3.3: Kinds of Arguments
Contemporary Western philosophy treats arguments as coming in two main types, deductive and inductive. The basic distinction and difference will be mentioned here.
Deductive arguments are arguments in which the premises (if true) guarantee the truth of the conclusion. The conclusion of a successful deductive argument cannot possibly be false, assuming its premises are true. This is what it means to label an argument as “valid” in logic. The form or structure of a deductive argument is the essential aspect to consider. Somewhat counter-intuitively, the premises do not need to be true for the conclusion to be true.
Arguments are a linguistic representation of an inference. So, using slightly different terminology, we can define deductive inferences . In a successful deductive inference, the premises and the denial of the conclusion constitute an inconsistent set of statements. An alternative way to describe the same relation: in a successful deductive inference, the truth of the premises makes the falsity of the conclusion logically impossible. A successful deductive inference is valid .
Deductive Example
1) All dogs are mammals.
2) All mammals breathe air.
_______________________________________________
SO: All dogs breathe air.
Inductive arguments are arguments with premises which make it likely that the conclusion is true but don’t absolutely guarantee its truth . Inductive arguments are by far the most common type of argument we see in our daily lives. We can assess inductive arguments along a spectrum of successful (stronger) to unsuccessful (weaker). The more successful (stronger) argument suggests that the premises mean the conclusion is probably true, with a high degree of likelihood. It is important to remember that inductive arguments can never fully guarantee the truth of the conclusion.
Using slightly different terminology, we can consider inductive inferences, referring to the actual thinking process in someone’s mind. In a successful inductive inference, the truth of the premises makes the falsity of the conclusion possible, but unlikely. Inductive inferences can be evaluated as “stronger” or “weaker” depending on the probability.
Inductive Example
1) The Interstate Bridge is regularly inspected by qualified engineers.
2) Vehicles have been driving over it for years.
_______________________________________________
SO: It will be safe to drive over it tomorrow.
One thing that makes applying the distinction between deductive and inductive arguments a bit tricky is this: we can’t look only at the premises OR only at the conclusion. Instead, we need to focus on the relationship between the premise(s) and the conclusion to tell what kind of argument we have.
A further contributor to trickiness: we can’t be distracted by the question of whether the statements are true or false. To classify an argument as deductive or inductive, we need to grant that the premises are true in a hypothetical way. We have to ask the question, “If those premises were true, would it be IMPOSSIBLE for the conclusion to be false?” If so, it is a deductive argument. Or “If those premises were true, would it be UNLIKELY, but still possible, that the conclusion is false? If so, it is an inductive argument.
As an example, consider this valid deductive argument:
1) All clouds are made out of spun sugar.
2) Anything made out of spun sugar is high in calories.
_______________________________________________
SO: All clouds are high in calories.
This argument is deductively successful because the truth of the premises would make the falsity of the conclusion impossible. Odd, isn’t it?
Some arguments are presented with premises missing. In those cases, the determination of deductive or inductive will depend on how that premise is filled in.
For example: I had an apple for lunch, so I had something healthy!
| Deductive | Inductive |
|
P1) I had an apple for lunch.
P2) All apples are healthy. (implied) SO I had something healthy |
P1) I had an apple for lunch.
P2) Most apples are healthy. (implied) SO I had something healthy |
Exercise: Deductive or Inductive?
Determine if the following arguments are deductive or inductive. It is a good idea to put the arguments in standard form first, so you are clear about the relation between premises and conclusion.
An interactive H5P element has been excluded from this version of the text. You can view it online here:
https://minnstate.pressbooks.pub/ctar/?p=670#h5p-5


4.5: Introduction to Describing Customers
What you’ll learn to do: Summarize the main factors used to describe customers
It’s important for retailers to understand their customers because their needs and desires shape their actions and purchases. Consumers are influenced by their life experiences, as well as demographic factors, so understanding how they make decisions can help retailers with promotions. Their decisions are also a result of their attitude and motivation, so even if two customers have similar life experiences and are members of the same demographics, they may make different choices.
Contributors and Attributions
CC licensed content, Original
- Introduction to Describing Customers. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


12: Mendel's Experiments and Heredity
Gregor Johann Mendel was a German-speaking Moravian scientist and Augustinian friar who gained posthumous fame as the founder of the modern science of genetics. Though farmers had known for centuries that crossbreeding of animals and plants could favor certain desirable traits, Mendel's experiments established many of the rules of heredity, now referred to as the laws of Mendelian inheritance.
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- 12.1: Introduction to Mendel's Experiments and Heredity
- Genetics is the study of heredity. Johann Gregor Mendel set the framework for genetics long before chromosomes or genes had been identified, at a time when meiosis was not well understood. Mendel selected a simple biological system and conducted methodical, quantitative analyses using large sample sizes. Because of Mendel’s work, the fundamental principles of heredity were revealed.
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- 12.2: Mendel’s Experiments and the Laws of Probability
- In 1865, Mendel presented the results of his experiments with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted faithfully from parents to offspring independently of other traits and in dominant and recessive patterns.
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- 12.3: Characteristics and Traits
- The genetic makeup of peas consists of two similar or homologous copies of each chromosome, one from each parent. Each pair of homologous chromosomes has the same linear order of genes; hence peas are diploid organisms. The same is true for many other plants and for virtually all animals. Diploid organisms utilize meiosis to produce haploid gametes, which contain one copy of each homologous chromosome that unite at fertilization to create a diploid zygote.
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- 12.4: Laws of Inheritance
- Mendel generalized the results of his pea-plant experiments into four postulates, some of which are sometimes called “laws,” that describe the basis of dominant and recessive inheritance in diploid organisms. As you have learned, more complex extensions of Mendelism exist that do not exhibit the same F2 phenotypic ratios (3:1). Nevertheless, these laws summarize the basics of classical genetics.
Thumbnail: Pea plants were used by Gregor Mendel to discover some fundamental laws of genetics. (Flicker-Christian Guthier-CC:A).
Contributors and Attributions
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Connie Rye (East Mississippi Community College), Robert Wise (University of Wisconsin, Oshkosh), Vladimir Jurukovski (Suffolk County Community College), Jean DeSaix (University of North Carolina at Chapel Hill), Jung Choi (Georgia Institute of Technology), Yael Avissar (Rhode Island College) among other contributing authors. Original content by OpenStax (CC BY 4.0; Download for free at<EMAIL_ADDRESS>).


