic of helplessness. They are constantly seeking advice from others and are unable to initiate any action on their own.
- Type V: People who display a marked degree of social naïveté. These individuals lack perception when it comes to the needs and feelings of others and may not realize that their behavior elicits reactions from and has an effect on others. Typically, these individuals are socially inept and unaware of appropriate behavior in ordinary social situations.
Several important points follow from this analysis.  Several deeply ingrained psychological problems keep people from making normal adjustments in many cases. Second, note that only one of the five types (Type I) exhibits a motivational problem. Managers must look beyond motivation for answers to the psychopathology of work. One type (Type V) exhibits a form of personality disorder, or at least social immaturity. But the remaining three types—those exhibiting anxiety, aggression, or dependency—all have problems relating not only to personality, but more importantly, to how the nature of the job affects that personality. In fact, anxiety, aggression, and dependency are major factors inherent in stressful jobs in organizations. Hence, it seems that at least three of the five reasons for failure to adjust to work relate to the extent to which the job is experienced as stressful and causes the individual to want to withdraw.
It has been wisely observed that “if, under stress, a man goes all to pieces, he will probably be told to pull himself together. It would be more effective to help him identify the pieces and to understand why they have come apart.” This is the role of the contemporary manager in dealing with stress. Managers cannot simply ignore the existence of stress on the job. Instead, they have a responsibility to understand stress and its causes.
We will explore the topic of work-related stress in several stages, first examining major organizational and personal influences on stress, then considering several outcomes of stress, and finally exploring methods for coping with stress on the job. Throughout, emphasis will be placed on how stress and its consequences affect people at work and what role managers can play in attempting to minimize the effects of stress on both the individual and the organization. We will make liberal use of practical examples, and, as usual, you will be given an opportunity to evaluate yourself on several aspects of stress and wellness in organizations.
Work-Related Stress
For our purposes here, stress will be defined as a physical and emotional reaction to potentially threatening aspects of the environment. This definition points to a poor fit between individuals and their environments. Either excessive demands are being made, or reasonable demands are being made that individuals are ill-equipped to handle. Under stress, individuals are unable to respond to environmental stimuli without undue psychological and/or physiological damage, such as chronic fatigue, tension, or high blood pressure. This damage resulting from experienced stress is usually referred to as strain .
Before we examine the concept of work-related stress in detail, several important points need to be made. First, stress is pervasive in the work environment. 4 Most of us experience stress at some time. For instance, a job may require too much or too little from us. In fact, almost any aspect of the work environment is capable of producing stress. Stress can result from excessive noise, light, or heat; too much or too little responsibility; too much or too little work to accomplish; or too much or too little supervision.
Second, it is important to note that all people do not react in the same way to stressful situations, even in the same occupation. One individual (a high-need achiever) may thrive on a certain amount of job-related tension; this tension may serve to activate the achievement motive. A second individual may respond to this tension by worrying about her inability to cope with the situation. Managers must recognize the central role of individual differences in the determination of experienced stress.
Often the key reason for the different reactions is a function of the different interpretations of a given event that different people make, especially concerning possible or probable consequences associated with the event. For example, the same report is required of student A and student B on the same day. Student A interprets the report in a very stressful way and imagines all the negative consequences of submitting a poor report. Student B interprets the report differently and sees it as an opportunity to demonstrate the things she has learned and imagines the positive consequences of turning in a high-quality report. Although both students face essentially the same event, they interpret and react to it differently.
Third, all stress is not necessarily bad. Although highly stressful situations invariably have dysfunctional consequences, moderate levels of stress often serve useful purposes. A moderate amount of job-related tension not only keeps us alert to environmental stimuli (possible dangers and opportunities), but in addition often provides a useful motivational function. Some experts argue that the best and most satisfying work that employees do is work performed under moderate stress. Some stress may be necessary for psychological growth, creative activities, and the acquisition of new skills. Learning to drive a car or play a piano or run a particular machine typically creates tension that is instrumental in skill development. It is only when the level of stress increases or when stress is prolonged that physical or psychological problems emerge.
General Adaptation Syndrome
The general physiological response to stressful events is believed to follow a fairly consistent pattern known as the general adaptation syndrome . General adaptation syndrome consists of three stages (see Figure \(\PageIndex{1}\)). The first stage, alarm, occurs at the first sign of stress. Here the body prepares to fight stress by releasing hormones from the endocrine glands. During this initial stage, heartbeat and respiration increase, blood sugar level rises, muscles tense up, pupils dilate, and digestion slows. At this stage the body prepares basically for a “fight or flight” response. That is, the body prepares to either get away from the threat or to combat it. Following this initial shock, the body moves into the second stage, resistance. The body attempts to repair any damage and return to a condition of stability and equilibrium. If successful, physical signs of stress will disappear. If the stress continues long enough, however, the body’s capacity for adaptation becomes exhausted. In this third stage, exhaustion , defenses wear away, and the individual experiences a variety of stress-related illnesses, including headaches, ulcers, and high blood pressure. This third stage is the most severe and presents the greatest threat both to individuals and to organizations.