Chapter 6: Chemical Kinetics
Introduction
In this chapter, we will discuss the fascinating field of chemical kinetics , which focuses on the rates at which chemical reactions occur and the factors that influence them. Unlike thermodynamics, which tells us whether a reaction is energetically favorable (as discussed in the following Chapter), kinetics answers the question of how fast a reaction will proceed. By understanding the principles of reaction rates, we can predict, measure, and even control the speed of chemical reactions, including understanding complexities in the mechanism of a reaction - the steps in which it occurs.
For example, catalytic converters accelerate the breakdown of harmful emissions like carbon monoxide (CO) and nitrogen oxides (NO x ) into safer products such as carbon dioxide (CO 2 ) and nitrogen gas (N 2 ). A typical reaction is:
2 CO (g) + O 2 (g) → 2 CO 2 (g)
The catalyst, typically platinum, palladium, or rhodium, lowers the activation energy required for these reactions, ensuring they occur rapidly even at engine exhaust temperatures.
Figure 6.1: The purpose of using metals like platinum in catalytic converters is to create a surface on which chemisorption can occur. Chemisorption is a process resulting from a chemical bond between adsorbate molecules (A and B) and specific surface locations on a material, known as active sites. (CC BY-SA, 2011: Brazosport College via WikiMedia Commons )
In the case of catalytic converters (and other pollutant-"scrubbing" technologies), clusters of metal-oxide molecules react with gases like CO and NO x such that they become oxidized to products like CO 2 . Since the reaction needs a surface cluster in order to occur, the rate of the reaction is dictated by the number of metallic sites, allowing for a controlled mechanism that is more expedient and thus efficient.
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- 6.2: Reaction Mechanisms
- A balanced chemical reaction does not necessarily reveal either the individual elementary reactions by which a reaction occurs or its rate law. A reaction mechanism is the microscopic path by which reactants are transformed into products. Each step is an elementary reaction. Species that are formed in one step and consumed in another are intermediates. Each elementary reaction can be described in terms of its molecularity. The slowest step in a reaction mechanism is the rate-determining step.
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- 6.3: Catalysis
- Catalysts participate in a chemical reaction and increase its rate. They do not appear in the reaction’s net equation and are not consumed during the reaction. Catalysts allow a reaction to proceed via a pathway that has a lower activation energy than the uncatalyzed reaction. In heterogeneous catalysis, catalysts provide a surface to which reactants bind in a process of adsorption. In homogeneous catalysis, catalysts are in the same phase as the reactants. Enzymes are biological catalysts.


1.2: Thoughts
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What is the difference between emotion, feeling, thought, logic, and intelligence? Use of any of them requires a lot of attention. Even when you are feeling something emotional your attention is directed toward that thing. The answer is that everything in life eventually results in a feeling. Even emotion results in a feeling. Emotion is unconscious thoughts about things, and thoughts are conscious thoughts about things. Thought results in feelings, so unconscious thought (emotion) is also going to result in feelings. [The question is, do the feelings come from the thoughts simultaneously, or later on, or both (and if later on, when exactly).]
If you think about it that way, thought and emotion are both in part feelings, that is, to some extent you feel them right away, in addition to them resulting in feelings later on. But that still means that feelings are always the end result. Then again, thoughts might be the result of current thoughts. That is like emotion, unconscious emotional thoughts are going to result in unconscious emotional thoughts later on. Even feelings could be called unconscious thoughts, because thought is just focusing on one thing for a brief period of time. [When thought about that way, what is the difference between an unconscious thought and an emotion? Is the unconscious thought stronger, more specific, or just something that has more of an influence on what you are thinking than feeling. Thoughts might have a better influence on other thoughts then they do on emotions (and emotions might have a better influence on emotion then they do on thought). Think of it this way, if you are doing something, but you "feel" like you don't want to do it, that isn't going to stop you from doing it as much as you thinking unconsciously over and over that you don't want to do it. The thinking in that instance seems like it is just more intense than the emotion, but not necessarily "felt" as much since it is just thought, not feeling. It has more of an influence over your actions, however, and maybe would generate anxiety instead of emotion. It is like the unconscious thought in that instead comes from your understanding that you don't want to do that activity, and since understanding is a function of thought, you would say that your unconscious thoughts are stopping you from doing it more than your emotions are stopping you.]
Therefore emotion, thought and feeling are really just periods of focus on certain things. With thought you just recognize what it is that you are focusing on. With emotions you feel deeply about what you are focusing on, and with feelings you are focusing on it less. Physical stimulus also results in feelings, and then you focus on those feelings, you aren’t necessarily focused on what caused the feelings (the physical stimulus itself) however. [This ties into the idea that someone can only pay attention to a few number of things at once (including emotion and thought) because if you are focused on one thing, you are probably going to be less focused on something else.]
Thus life is really just different types of feelings; you could categorize all of life as feeling. Even when you think you are in a period when you’re not feeling anything, you really are feeling something; you just don’t recognize what it is that you are feeling. Remember that feelings are thoughts you can’t identify. And since a thought is going to be about something, another way to think about life is just stuff happening. Stuff happening results in feelings in your brain, where more stuff happens. It is all-concrete. [And stuff happens all the time, so you are probably going to be feeling something more than you can recognize when you are feeling something.]
The definition of intellect and thoughts is almost understanding (those concrete things). Emotion is feeling, completely separate from facts or information. All facts and information are going to be about things that cause feeling, however, since all things that happen cause feelings and all facts and information are about things that happen. So facts and information are just feelings organized in a logical manner. [Unless the fact doesn't generate feeling, but most things cause feelings.] Intellect and thought also generates feelings when those thoughts are processed in your mind. Since thought is really only about feelings, it is logical that thought actually has root in feelings. For example, all events are really feelings in the mind, so thoughts are actually just comparing feelings. You take two feelings and can arrive at one thought. Take the feeling of a frog moving and the feeling of a threat of danger. The two feelings combined equal the idea or thought that the frog needs to move when there is danger – the thought is actually just understanding how feelings interact. All thought is is the understanding of how feelings and real events interact with themselves. Feeling is what provides the motivation to arrive at the answer (the thought). If you just had the facts, there is a threat, and the frog can jump, you aren’t going to arrive at the conclusion that the frog should jump away. You need to take the feeling that there is a threat and the feeling that the frog can jump and then combine the two sensory images in your head to arrive at the answer. [It is like the feeling provides the motivation, without emotion thought really wouldn't be possible because there would be no need to arrive at any conclusions.]