Types of Stress: Frustration and Anxiety
There are several different ways to categorize stress. However, from a managerial perspective, it is useful to focus on only two forms: frustration and anxiety. Frustration refers to a psychological reaction to an obstruction or impediment to goal-oriented behavior. Frustration occurs when an individual wishes to pursue a certain course of action but is prevented from doing so. This obstruction may be externally or internally caused. Examples of people experiencing obstacles that lead to frustration include a salesperson who continually fails to make a sale, a machine operator who cannot keep pace with the machine, or even a person ordering coffee from a machine that fails to return the correct change. The prevalence of frustration in work organizations should be obvious from this and other examples.
Whereas frustration is a reaction to an obstruction in instrumental activities or behavior, anxiety is a feeling of inability to deal with anticipated harm. Anxiety occurs when people do not have appropriate responses or plans for coping with anticipated problems. It is characterized by a sense of dread, a foreboding, and a persistent apprehension of the future for reasons that are sometimes unknown to the individual.
What causes anxiety in work organizations? Hamner and Organ suggest several factors:
“Differences in power in organizations which leave people with a feeling of vulnerability to administrative decisions adversely affecting them; frequent changes in organizations, which make existing behavior plans obsolete; competition, which creates the inevitability that some persons lose ‘face,’ esteem, and status; and job ambiguity (especially when it is coupled with pressure). To these may be added some related factors, such as lack of job feedback, volatility in the organization’s economic environment, job insecurity, and high visibility of one’s performance (successes as well as failures). Obviously, personal, non-organizational factors come into play as well, such as physical illness, problems at home, unrealistically high personal goals, and estrangement from one’s colleagues or one’s peer group.”
Concept Check
- What is work related stress?


1.7: Lame's Theorem
In this section, we give an estimate to the number of steps needed to find the greatest common divisor of two integers using the Euclidean algorithm. To do this, we have to introduce the Fibonacci numbers for the sake of proving a lemma that gives an estimate on the growth of Fibonacci numbers in the Fibonacci sequence. The lemma that we prove will be used in the proof of Lame’s theorem.
The Fibonacci sequence is defined recursively by \(f_1=1\), \(f_2=1\), and \[f_{n}=f_{n-1}+f_{n-2} \mbox{for} \ \ n\geq 3.\] The terms in the sequence are called Fibonacci numbers.
In the following lemma, we give a lower bound on the growth of Fibonacci numbers. We will show that Fibonacci numbers grow faster than a geometric series with common ratio \(\alpha=(1+\sqrt{5})/2\).
[lem2] For \(n\geq 3\), we have \(f_n>\alpha^{n-2}\) where \(\alpha=(1+\sqrt{5})/2\).
We use the second principle of mathematical induction to prove our result. It is easy to see that this is true for \(n=3\) and \(n=4\). Assume that \(\alpha^{k-2}<f_k\) for all integers \(k\) where \(k\leq n\). Now since \(\alpha\) is a solution of the polynomial \(x^2-x-1=0\), we have \(\alpha^2=\alpha+1\). Hence \[\alpha^{n-1}=\alpha^2.\alpha^{n-3}=(\alpha+1).\alpha^{n-3}=\alpha^{n-2}+\alpha^{n-3}.\] By the inductive hypothesis, we have \[\alpha^{n-2}<f_n, \ \ \ \alpha^{n-3}<f_{n-1}.\] After adding the two inequalities, we get \[\alpha^{n-1}<f_{n}+f_{n-1}=f_{n+1}.\]
We now present Lame’s theorem.
using the Euclidean algorithm to find the greatest common divisor of two positive integers has number of divisions less than or equal five times the number of decimal digits in the minimum of the two integers.
Let \(a\) and \(b\) be two positive integers where \(a>b\).  Thus we have \[\begin{aligned} r_n&\geq& 1=f_2,\\ r_{n-1}&\geq& 2r_n\geq 2f_2=f_3,\\ r_{n-2}&\geq& r_{n-1}+r_n\geq f_3+f_2=f_4,\\ r_{n-3}&\geq& r_{n-2}+r_{n-1}\geq f_4+f_3=f_5,\\ &.&\\ &.&\\ &.&\\ r_2&\geq& r_3+r_4\geq f_{n-1}+f_{n-2}=f_n,\\ b&=& r_1\geq r_2+r_3\geq f_n+f_{n-1}=f_{n+1}.\end{aligned}\] Thus notice that \(b\geq f_{n+1}\). By Lemma [lem2] , we have \(f_{n+1}>\alpha^{n-1}\) for \(n>2\). As a result, we have \(b>\alpha^{n-1}\). Now notice since \[\log_{10}\alpha>\frac{1}{5},\] we see that \[log_{10}b>(n-1)/5.\] Thus we have \[n-1<5log_{10}b.\] Now let \(b\) has \(k\) decimal digits. As a result, we have \(b<10^k\) and thus \(log_{10}b<k\). Hence we conclude that \(n-1<5k\). Since \(k\) is an integer, we conclude that \(n\leq 5k\).
Exercises
- Find an upper bound for the number of steps in the Euclidean algorithm that is used to find the greatest common divisor of 38472 and 957748838.