That shows how all intellect is powered and motivated by emotion. It also shows that frogs have thoughts; the frog has to have the thought to jump away when it sees a threat, as a thought is just the combination of two feelings resulting in the resulting feeling of wanting to move away. That process of feelings is like a thought process. Thoughts are a little different for humans, however, because humans have such a large memory that they are able to compare this experience to all the other experiences in their life while the frog only remembers the current situation and is programmed (brain wiring) to jump away. The frog doesn’t have a large enough memory to learn from new information and change its behavior. That shows how humans are very similar to frogs in how they process data (in one way at least), and that one thing that separates a human from a frog is a larger memory which can store lots of useful information and potential behavioral patterns. [That brings up the question, exactly how good is a frogs memory? It can make its way around a pool without hitting the same spot over and over, so it lasts at least a few minutes. But that is memory for simple things, it isn't smart enough (remember from the logic chapter) to process more complicated data to remember. On the other hand, humans really don't do complicated stuff most of the time, so we are very similar to frogs).]
It is clear that emotion motivates thoughts to occur at all, for instance you want something, and then that brings up the thought that you want that thing. That same thing happens almost instantaneously for some things, like the frog jumping away. It must have been an emotion that caused the frog to jump away because that is similar to how a human would respond if there is a danger, the feeling of danger causes you to jump away. If emotions can influence behavior like that, it could be that a different emotion arises from word to word in your own thought process. Like thinking of a loved one might bring up the emotion love, which might influence what you say next. So emotions have a clear motivating role for simple actions and thoughts, and that is because emotion is simple. One emotion cannot be a complicated thought process that you understand, it could bring up a complicated thought process, and you may have an emotion for the thought process - but the emotion itself isn't that informative. Izard said "Feeling in basic emotion affects action but not higher-order cognition, which has little or no presence in basic emotion processes." (Izard)
Thoughts, especially in humans, are not that independent – they can be much more complicated and it can appear to be that nothing is as it seems. If someone says to you, “I know x”. He isn’t just saying that he knows x, but there is a chain of other thoughts that also occur in your mind. You analyze the statement he made and it causes you to think automatically, “Do I know x too?” “Why does he think I care that he knows x?” “Is there anything else about x that is significant that I am missing?” “What if this other person is smarter than me?” that doesn’t lead to a feeling of being dumb (it might), instead it leads to another concrete thing “maybe I am stupid” or the thought “maybe that person is stupid” interacting with the thought “because that thing he said was wrong”. So one simple thought for a human can mean much much more than that one thought. That example shows another way in which humans are different from frogs – they are capable of more simultaneous thoughts. It is also the memory working hand in hand with that capacity of simultaneous thought as well, if you had no memory then you wouldn’t have information to compare and bring up those simultaneous thoughts. [Remember that thoughts can lead to and are sometimes emotions (unconscious ones), so that example of the unconscious thought process was really an emotion - worry. It was worse than just worrying though, it was worrying about specific things, so the emotion was of a more specific type then just worry. Thus there isn't just one emotion of "worry" but there is "worry about your intelligence, etc".]
They can all be moving at the same time as well, not only does one thought follow another; but it occurs instantaneously. If the thing the person said was something you didn’t know, it might make you feel stupid, thus the thought results in a feeling. But that feeling can be translated to a thought. So it isn’t the feeling, “I am stupid” it is the thought “I am stupid”.   Thus thought, feeling, and emotion is just paying attention to different things in your head. Concrete things. [In other words, all emotions are not only real things, but it could be said that all emotions have a source, since they are real things.]
It is a little more complicated than that, however. It is going to be a mix of a lot of concrete thoughts interacting with each other, not just the thought “I am stupid” repeated over and over but maybe also a less intense idea of “well I know x and y that that person doesn’t, maybe this was just one event”. So anything that is said or done is possibly followed by a long series of unconscious thoughts and thought processes. [Or, there might be many implications to any one thought. (there might not be, however, like you could relate almost anything to sex, but in reality that isn't necessary).]
There were two examples of thoughts, one was with the frog and the danger of a threat, and the other was a questioning of ones intellect relative to someone else. The example with the frog was an example of a thought process that was simple, while the example with the person showed how some thought processes can be much more complicated than they appear.
A good example of how feelings are mixed in with emotions and physical reactions, and how feelings help motivate thoughts; can be found in this explanation of Wundt’s ideas (by organic they mean bodily) “Wundt starts with the unanalyzable feelings that alter the stream of ideas. For example, the unanalyzable feelings of “fear” or “joy” can influence the current stream of ideation, encouraging some, discouraging some, or inhibiting other ideas. This altered stream of ideas produces a secondary feeling as well as organic reactions. And the organic reactions produce sensory feelings that are added to or fused with the preceding feeling (or sensation) and thus intensify the conscious feeling.” (Mandler 2003)
How thoughts and feelings interact delicately was also pondered by Titchener, the following is an explanation of his theory: “Titchener postulated that a train of ideas need be interrupted by a vivid feeling, that this feeling shall reflect the situation in the outside world (as distinct from inner experience), and that the feeling shall be enriched by organic sensations, set up in the course of bodily adjustment to the incident. The emotion itself, as experienced, consists of the stimulus association of ideas, some part of which are always organic sensations.” (Mandler 2003) That shows how thoughts and feelings can occur in sequence or simultaneously, and that the feelings you have can also be physical ones that interrupt or encourage your thoughts.
How This Chapter shows how Intelligence is intertwined with Emotion
- It is stated first that use of emotion and thought requires attention, and therefore they both cause feelings, and if they both cause feelings then they are going to be similar in nature. Your intellect (or ability to do things which are real) is going to generate feelings just like emotions do.
- Feelings can result in thoughts – this was shown with the frog example, the frog has the thought “jump” which comes from the feeling of a threat of danger, and the feeling of it’s understanding that it can jump. That shows how thoughts can be encouraged by feelings and mixed in with them.
 Those thoughts are a function of intelligence, because they are causing you to think about real things, which is what intelligence is. [Therefore feeling is also a function of intelligence, since feelings are about real things, and intellect causes you to think real things, and feeling is unconscious thought.]
References
Izzard, Carroll (2009). Emotion Theory and Research: Highlights, Unanswered Questions,and Emerging Issues. Annual Review of Psychology: 60, 7.
Mandler, George (2003). Emotion. In D. Freedheim & I. Weiner (Eds.) Handbook of Psychology: Volume 1, History of Psychology, p 161.


19.5: The p-value Is a Lie.
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The cake is a lie.
The cake is a lie.
The cake is a lie.
The cake is a lie.
– Portal 262
Okay, at this point you might be thinking that the real problem is not with orthodox statistics, just the p<.05 standard. In one sense, that’s true. The recommendation that Johnson (2013) gives is not that “everyone must be a Bayesian now”. Instead, the suggestion is that it would be wiser to shift the conventional standard to something like a p<.01 level. That’s not an unreasonable view to take, but in my view the problem is a little more severe than that. In my opinion, there’s a fairly big problem built into the way most (but not all) orthodox hypothesis tests are constructed. They are grossly naive about how humans actually do research, and because of this most p-values are wrong.