- Find an upper bound for the number of steps in the Euclidean algorithm that is used to find the greatest common divisor of 15 and 75. Verify your result by using the Euclidean algorithm to find the greatest common divisor of the two integers.
Contributors and Attributions
-
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


7.3.1: Slicing Solids
Lesson
Let's see what shapes you get when you slice a three-dimensional object.
Exercise \(\PageIndex{1}\): Prisms, Pyramids, and Polyhedra
Describe each shape as precisely as you can. Click on the applet and drag the mouse to show the object turning in 3D.
Exercise \(\PageIndex{2}\): What's the Cross Section
Here are a rectangular prism and a pyramid with the same base and same height. Drag the large red point up and down to move the plane through the solids.
- If we slice each solid parallel to its base halfway up, what shape cross sections would we get? What is the same about the cross sections? What is different?
- If we slice each solid parallel to its base near the top, what shape cross sections would we get? What is the same about the cross sections? What is different?
Are you ready for more?
Describe the cross sections that would result from slicing each solid perpendicular to its base.
Exercise \(\PageIndex{3}\): Card Sort: Cross Sections
Your teacher will give you a set of cards. Sort the images into groups that make sense to you. Be prepared to explain your reasoning.
Exercise \(\PageIndex{4}\): Drawing Cross Sections
Use the applet to draw each cross section and describe it in words. Start each one by dragging the gray bar on the left side across the screen until you see the whole 3D image on the left and the controls on the right.
-
Here is an applet with a rectangular prism, 4 units by 2 units by 3 units.
- A plane cuts the prism parallel to the bottom and top faces.
- The plane moves up and cuts the prism at a different height.
- A vertical plane cuts the prism diagonally.
-
A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.
- A plane cuts the pyramid parallel to the base.
- A vertical plane cuts the prism.
-
A cube has an edge of length 4.
- A plane cuts off the corner of the cube.
- The plane moves farther from the corner and makes a cut through the middle of the cube.
Summary
When we slice a three-dimensional object, we expose new faces that are two dimensional. The two-dimensional face is a cross section . Many different cross sections are possible when slicing the same three-dimensional object.
Here are two peppers. One is sliced horizontally, and the other is sliced vertically, producing different cross sections.
The imprints of the slices represent the two-dimensional faces created by each slice.
It takes practice imagining what the cross section of a three-dimensional object will be for different slices. It helps to experiment and see for yourself what happens!
Glossary Entries
Definition: Base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
Definition: Cross Section
A cross section is the new face you see when you slice through a three-dimensional figure.
For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.
Definition: Prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
Definition: Pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
Practice
Exercise \(\PageIndex{5}\)
A cube is cut into two pieces by a single slice that passes through points \(A\), \(B\), and \(C\). What shape is the cross section?
Exercise \(\PageIndex{6}\)
Describe how to slice the three-dimensional figure to result in each cross section.
Three-dimensional figure:
Cross sections:
Exercise \(\PageIndex{7}\)
Here are two three-dimensional figures.
Describe a way to slice one of the figures so that the cross section is a rectangle.
Exercise \(\PageIndex{8}\)
Each row contains the degree measures of two supplementary angles. Complete the table.
| measure of an angle | measure of its supplement |
|---|---|
| \(80^{\circ}\) | |
| \(25^{\circ}\) | |
| \(119^{\circ}\) | |
| \(x\) |
(From Unit 7.1.2)
Exercise \(\PageIndex{9}\)
Two months ago, the price, in dollars, of a cell phone was \(c\).
- Last month, the price of the phone increased by 10%. Write an expression for the price of the phone last month.
- This month, the price of the phone decreased by 10%. Write an expression for the price of the phone this month.
- Is the price of the phone this month the same as it was two months ago? Explain your reasoning.
(From Unit 4.2.3)


2.2: Atomic Structure, Isotopes, and Atomic Mass
-
- Last updated
- Save as PDF
- Scott Van Bramer
- Widener University
Learning Objectives
- Write and interpret symbols that depict the atomic number, mass number, and charge of an atom or ion
- Define the unified atomic mass unit and average atomic mass
- Calculate average atomic mass and isotopic abundance
- State the periodic law and explain the organization of elements in the periodic table
- Predict the general properties of elements based on their location within the periodic table
- Identify metals, nonmetals, and metalloids by their properties and/or location on the periodic table
Structure of the Atom
The development of modern atomic theory revealed much about the inner structure of atoms. It was learned that an atom contains a very small nucleus composed of positively charged protons and uncharged neutrons, surrounded by a much larger volume of space containing negatively charged electrons. The nucleus contains the majority of an atom’s mass because protons and neutrons are much heavier than electrons, whereas electrons occupy almost all of an atom’s volume. The diameter of an atom is on the order of 10 −10 m, whereas the diameter of the nucleus is roughly 10 −15 m—about 100,000 times smaller. For a perspective about their relative sizes, consider this: If the nucleus were the size of a blueberry, the atom would be about the size of a football stadium (Figure \(\PageIndex{1}\)).
Atoms—and the protons, neutrons, and electrons that compose them—are extremely small. For example, a carbon atom weighs less than 2 \(\times\) 10 −23 g, and an electron has a charge of less than 2 \(\times\) 10 −19 C (coulomb). When describing the properties of tiny objects such as atoms, we use appropriately small units of measure, such as the atomic mass unit (amu) and the fundamental unit of charge (e) . The atomic mass unit (amu) was originally defined based on hydrogen, the lightest element, and oxygen. Because of this ambiguity it was redefined in 1961 as the unified atomic mass unit (u) with regard to the most abundant isotope of carbon, atoms of which are assigned masses of exactly 12 u. Thus, one u is exactly \(1/12\) of the mass of one carbon-12 atom. The Dalton (Da) and the atomic mass unit (amu) are alternative units that are almost identical to the u. The fundamental unit of charge (also called the elementary charge) equals the magnitude of the charge of an electron (e) with e = 1.602 \(\times\) 10 −19 C.
A proton has a mass of 1.0073 u and a charge of 1+. A neutron is a slightly heavier particle with a mass 1.0087 u and a charge of zero; as its name suggests, it is neutral. The electron has a charge of 1− and is a much lighter particle with a mass of about 0.00055 u (it would take about 1800 electrons to equal the mass of one proton. The properties of these fundamental particles are summarized in Table \(\PageIndex{1}\). You may notice the sum of an atom’s subatomic particles does not equal the atom’s actual mass: The total mass of six protons, six neutrons, and six electrons is 12.0993 u, slightly larger than the 12.00 u of an actual carbon-12 atom. This “missing” mass is known as the mass defect, it is related to the energy holding the atom together.
| Name | Location | Charge (C) | Unit Charge | Mass (u) | Significance |
|---|---|---|---|---|---|
| electron | outside nucleus | \(−1.602 \times 10^{−19}\) | 1− | 0.00055 | charge = proton - electron |
| proton | nucleus | \(1.602 \times 10^{−19}\) | 1+ | 1.00727 | atomic number = proton |
| neutron | nucleus | 0 | 0 | 1.00866 | mass = proton + neutron |
The number of protons in the nucleus of an atom is its atomic number (\(Z\)) . This is the defining trait of an element: Its value determines the identity of the atom. For example, any atom that contains six protons is the element carbon and has the atomic number 6, regardless of how many neutrons or electrons it may have. A neutral atom must contain the same number of positive and negative charges, so the number of protons equals the number of electrons. Therefore, the atomic number also indicates the number of electrons in an uncharged atom. The total number of protons and neutrons in an atom is called its mass number (A) . The number of neutrons is therefore the difference between the mass number and the atomic number: A – Z = number of neutrons.
\[\begin{align*}
\ce{atomic\: number\:(Z)\: &= \:number\: of\: protons\\
mass\: number\:(A)\: &= \:number\: of\: protons + number\: of\: neutrons\\
A-Z\: &= \:number\: of\: neutrons}
\end{align*}\]
Atoms are electrically neutral if they contain the same number of positively charged protons and negatively charged electrons. When the numbers of these subatomic particles are not equal, the atom is electrically charged and is called an ion . The charge of an atom is defined as follows:
Atomic charge = number of protons − number of electrons
As will be discussed in more detail later in this chapter, atoms (and molecules) typically acquire charge by gaining or losing electrons. An atom that gains one or more electrons will exhibit a negative charge and is called an anion . Positively charged atoms called cations are formed when an atom loses one or more electrons. For example, a neutral sodium atom (Z = 11) has 11 electrons. If this atom loses one electron, it will become a cation with a 1+ charge (11 − 10 = 1+). A neutral oxygen atom (Z = 8) has eight electrons, and if it gains two electrons it will become an anion with a 2− charge (8 − 10 = 2−).
Example \(\PageIndex{1}\): Composition of an Atom
Iodine is an essential trace element in our diet; it is needed to produce thyroid hormone. Insufficient iodine in the diet can lead to the development of a goiter, an enlargement of the thyroid gland (Figure \(\PageIndex{2}\)).
The addition of small amounts of iodine to table salt (iodized salt) has essentially eliminated this health concern in the United States, but as much as 40% of the world’s population is still at risk of iodine deficiency. The iodine atoms are added as anions, and each has a 1− charge and a mass number of 127. Determine the numbers of protons, neutrons, and electrons in one of these iodine anions.
Solution
The atomic number of iodine (53) tells us that a neutral iodine atom contains 53 protons in its nucleus and 53 electrons outside its nucleus. Because the sum of the numbers of protons and neutrons equals the mass number, 127, the number of neutrons is 74 (127 − 53 = 74). Since the iodine is added as a 1− anion, the number of electrons is 54 [53 – (1–) = 54].
Exercise \(\PageIndex{1}\)
An ion of platinum has a mass number of 195 and contains 74 electrons. How many protons and neutrons does it contain, and what is its charge?
- Answer
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78 protons; 117 neutrons; charge is 4+
Chemical Symbols
A chemical symbol is an abbreviation that we use to indicate an element or an atom of an element. For example, the symbol for mercury is Hg (Figure \(\PageIndex{3}\)). We use the same symbol to indicate one atom of mercury (microscopic domain) or to label a container of many atoms of the element mercury (macroscopic domain).