Sounds like an absurd claim, right? Well, consider the following scenario. You’ve come up with a really exciting research hypothesis and you design a study to test it. You’re very diligent, so you run a power analysis to work out what your sample size should be, and you run the study. You run your hypothesis test and out pops a p-value of 0.072.
What should you do? Here are some possibilities:
- You conclude that there is no effect, and try to publish it as a null result
- You guess that there might be an effect, and try to publish it as a “borderline significant” result
- You give up and try a new study
- You collect some more data to see if the p value goes up or (preferably!) drops below the “magic” criterion of p<.05
Which would you choose? Before reading any further, I urge you to take some time to think about it. Be honest with yourself. But don’t stress about it too much, because you’re screwed no matter what you choose. Based on my own experiences as an author, reviewer and editor, as well as stories I’ve heard from others, here’s what will happen in each case:
- Let’s start with option 1. If you try to publish it as a null result, the paper will struggle to be published. Some reviewers will think that p=.072 is not really a null result. They’ll argue it’s borderline significant. Other reviewers will agree it’s a null result, but will claim that even though some null results are publishable, yours isn’t. One or two reviewers might even be on your side, but you’ll be fighting an uphill battle to get it through.
- Okay, let’s think about option number 2. Suppose you try to publish it as a borderline significant result. Some reviewers will claim that it’s a null result and should not be published. Others will claim that the evidence is ambiguous, and that you should collect more data until you get a clear significant result. Again, the publication process does not favour you.
- Given the difficulties in publishing an “ambiguous” result like p=.072, option number 3 might seem tempting: give up and do something else. But that’s a recipe for career suicide. If you give up and try a new project else every time you find yourself faced with ambiguity, your work will never be published. And if you’re in academia without a publication record you can lose your job. So that option is out.
- It looks like you’re stuck with option 4.  Seems sensible, but unfortunately for you, if you do this all of your p-values are now incorrect. All of them. Not just the p-values that you calculated for this study. All of them. All the p-values you calculated in the past and all the p-values you will calculate in the future. Fortunately, no-one will notice. You’ll get published, and you’ll have lied.
Wait, what? How can that last part be true?  You collected some data, the results weren’t conclusive, so now what you want to do is collect more data until the the results are conclusive. What’s wrong with that?
Honestly, there’s nothing wrong with it. It’s a reasonable, sensible and rational thing to do. In real life, this is exactly what every researcher does. Unfortunately, the theory of null hypothesis testing as I described it in Chapter 11 forbids you from doing this. 263 The reason is that the theory assumes that the experiment is finished and all the data are in. And because it assumes the experiment is over, it only considers two possible decisions. If you’re using the conventional p<.05 threshold, those decisions are:
| Outcome | Action |
|---|---|
| p less than .05 | Reject the null |
| p greater than .05 | Retain the null |
What you’re doing is adding a third possible action to the decision making problem. Specifically, what you’re doing is using the p-value itself as a reason to justify continuing the experiment. And as a consequence you’ve transformed the decision-making procedure into one that looks more like this:
| Outcome | Action |
|---|---|
| p less than .05 | Stop the experiment and reject the null |
| p between .05 and .1 | Continue the experiment |
| p greater than .1 | Stop the experiment and retain the null |
The “basic” theory of null hypothesis testing isn’t built to handle this sort of thing, not in the form I described back in Chapter 11. If you’re the kind of person who would choose to “collect more data” in real life, it implies that you are not making decisions in accordance with the rules of null hypothesis testing. Even if you happen to arrive at the same decision as the hypothesis test, you aren’t following the decision process it implies, and it’s this failure to follow the process that is causing the problem. 264 Your p-values are a lie.
Worse yet, they’re a lie in a dangerous way, because they’re all too small . To give you a sense of just how bad it can be, consider the following (worst case) scenario. Imagine you’re a really super-enthusiastic researcher on a tight budget who didn’t pay any attention to my warnings above. You design a study comparing two groups. You desperately want to see a significant result at the p<.05 level, but you really don’t want to collect any more data than you have to (because it’s expensive). In order to cut costs, you start collecting data, but every time a new observation arrives you run a t-test on your data. If the t-tests says p<.05 then you stop the experiment and report a significant result. If not, you keep collecting data. You keep doing this until you reach your pre-defined spending limit for this experiment. Let’s say that limit kicks in at N=1000 observations. As it turns out, the truth of the matter is that there is no real effect to be found: the null hypothesis is true. So, what’s the chance that you’ll make it to the end of the experiment and (correctly) conclude that there is no effect? In an ideal world, the answer here should be 95%. After all, the whole point of the p<.05 criterion is to control the Type I error rate at 5%, so what we’d hope is that there’s only a 5% chance of falsely rejecting the null hypothesis in this situation. However, there’s no guarantee that will be true.
Figure 17.1: How badly can things go wrong if you re-run your tests every time new data arrive? If you are a frequentist, the answer is “very wrong”.
So how bad is it? The answer is shown as the solid black line in Figure 17.1, and it’s astoundingly bad. If you peek at your data after every single observation, there is a 49% chance that you will make a Type I error. That’s, um, quite a bit bigger than the 5% that it’s supposed to be. By way of comparison, imagine that you had used the following strategy. Start collecting data. Every single time an observation arrives, run a Bayesian t-test (Section 17.7 and look at the Bayes factor. I’ll assume that Johnson (2013) is right, and I’ll treat a Bayes factor of 3:1 as roughly equivalent to a p-value of .05. 265 This time around, our trigger happy researcher uses the following procedure: if the Bayes factor is 3:1 or more in favour of the null, stop the experiment and retain the null. If it is 3:1 or more in favour of the alternative, stop the experiment and reject the null. Otherwise continue testing. Now, just like last time, let’s assume that the null hypothesis is true. What happens? As it happens, I ran the simulations for this scenario too, and the results are shown as the dashed line in Figure 17.1. It turns out that the Type I error rate is much much lower than the 49% rate that we were getting by using the orthodox t-test.
In some ways, this is remarkable. The entire point of orthodox null hypothesis testing is to control the Type I error rate. Bayesian methods aren’t actually designed to do this at all. Yet, as it turns out, when faced with a “trigger happy” researcher who keeps running hypothesis tests as the data come in, the Bayesian approach is much more effective. Even the 3:1 standard, which most Bayesians would consider unacceptably lax, is much safer than the p<.05 rule.
really this bad?