The symbols for several common elements and their atoms are listed in Table \(\PageIndex{2}\). Some symbols are derived from the common name of the element; others are abbreviations of the name in another language. Symbols have one or two letters, for example, H for hydrogen and Cl for chlorine. To avoid confusion with other notations, only the first letter of a symbol is capitalized. For example, Co is the symbol for the element cobalt, but CO is the notation for the compound carbon monoxide, which contains atoms of the elements carbon (C) and oxygen (O). All known elements and their symbols are in the periodic table.
| Element | Symbol | Element | Symbol |
|---|---|---|---|
| aluminum | Al | iron | Fe (from ferrum ) |
| bromine | Br | lead | Pb (from plumbum ) |
| calcium | Ca | magnesium | Mg |
| carbon | C | mercury | Hg (from hydrargyrum ) |
| copper | Cu (from cuprum ) | silicon | Si |
| gold | Au (from aurum ) | sodium | Na (from natrium ) |
Traditionally, the discoverer (or discoverers) of a new element names the element. However, until the name is recognized by the International Union of Pure and Applied Chemistry (IUPAC), the recommended name of the new element is based on the Latin word(s) for its atomic number. For example, element 106 was called unnilhexium (Unh), element 107 was called unnilseptium (Uns), and element 108 was called unniloctium (Uno) for several years. These elements are now named after scientists or locations; for example, element 106 is now known as seaborgium (Sg) in honor of Glenn Seaborg, a Nobel Prize winner who was active in the discovery of several heavy elements.
Isotopes
The symbol for a specific isotope of any element is written by placing the mass number as a superscript to the left of the element symbol (Figure \(\PageIndex{4}\)). The atomic number is sometimes written as a subscript preceding the symbol, but since this number defines the element’s identity, as does its symbol, it is often omitted. For example, magnesium exists as a mixture of three isotopes, each with an atomic number of 12 and with mass numbers of 24, 25, and 26, respectively. These isotopes can be identified as 24 Mg, 25 Mg, and 26 Mg. These isotope symbols are read as “element, mass number” and can be symbolized consistent with this reading. For instance, 24 Mg is read as “magnesium 24,” and can be written as “magnesium-24” or “Mg-24.” 25 Mg is read as “magnesium 25,” and can be written as “magnesium-25” or “Mg-25.” All magnesium atoms have 12 protons in their nucleus. They differ only because a 24 Mg atom has 12 neutrons in its nucleus, a 25 Mg atom has 13 neutrons, and a 26 Mg has 14 neutrons.
Figure \(\PageIndex{4}\) also shows how the charge of an atom is represented with the symbol for an element. In this example, the helium has a charge of +2 which is written as a superscript following the element symbol: 4 He 2 + . The magnesium isotope has a mass of 24 and a charge of +2, so it is written as 24 Mg 2 + .
Information about the naturally occurring isotopes of elements with atomic numbers 1 through 5 is given in Table \(\PageIndex{2}\). Note that in addition to standard names and symbols, the isotopes of hydrogen are often referred to using common names and accompanying symbols. Hydrogen-2, symbolized 2 H, is also called deuterium and sometimes symbolized D. Hydrogen-3, symbolized 3 H, is also called tritium and sometimes symbolized T.
| Element | Symbol | Atomic Number | Number of Protons | Number of Neutrons | Mass (u) | % Natural Abundance |
|---|---|---|---|---|---|---|
| hydrogen |
\(\ce{^1_1H}\)
(protium) |
1 | 1 | 0 | 1.0078 | 99.989 |
|
\(\ce{^2_1H}\)
(deuterium) |
1 | 1 | 1 | 2.0141 | 0.0115 | |
|
\(\ce{^3_1H}\)
(tritium) |
1 | 1 | 2 | 3.01605 | — (trace) | |
| helium | \(\ce{^3_2He}\) | 2 | 2 | 1 | 3.01603 | 0.00013 |
| \(\ce{^4_2He}\) | 2 | 2 | 2 | 4.0026 | 100 | |
| lithium | \(\ce{^6_3Li}\) | 3 | 3 | 3 | 6.0151 | 7.59 |
| \(\ce{^7_3Li}\) | 3 | 3 | 4 | 7.0160 | 92.41 | |
| beryllium | \(\ce{^9_4Be}\) | 4 | 4 | 5 | 9.0122 | 100 |
| boron | \(\ce{^{10}_5B}\) | 5 | 5 | 5 | 10.0129 | 19.9 |
| \(\ce{^{11}_5B}\) | 5 | 5 | 6 | 11.0093 | 80.1 |
Atomic Mass and Isotopes
Because each proton and each neutron contribute approximately one u to the mass of an atom, and each electron contributes far less, the atomic mass of a single atom is approximately equal to its mass number (a whole number). However, the average masses of atoms of most elements are not whole numbers because most elements exist naturally as mixtures of two or more isotopes.