The example I gave in the previous section is a pretty extreme situation. In real life, people don’t run hypothesis tests every time a new observation arrives. So it’s not fair to say that the p<.05 threshold “really” corresponds to a 49% Type I error rate (i.e., p=.49). But the fact remains that if you want your p-values to be honest, then you either have to switch to a completely different way of doing hypothesis tests, or you must enforce a strict rule: no peeking . You are not allowed to use the data to decide when to terminate the experiment. You are not allowed to look at a “borderline” p-value and decide to collect more data. You aren’t even allowed to change your data analyis strategy after looking at data.
And yes, these rules are surprisingly strict. As a class exercise a couple of years back, I asked students to think about this scenario. Suppose you started running your study with the intention of collecting N=80 people. When the study starts out you follow the rules, refusing to look at the data or run any tests. But when you reach N=50 your willpower gives in… and you take a peek. Guess what? You’ve got a significant result! Now, sure, you know you said that you’d keep running the study out to a sample size of N=80, but it seems sort of pointless now, right? The result is significant with a sample size of N=50, so wouldn’t it be wasteful and inefficient to keep collecting data? Aren’t you tempted to stop? Just a little? Well, keep in mind that if you do, your Type I error rate at p<.05 just ballooned out to 8%. When you report p<.05 in your paper, what you’re really saying is p<.08. That’s how bad the consequences of “just one peek” can be.
Now consider this … the scientific literature is filled with t-tests, ANOVAs, regressions and chi-square tests. When I wrote this book I didn’t pick these tests arbitrarily. The reason why these four tools appear in most introductory statistics texts is that these are the bread and butter tools of science. None of these tools include a correction to deal with “data peeking”: they all assume that you’re not doing it. But how realistic is that assumption? In real life, how many people do you think have “peeked” at their data before the experiment was finished and adapted their subsequent behaviour after seeing what the data looked like? Except when the sampling procedure is fixed by an external constraint, I’m guessing the answer is “most people have done it”. If that has happened, you can infer that the reported p-values are wrong. Worse yet, because we don’t know what decision process they actually followed, we have no way to know what the p-values should have been. You can’t compute a p-value when you don’t know the decision making procedure that the researcher used. And so the reported p-value remains a lie.
Given all of the above, what is the take home message? It’s not that Bayesian methods are foolproof. If a researcher is determined to cheat, they can always do so. Bayes’ rule cannot stop people from lying, nor can it stop them from rigging an experiment. That’s not my point here. My point is the same one I made at the very beginning of the book in Section 1.1: the reason why we run statistical tests is to protect us from ourselves. And the reason why “data peeking” is such a concern is that it’s so tempting, even for honest researchers . A theory for statistical inference has to acknowledge this. Yes, you might try to defend p-values by saying that it’s the fault of the researcher for not using them properly. But to my mind that misses the point. A theory of statistical inference that is so completely naive about humans that it doesn’t even consider the possibility that the researcher might look at their own data isn’t a theory worth having. In essence, my point is this:
Good laws have their origins in bad morals.
– Ambrosius Macrobius 266
Good rules for statistical testing have to acknowledge human frailty. None of us are without sin. None of us are beyond temptation. A good system for statistical inference should still work even when it is used by actual humans. Orthodox null hypothesis testing does not. 267


9.4: Gaussian Quadratures
Previously, we have assumed in our analysis of numerical integration schemes that the discretization points are equally-spaced. This assumption is not strictly necessary; for instance, we can easily modify the mid-point rule and trapezium rule formulas to work with non-equally-spaced points.
However, if we are free to choose the discretization points for performing numerical integration, and these points need not be equally spaced, then it is possible to exploit this freedom to further improve the accuracy of the numerical integration. The idea is to optimize the placement of the discretization points, so as to minimize the resulting numerical error. This is the basic idea behind the method of integration by Gaussian quadratures .
We will not discuss the details of this numerical integration method. To use it, you can call the
scipy.integrate.quad
function. This function uses a low-level numerical library named
QUADPACK
, which performs quadrature integration with
adaptive quadratures
—i.e., it automatically figures out how many discretization points should be used, and where they should be located, in order to produce a result with the desired numericaly accuracy.
Because QUADPACK figures out the discretization points for itself, you have to pass
quad
a
function
representing the integrand, rather than an array of integrand values as with
trapz
or
simps
. The standard way to call the function is
t = quad(f, a, b)
which calculates the integral
\[t = \int_a^b f(x) dx.\]
The return value is a tuple of the form
(t,err)
, where
t
is the value of the integral and
err
is an estimate of the numerical error. The
quad
function also accepts many other optional inputs, which can be used to specify additional inputs for passing to the integrand function, the error tolerance, the number of subintervals to use, etc. See the
documentation
for details.
The choice of whether to perform numerical integration using Simpson's rule (
simps
) or Gaussian quadratures (
quad
) is situational. If you already know the values of the integrands at a pre-determined set of discretization points (e.g., from the result of a finite-difference calculation), then use
simps
. If you can define a function that can quickly calculate the value of the integrand at any point, use
quad
.
Here is an example of using
quad
to compute the integral \(\int_{0}^\infty \frac{dx}{x^2 + 1}\):
from scipy import * from scipy.integrate import quad def f(x): return 1.0 / (x*x+1) integ, _ = quad(f, 0, 1000) print(integ)
(Note that
quad
returns two values; the first is the computed value of the integral, and the other is the absolute error, which we're not interested in, so we toss it in the "throwaway" variable named
_
. See the
documentation
for details.) Running the above program prints the result \(1.569796\dots,\) which differs from the exact result, \(\pi/2 = 1.570796\dots,\) by a relative error of \(0.06\%\).
Here is another example, where the integrand takes an additional parameter: \(\int_0^\infty x e^{-\lambda x}\, dx\):
from scipy import * from scipy.integrate import quad def f(x, lambd): return x * exp(-lambd * x) integ, _ = quad(f, 0, 100, args=(0.5)) print(integ)
Running the program prints the result \(4.0\), which agrees with the exact result of \(1/\lambda^2\) for the chosen value of the parameter \(\lambda =0.5\).


2.5: Assessment
True/False
Indicate whether the statement is true or false.
____ 1. Radio telemetry is not used much to communicate field data as it is expensive and limited in range to only several hundred feet from the device.
____ 2. Remote Telemetry Units (RTUs) and Programmable Logic Controllers (PLCs) serve roughly the same purpose, monitoring process feedback and sending the data to a centralized computer.
____ 3. A logic programming language that is similar to an AC wiring diagram is called a Function Block Diagram.
____ 4. SCADA systems can be networked in either a LAN or a WAN.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 5. Which of the following is not a part of industrial control systems?
| a. Supervisory Control and Data Acquisition | c. Production Control Systems |
| b. Distributed Control Systems | d. Programmable Logic Controllers |
____ 6. Which of the following is not one of the main methods by which measurement data is communicated to a system?
| a. Computer protocols, such as serial communications | c. Binary alarms |
| b. Analog devices | d. Digital devices |
____ 7. Which of the following is not a function of SCADA data?
| a. Providing information from which reports and trending data can be generated | c. Providing meaningful displays for operators |
| b. Monitoring and annunciating alarm conditions | d. All of the above |
Completion
Complete each sentence.
8. A system that provides for remote monitoring and control of industrial devices and equipment (bringing plant/process data into a computer systems) is known as ________________________.
9. ______________________ can be either digital or analog devices that provide an audible warning of a condition.
Short Answer
10. Discuss some of the useful information that SCADA reports can provide.
For the answers to these questions, email your name, the name of your college or other institution, and your position there to<EMAIL_ADDRESS>CyberWatch West will email you a copy of the answer key.


9.13: Reading- Dikes and Sills
Dikes
A dike is a sheet of rock that formed in a fracture in a pre-existing rock body. Dikes can be either magmatic or sedimentary in origin. Magmatic dikes form when magma intrudes into a crack then crystallizes as a sheet intrusion, either cutting across layers of rock or through an unlayered mass of rock. Clastic dikes are formed when sediment fills a pre-existing crack. [1]
Magmatic Dikes
An intrusive dike is an igneous body with a very high aspect ratio, which means that its thickness is usually much smaller than the other two dimensions. Thickness can vary from sub-centimeter scale to many meters, and the lateral dimensions can extend over many kilometres. A dike is an intrusion into an opening cross-cutting fissure, shouldering aside other pre-existing layers or bodies of rock; this implies that a dike is always younger than the rocks that contain it. Dikes are usually high-angle to near-vertical in orientation, but subsequent tectonic deformation may rotate the sequence of strata through which the dike propagates so that the dike becomes horizontal. Near-horizontal, or conformable intrusions, along bedding planes between strata are called intrusive sills.
Sometimes dikes appear in swarms, consisting of several to hundreds of dikes emplaced more or less contemporaneously during a single intrusive event. The world’s largest dike swarm is the Mackenzie dike swarm in the Northwest Territories, Canada. [2]
Dikes often form as either radial or concentric swarms around plutonic intrusives, volcanic necks or feeder vents in volcanic cones. The latter are known as ring dikes.
Dikes can vary in texture and their composition can range from diabase or basaltic to granitic or rhyolitic, but on a global perspective the basaltic composition prevails, manifesting ascent of vast volumes of mantle-derived magmas through fractured lithosphere throughout Earth history. Pegmatite dikes comprise extremely coarse crystalline granitic rocks—often associated with late-stage granite intrusions or metamorphic segregations. Aplite dikes are fine-grained or sugary-textured intrusives of granitic composition.
Sedimentary Dikes
Sedimentary dikes or clastic dikes are vertical bodies of sedimentary rock that cut off other rock layers. They can form in two ways:
- When a shallow unconsolidated sediment is composed of alternating coarse grained andimpermeable clay layers the fluid pressure inside the coarser layers may reach a critical value due to lithostatic overburden. Driven by the fluid pressure the sediment breaks through overlying layers and forms a dike.
- When a soil is under permafrost conditions the pore water is totally frozen. When cracks are formed in such rocks, they may fill up with sediments that fall in from above. The result is a vertical body of sediment that cuts through horizontal layers: a dike.
Sills
In geology, a sill is a tabular sheet intrusion that has intruded between older layers of sedimentary rock, beds of volcanic lava or tuff, or even along the direction of foliation in metamorphic rock. The term sill is synonymous with concordant intrusive sheet. This means that the sill does not cut across preexisting rocks, in contrast to dikes, discordant intrusive sheets which do cut across older rocks. Sills are fed by dikes, except in unusual locations where they form in nearly vertical beds attached directly to a magma source. The rocks must be brittle and fracture to create the planes along which the magma intrudes the parent rock bodies, whether this occurs along preexisting planes between sedimentary or volcanic beds or weakened planes related to foliation in metamorphic rock. These planes or weakened areas allow the intrusion of a thin sheet-like body of magma paralleling the existing bedding planes, concordant fracture zone, or foliations.
Sills parallel beds (layers) and foliations in the surrounding country rock. They can be originally emplaced in a horizontal orientation, although tectonic processes may cause subsequent rotation of horizontal sills into near vertical orientations. Sills can be confused with solidified lava flows; however, there are several differences between them. Intruded sills will show partial melting and incorporation of the surrounding country rock. On both contact surfaces of the country rock into which the sill has intruded, evidence of heating will be observed (contact metamorphism). Lava flows will show this evidence only on the lower side of the flow. In addition, lava flows will typically show evidence of vesicles (bubbles) where gases escaped into the atmosphere. Because sills generally form at shallow depths (up to many kilometers) below the surface, the pressure of overlying rock prevents this from happening much, if at all. Lava flows will also typically show evidence of weathering on their upper surface, whereas sills, if still covered by country rock, typically do not.
Associated Ore Deposits
Certain layered intrusions are a variety of sill that often contain important ore deposits. Precambrian examples include the Bushveld, Insizwa and the Great Dyke complexes of southern Africa, the Duluth intrusive complex of the Superior District, and the Stillwater igneous complex of the United States. Phanerozoic examples are usually smaller and include the Rùm peridotite complex of Scotland and the Skaergaard igneous complex of east Greenland. These intrusions often contain concentrations of gold, platinum, chromium and other rare elements.
Transgressive Sills
Despite their concordant nature, many large sills change stratigraphic level within the intruded sequence, with each concordant part of the intrusion linked by relatively short dike-like segments. Such sills are known as transgressive, examples include the Whin Sill and sills within the Karoo basin. [3] The geometry of large sill complexes in sedimentary basins has become clearer with the availability of 3D seismic reflection data. [4] Such data has shown that many sills have an overall saucer shape and that many others are at least in part transgressive. [5]
Other Meanings
“Sill” may also refer to the rise in depth near the mouth of a fjord caused by the terminal moraine of the previous glacier.