The mass of an element shown in a periodic table or listed in a table of atomic masses is a weighted, average mass of all the isotopes present in a naturally occurring sample of that element. This is equal to the sum of each individual isotope’s mass multiplied by its fractional abundance.
\[\mathrm{average\: mass}=\sum_{i}(\mathrm{fractional\: abundance\times isotopic\: mass})_i\]
For example, the element boron is composed of two isotopes: About 19.9% of all boron atoms are 10 B with a mass of 10.0129 u, and the remaining 80.1% are 11 B with a mass of 11.0093 u. The average atomic mass for boron is calculated to be:
\[\begin{align*}
\textrm{boron average mass} &=\mathrm{(0.199\times10.0129\: u)+(0.801\times11.0093\: u)}\\
&=\mathrm{1.99\: u+8.82\: u}\\
&=\mathrm{10.81\: u}
\end{align*}\]
It is important to understand that no single boron atom weighs exactly 10.8 u; 10.8 u is the average mass of a large number of naturally occurring boron atoms. The individual boron atoms weigh either approximately 10 u or 11 u.
Example \(\PageIndex{2}\): Calculation of Average Atomic Mass
A meteorite found in central Indiana contains traces of the noble gas neon picked up from the solar wind during the meteorite’s trip through the solar system. Analysis of a sample of the gas showed that it consisted of 91.84% 20 Ne (mass 19.9924 u), 0.47% 21 Ne (mass 20.9940 u), and 7.69% 22 Ne (mass 21.9914 u). What is the average mass of the neon in the solar wind?
Solutio n
\[\begin{align*}
\mathrm{average\: mass} &=\mathrm{(0.9184\times19.9924\: u)+(0.0047\times20.9940\: u)+(0.0769\times21.9914\: u)}\\
&=\mathrm{(18.36+0.099+1.69)\:u}\\
&=\mathrm{20.15\: u}
\end{align*}\]
The average mass of a neon atom in the solar wind is 20.15 u. The average mass of a terrestrial neon atom is 20.1796 u. This result demonstrates that we may find slight differences in the natural abundance of isotopes, depending on their origin. For some elements the differences in these isotope abundances are used to identify where the element came from. Neon from a meteorite has a different isotope distribution thane neon from the earth's crust. Lead from different mines can have different isotope distributions depending on the geology of the locations.
Exercise \(\PageIndex{2}\)
A sample of magnesium is found to contain 78.70% of 24 Mg atoms (mass 23.98 u), 10.13% of 25 Mg atoms (mass 24.99 u), and 11.17% of 26 Mg atoms (mass 25.98 u). Calculate the average mass of a Mg atom.
- Answer
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24.31 u
We can also do variations of this type of calculation, as shown in the next example.
Example \(\PageIndex{3}\): Calculation of Percent Abundance
Naturally occurring chlorine consists of 35 Cl (mass 34.96885 u) and 37 Cl (mass 36.96590 u), with an average mass of 35.453 u. What is the percent composition of Cl in terms of these two isotopes?
Solution
The average mass of chlorine is the fraction that is 35 Cl times the mass of 35 Cl plus the fraction that is 37 Cl times the mass of 37 Cl.
\[\mathrm{average\: mass=(fraction\: of\: ^{35}Cl\times mass\: of\: ^{35}Cl)+(fraction\: of\: ^{37}Cl\times mass\: of\: ^{37}Cl)}\]
If we let x represent the fraction that is 35 Cl, then the fraction that is 37 Cl is represented by 1.00 − x .
(The fraction that is 35 Cl + the fraction that is 37 Cl must add up to 1, so the fraction of 37 Cl must equal 1.00 − the fraction of 35 Cl.)
Substituting this into the average mass equation, we have:
\[\begin{align*}
\mathrm{35.453\: u} &=(x\times 34.96885\: \ce{u})+[(1.00-x)\times 36.96590\: \ce{u}]\\
35.453 &=34.96885x+36.96590-36.96590x\\
1.99705x &=1.513\\
x&=\dfrac{1.513}{1.99705}=0.7576
\end{align*}\]
So solving yields: x = 0.7576, which means that 1.00 − 0.7576 = 0.2424. Therefore, chlorine consists of 75.76% 35 Cl and 24.24% 37 Cl.
Exercise \(\PageIndex{3}\)
Naturally occurring copper consists of 63 Cu (mass 62.9296 u) and 65 Cu (mass 64.9278 u), with an average mass of 63.546 u. What is the percent composition of Cu in terms of these two isotopes?
- Answer
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69.15% Cu-63 and 30.85% Cu-65
The occurrence and natural abundances of isotopes can be experimentally determined using an instrument called a mass spectrometer. Mass spectrometry (MS) is widely used in chemistry, forensics, medicine, environmental science, and many other fields to analyze and help identify the substances in a sample of material. In a typical mass spectrometer (Figure \(\PageIndex{5}\)), the sample is vaporized and exposed to a high-energy electron beam that causes the sample’s atoms (or molecules) to become electrically charged, typically by losing one or more electrons. These cations then pass through a (variable) electric or magnetic field that deflects each cation’s path to an extent that depends on both its mass and charge (similar to how the path of a large steel ball bearing rolling past a magnet is deflected to a lesser extent that that of a small steel BB). The ions are detected, and a plot of the relative number of ions generated versus their mass-to-charge ratios (a mass spectrum ) is made. The height of each vertical feature or peak in a mass spectrum is proportional to the fraction of cations with the specified mass-to-charge ratio. Since its initial use during the development of modern atomic theory, MS has evolved to become a powerful tool for chemical analysis in a wide range of applications.
Video \(\PageIndex{1}\) : Watch this video from the Royal Society for Chemistry for a brief description of the rudiments of mass spectrometry.
Finding the Averaged Atomic Mass of an Element - Video
This project was preformed to supply Libretext authors with videos on General Chemistry topics which can be used to enhance their projects. Also, these videos are meant to act as a learning resource for all General Chemistry students .
Video Topics
Averaged Atomic Weight of an element is determined by the number of isotopes of the element and their relative abundance.
Link to Video
Finding the Averaged Atomic Weight of an Element: https://youtu.be/bmP6Gr9zJiQ
The Periodic Table
As early chemists worked to purify ores and discovered more elements, they realized that various elements could be grouped together by their similar chemical behaviors. One such grouping includes lithium (Li), sodium (Na), and potassium (K): These elements all are shiny, conduct heat and electricity well, and have similar chemical properties. A second grouping includes calcium (Ca), strontium (Sr), and barium (Ba), which also are shiny, good conductors of heat and electricity, and have chemical properties in common. However, the specific properties of these two groupings are notably different from each other. For example: Li, Na, and K are much more reactive than are Ca, Sr, and Ba; Li, Na, and K form compounds with oxygen in a ratio of two of their atoms to one oxygen atom, whereas Ca, Sr, and Ba form compounds with one of their atoms to one oxygen atom. Fluorine (F), chlorine (Cl), bromine (Br), and iodine (I) also exhibit similar properties to each other, but these properties are drastically different from those of any of the elements above.
Dimitri Mendeleev in Russia (1869) and Lothar Meyer in Germany (1870) independently recognized that there was a periodic relationship among the properties of the elements known at that time. Both published tables with the elements arranged according to increasing atomic mass. But Mendeleev went one step further than Meyer: He used his table to predict the existence of elements that would have the properties similar to aluminum and silicon, but were yet unknown. The discoveries of gallium (1875) and germanium (1886) provided great support for Mendeleev’s work. Although Mendeleev and Meyer had a long dispute over priority, Mendeleev’s contributions to the development of the periodic table are now more widely recognized (Figure \(\PageIndex{6}\)).
By the twentieth century, it became apparent that the periodic relationship involved atomic numbers rather than atomic masses. The modern statement of this relationship, the periodic law , is as follows: the properties of the elements are periodic functions of their atomic numbers . A modern periodic table arranges the elements in increasing order of their atomic numbers and groups atoms with similar properties in the same vertical column (Figure \(\PageIndex{2}\)). Each box represents an element and contains its atomic number, symbol, average atomic mass, and (sometimes) name. The elements are arranged in seven horizontal rows, called periods or series , and 18 vertical columns, called groups . Groups are labeled at the top of each column. In the United States, the labels traditionally were numerals with capital letters. However, IUPAC recommends that the numbers 1 through 18 be used, and these labels are more common. For the table to fit on a single page, parts of two of the rows, a total of 14 columns, are usually written below the main body of the table.
Many elements differ dramatically in their chemical and physical properties, but some elements are similar in their behaviors. For example, many elements appear shiny, are malleable (able to be deformed without breaking) and ductile (can be drawn into wires), and conduct heat and electricity well. Other elements are not shiny, malleable, or ductile, and are poor conductors of heat and electricity. We can sort the elements into large classes with common properties: metals (elements that are shiny, malleable, good conductors of heat and electricity—shaded yellow); nonmetals (elements that appear dull, poor conductors of heat and electricity—shaded green); and metalloids (elements that conduct heat and electricity moderately well, and possess some properties of metals and some properties of nonmetals—shaded purple).
The elements can also be classified into the main-group elements (or representative elements ) in the columns labeled 1, 2, and 13–18; the transition metals in the columns labeled 3–12; and inner transition metals in the two rows at the bottom of the table (the top-row elements are called lanthanides and the bottom-row elements are actinides ; Figure \(\PageIndex{7}\)). The elements can be subdivided further by more specific properties, such as the composition of the compounds they form. For example, the elements in group 1 (the first column) form compounds that consist of one atom of the element and one atom of hydrogen. These elements (except hydrogen) are known as alkali metals , and they all have similar chemical properties. The elements in group 2 (the second column) form compounds consisting of one atom of the element and two atoms of hydrogen: These are called alkaline earth metals , with similar properties among members of that group. Other groups with specific names are the pnictogens (group 15), chalcogens (group 16), halogens (group 17), and the noble gases (group 18, also known as inert gases ). The groups can also be referred to by the first element of the group: For example, the chalcogens can be called the oxygen group or oxygen family. Hydrogen is a unique, nonmetallic element with properties similar to both group 1 and group 17 elements. For that reason, hydrogen may be shown at the top of both groups, or by itself.