- Essentials of Geology, 3rd Ed, Stephen Marshak ↵
- Pilkington, Mark and Walter R. Roest; Removing varying directional trends in aeromagnetic data, Geophysics, vol. 63 no. 2 (1998), pp. 446–453. abstract ↵
- Holmes, A. 1978. Principles of Physical Geology, Van Nostrand Reinhold (UK) Co.Ltd., 720 pp. See also, Polteau, S., Mazzini, A., Galland, O., Planke, S. & Malthe-Sørenssen, A. 2008. Saucer-shaped intrusions: Occurrences, emplacement and implications , Earth and Planetary Science Letters 266, 195–204. ↵
- Thomson, K. & Hutton, D. 2004. Geometry and growth of sill complexes: insights using 3D seismic from the North Rockall Trough , Bulletin of Volcanology, 66 , 364–375. ↵
- Planke, S., Rasmussen, T., Rey, S.S. & Myklebust, R. 2005. Seismic characteristics and distribution of volcanic intrusions and hydrothermal vent complexes in the Vøring and Møre basins , In: Doré, A. G. & Vining, B. A. (eds) Petroleum Geology: North-West Europe and Global Perspectives — Proceedings of the 6th Petroleum Geology Conference , 833–844. Petroleum Geology Conferences Ltd. Published by the Geological Society, London. ↵
Contributors and Attributions
- Dike (geology). Provided by : Wikipedi. Located at : https://en.Wikipedia.org/wiki/Dike_%28geology%29 . License : CC BY-SA: Attribution-ShareAlike
- Sill (geology). Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Sill_%28geology%29 . License : CC BY-SA: Attribution-ShareAlike


9.3: Fourier Series
Our solution of the diffusion and wave equations will require use of a Fourier series. A periodic function \(f(x)\) with period \(2L\), can be represented as a Fourier series in the form \[\label{eq:1}f(x)=\frac{a_0}{2}+\sum\limits_{n=1}^\infty\left(a_n\cos\frac{n\pi x}{L}+b_n\sin\frac{n\pi x}{L}\right).\]
Determination of the coefficients \(a_0,\: a_1,\: a_2,\ldots\) and \(b_1,\: b_2,\: b_3,\ldots\) makes use of orthogonality relations for sine and cosine. We first define the widely used Kronecker delta \(\delta_{nm}\) as \[\delta_{nm}=\left\{\begin{array}{rl}1&\text{if }n=m; \\ 0&\text{otherwise.}\end{array}\right.\nonumber\]
The orthogonality relations for \(n\) and \(m\) positive integers are then given with compact notation as the integration formulas \[\label{eq:2} \int_{-L}^L\cos\left(\frac{m\pi x}{L}\right)\cos\left(\frac{n\pi x}{L}\right)dx=L\delta_{nm},\] \[\label{eq:3}\int_{-L}^L\sin\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi x}{L}\right)dx=L\delta_{nm},\] \[\label{eq:4}\int_{-L}^L\cos\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi x}{L}\right)dx=0.\]
To illustrate the integration technique used to obtain these results, we derive \(\eqref{eq:3}\) assuming that \(n\) and \(m\) are positive integers with \(n\neq m\). Changing variables to \(ξ = \pi x/L\), we obtain \[\begin{aligned}\int_{-L}^L\sin&\left(\frac{m\pi x}{L}\right)\sin\left(\frac{n\pi x}{L}\right)dx \\ &=\frac{L}{\pi}\int_{-\pi}^\pi\sin(m\xi)\sin(n\xi)d\xi \\ &=\frac{L}{2\pi}\int_{-\pi}^\pi [\cos ((m-n)\xi)-\cos((m+n)\xi)]d\xi \\ &=\frac{L}{2\pi}\left[\frac{1}{m-n}\sin((m-n)\xi)-\frac{1}{m+n}\sin((m+n)\xi)\right]_{-\pi}^\pi \\ &=0.\end{aligned}\]
For \(m=n\), however, \[\begin{aligned}\int_{-L}^L\sin^2\left(\frac{n\pi x}{L}\right)dx&=\frac{L}{\pi}\int_{-\pi}^\pi\sin^2(n\xi)d\xi \\ &=\frac{L}{2\pi}\int_{-\pi}^\pi(1-\cos(2n\xi))d\xi \\ &=\frac{L}{2\pi}\left[\xi-\frac{1}{2n}\sin 2n\xi\right]_{-\pi}^\pi \\ &=L.\end{aligned}\]
Integration formulas given by \(\eqref{eq:2}\) and \(\eqref{eq:4}\) can be similarly derived.
To determine the coefficient \(a_n\), we multiply both sides of \(\eqref{eq:1}\) by \(\cos (n\pi x/L)\) with \(n\) a nonnegative integer, and change the dummy summation variable from \(n\) to \(m\). Integrating over \(x\) from \(−L\) to \(L\) and assuming that the integration can be done term by term in the infinite sum, we obtain \[\begin{aligned}\int_{-L}^Lf(x)\cos &\frac{n\pi x}{L}dx=\frac{a_0}{2}\int_{-L}^L\cos\frac{n\pi x}{L}dx \\ &+\sum\limits_{m=1}^\infty\left\{a_m\int_{-L}^L\cos\frac{n\pi x}{L}\cos\frac{m\pi x}{L}dx+b_m\int_{-L}^L\cos\frac{n\pi x}{L}\sin\frac{m\pi x}{L}dx\right\}.\end{aligned}\]
If \(n = 0\), then the second and third integrals on the right-hand-side are zero and the first integral is \(2L\) so that the right-hand-side becomes \(La_0\). If \(n\) is a positive integer, then the first and third integrals on the right-hand-side are zero, and the second integral is \(L\delta_{nm}\). For this case, we have \[\begin{aligned}\int_{-L}^Lf(x)\cos\frac{n\pi x}{L}dx&=\sum\limits_{m=1}^\infty La_m\delta_{nm} \\ &=La_n,\end{aligned}\] where all the terms in the summation except \(m = n\) are zero by virtue of the Kronecker delta. We therefore obtain for \(n = 0, 1, 2,\ldots\) \[\label{eq:5}a_n=\frac{1}{L}\int_{-L}^L f(x)\cos\frac{n\pi x}{L}dx.\]
To determine the coefficients \(b_1,\: b_2,\: b_3,\ldots\), we multiply both sides of \(\eqref{eq:1}\) by \(\sin (n\pi x/L)\), with \(n\) a positive integer, and again change the dummy summation variable from \(n\) to \(m\). Integrating, we obtain \[\begin{aligned}\int_{-L}^L f(x)\sin &\frac{n\pi x}{L}dx=\frac{a_0}{2}\int_{-L}^L\sin\frac{n\pi x}{L}dx \\ &+\sum\limits_{m=1}^\infty\left\{a_m\int_{-L}^L\sin\frac{n\pi x}{L}\cos\frac{m\pi x}{L}dx+b_m\int_{-L}^L\sin\frac{n\pi x}{L}\sin\frac{m\pi x}{L}dx\right\}.\end{aligned}\]
Here, the first and second integrals on the right-hand-side are zero, and the third integral is \(L\delta_{nm}\) so that \[\begin{aligned}\int_{-L}^L f(x)\sin\frac{n\pi x}{L}dx&=\sum\limits_{m=1}^\infty Lb_m\delta_{nm} \\ &=Lb_n.\end{aligned}\]
Hence, for \(n = 1,\: 2,\: 3,\ldots\), \[\label{eq:6}b_n=\frac{1}{L}\int_{-L}^L f(x)\sin\frac{n\pi x}{L}dx.\]
Our results for the Fourier series of a function \(f(x)\) with period \(2L\) are thus given by \(\eqref{eq:1}\), \(\eqref{eq:5}\) and \(\eqref{eq:6}\).