Example \(\PageIndex{4}\): Naming Groups of Elements
Atoms of each of the following elements are essential for life. Give the group name for the following elements:
- chlorine
- calcium
- sodium
- sulfur
Solution
The family names are as follows:
- halogen
- alkaline earth metal
- alkali metal
- chalcogen
Exercise \(\PageIndex{4}\)
Give the group name for each of the following elements:
- krypton
- selenium
- barium
- lithium
- Answer a
-
noble gas
- Answer b
-
chalcogen
- Answer c
-
alkaline earth metal
- Answer d
-
alkali metal
In studying the periodic table, you might have noticed something about the atomic masses of some of the elements. Element 43 (technetium), element 61 (promethium), and most of the elements with atomic number 84 (polonium) and higher have their atomic mass given in square brackets. This is done for elements that consist entirely of unstable, radioactive isotopes (you will learn more about radioactivity in the nuclear chemistry chapter). An average atomic weight cannot be determined for these elements because their radioisotopes may vary significantly in relative abundance, depending on the source, or may not even exist in nature. The number in square brackets is the atomic mass number (and approximate atomic mass) of the most stable isotope of that element.
Summary
An atom consists of a small, positively charged nucleus surrounded by electrons. The nucleus contains protons and neutrons; its diameter is about 100,000 times smaller than that of the atom. The mass of one atom is usually expressed in unified atomic mass units (u), which is referred to as the atomic mass. A u is defined as exactly \(1/12\) of the mass of a carbon-12 atom and is equal to 1.6605 \(\times\) 10 −24 g.
Protons are relatively heavy particles with a charge of 1+ and a mass of 1.0073 u. Neutrons are relatively heavy particles with no charge and a mass of 1.0087 u. Electrons are light particles with a charge of 1− and a mass of 0.00055 u. The number of protons in the nucleus is called the atomic number (Z) and is the property that defines an atom’s elemental identity. The sum of the numbers of protons and neutrons in the nucleus is called the mass number and, expressed in u, is approximately equal to the mass of the atom. An atom is neutral when it contains equal numbers of electrons and protons.
Isotopes of an element are atoms with the same atomic number but different mass numbers; isotopes of an element, therefore, differ from each other only in the number of neutrons within the nucleus. When a naturally occurring element is composed of several isotopes, the atomic mass of the element represents the average of the masses of the isotopes involved. A chemical symbol identifies the atoms in a substance using symbols, which are one-, two-, or three-letter abbreviations for the atoms.
The discovery of the periodic recurrence of similar properties among the elements led to the formulation of the periodic table, in which the elements are arranged in order of increasing atomic number in rows known as periods and columns known as groups. Elements in the same group of the periodic table have similar chemical properties. Elements can be classified as metals, metalloids, and nonmetals, or as a main-group elements, transition metals, and inner transition metals. Groups are numbered 1–18 from left to right. The elements in group 1 are known as the alkali metals; those in group 2 are the alkaline earth metals; those in 15 are the pnictogens; those in 16 are the chalcogens; those in 17 are the halogens; and those in 18 are the noble gases.
Key Equations
- \(\mathrm{average\: mass}=\sum_{i}(\mathrm{fractional\: abundance \times isotopic\: mass})_i\)
Glossary
- actinide
- inner transition metal in the bottom of the bottom two rows of the periodic table
- alkali metal
- element in group 1
- alkaline earth metal
- element in group 2
- anion
- negatively charged atom or molecule (contains more electrons than protons)
- atomic mass
- average mass of atoms of an element, expressed in u
- atomic number (Z)
- number of protons in the nucleus of an atom
- cation
- positively charged atom or molecule (contains fewer electrons than protons)
- chalcogen
- element in group 16
- chemical symbol
- one-, two-, or three-letter abbreviation used to represent an element or its atoms
- Dalton (Da)
- alternative unit equivalent to the atomic mass unit
- fundamental unit of charge
- (also called the elementary charge) equals the magnitude of the charge of an electron (e) with e = 1.602 \(\times\) 10 −19 C
- group
- vertical column of the periodic table
- halogen
- element in group 17
- inert gas
- (also, noble gas) element in group 18
- inner transition metal
- (also, lanthanide or actinide) element in the bottom two rows; if in the first row, also called lanthanide, or if in the second row, also called actinide
- ion
- electrically charged atom or molecule (contains unequal numbers of protons and electrons)
- lanthanide
- inner transition metal in the top of the bottom two rows of the periodic table
- main-group element
- (also, representative element) element in columns 1, 2, and 12–18
- mass number (A)
- sum of the numbers of neutrons and protons in the nucleus of an atom
- metal
- element that is shiny, malleable, good conductor of heat and electricity
- metalloid
- element that conducts heat and electricity moderately well, and possesses some properties of metals and some properties of nonmetals
- noble gas
- (also, inert gas) element in group 18
- nonmetal
- element that appears dull, poor conductor of heat and electricity
- period
- (also, series) horizontal row of the periodic table
- periodic law
- properties of the elements are periodic function of their atomic numbers.
- periodic table
- table of the elements that places elements with similar chemical properties close together
- pnictogen
- element in group 15
- representative element
- (also, main-group element) element in columns 1, 2, and 12–18
- transition metal
- element in columns 3–11
- series
- (also, period) horizontal row of the period table
- unified atomic mass unit (u)
- (also, atomic mass unit (amu) or Dalton, Da) unit of mass equal to \(\dfrac{1}{12}\) of the mass of a 12 C atom
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).