20.7: Interstellar Matter around the Sun
By the end of this section, you will be able to:
- Describe how interstellar matter is arranged around our solar system
- Explain why scientists think that the Sun is located in a hot bubble
We want to conclude our discussion of interstellar matter by asking how this material is organized in our immediate neighborhood. As we discussed above, orbiting X-ray observatories have shown that the Galaxy is full of bubbles of hot, X-ray-emitting gas. They also revealed a diffuse background of X-rays that appears to fill the entire sky from our perspective (Figure 20.19). While some of this emission comes from the interaction of the solar wind with the interstellar medium, a majority of it comes from beyond the solar system. The natural explanation for why there is X-ray-emitting gas all around us is that the Sun is itself inside one of the bubbles. We therefore call our “neighborhood” the Local Hot Bubble, or Local Bubble for short. The Local Bubble is much less dense—an average of approximately 0.01 atoms per cm 3 —than the average interstellar density of about 1 atom per cm 3 . This local gas has a temperature of about a million degrees, just like the gas in the other superbubbles that spread throughout our Galaxy, but because there is so little hot material, this high temperature does not affect the stars or planets in the area in any way.
What caused the Local Bubble to form? Scientists are not entirely sure, but the leading candidate is winds from stars and supernova explosions. In a nearby region in the direction of the constellations Scorpius and Centaurus, a lot of star formation took place about 15 million years ago. The most massive of these stars evolved very quickly until they produced strong winds, and some ended their lives by exploding. These processes filled the region around the Sun with hot gas, driving away cooler, denser gas. The rim of this expanding superbubble reached the Sun about 7.6 million years ago and now lies more than 200 light-years past the Sun in the general direction of the constellations of Orion, Perseus, and Auriga.
A few clouds of interstellar matter do exist within the Local Bubble. The Sun itself seems to have entered a cloud about 10,000 years ago. This cloud is warm (with a temperature of about 7000 K) and has a density of 0.3 hydrogen atom per cm 3 —higher than most of the Local Bubble but still so tenuous that it is also referred to as Local Fluff (Figure 20.20). (Aren’t these astronomical names fun sometimes?)
While this is a pretty thin cloud, we estimate that it contributes 50 to 100 times more particles than the solar wind to the diffuse material between the planets in our solar system. These interstellar particles have been detected and their numbers counted by the spacecraft traveling between the planets. Perhaps someday, scientists will devise a way to collect them without destroying them and to return them to Earth, so that we can touch—or at least study in our laboratories—these messengers from distant stars.


1.16: Book XVI
ULYSSES REVEALS HIMSELF TO TELEMACHUS.
Meanwhile Ulysses and the swineherd had lit a fire in the hut and were were getting breakfast ready at daybreak, for they had sent the men out with the pigs. When Telemachus came up, the dogs did not bark but fawned upon him, so Ulysses, hearing the sound of feet and noticing that the dogs did not bark, said to Eumaeus:
“Eumaeus, I hear footsteps; I suppose one of your men or some one of your acquaintance is coming here, for the dogs are fawning upon him and not barking.”
The words were hardly out of his mouth before his son stood at the door. Eumaeus sprang to his feet, and the bowls in which he was mixing wine fell from his hands, as he made towards his master. He kissed his head and both his beautiful eyes, and wept for joy. A father could not be more delighted at the return of an only son, the child of his old age, after ten years’ absence in a foreign country and after having gone through much hardship.  When I heard you had gone to Pylos I made sure I was never going to see you any more. Come in, my dear child, and sit down, that I may have a good look at you now you are home again; it is not very often you come into the country to see us herdsmen; you stick pretty close to the town generally. I suppose you think it better to keep an eye on what the suitors are doing.”
“So be it, old friend,” answered Telemachus, “but I am come now because I want to see you, and to learn whether my mother is still at her old home or whether some one else has married her, so that the bed of Ulysses is without bedding and covered with cobwebs.”
“She is still at the house,” replied Eumaeus, “grieving and breaking her heart, and doing nothing but weep, both night and day continually.”
As he spoke he took Telemachus’ spear, whereon he crossed the stone threshold and came inside. Ulysses rose from his seat to give him place as he entered, but Telemachus checked him; “Sit down, stranger,” said he, “I can easily find another seat, and there is one here who will lay it for me.”
Ulysses went back to his own place, and Eumaeus strewed some green brushwood on the floor and threw a sheepskin on top of it for Telemachus to sit upon. Then the swineherd brought them platters of cold meat, the remains from what they had eaten the day before, and he filled the bread baskets with bread as fast as he could. He mixed wine also in bowls of ivy-wood, and took his seat facing Ulysses. Then they laid their hands on the good things that were before them, and as soon as they had had enough to eat and drink Telemachus said to Eumaeus, “Old friend, where does this stranger come from? How did his crew bring him to Ithaca, and who were they?—for assuredly he did not come here by land.”
To this you answered, O swineherd Eumaeus, “My son, I will tell you the real truth. He says he is a Cretan, and that he has been a great traveller. At this moment he is running away from a Thesprotian ship, and has taken refuge at my station, so I will put him into your hands. Do whatever you like with him, only remember that he is your suppliant.”
“I am very much distressed,” said Telemachus, “by what you have just told me. How can I take this stranger into my house? I am as yet young, and am not strong enough to hold my own if any man attacks me. My mother cannot make up her mind whether to stay where she is and look after the house out of respect for public opinion and the memory of her husband, or whether the time is now come for her to take the best man of those who are wooing her, and the one who will make her the most advantageous offer; still, as the stranger has come to your station I will find him a cloak and shirt of good wear, with a sword and sandals, and will send him wherever he wants to go. Or if you like you can keep him here at the station, and I will send him clothes and food that he may be no burden on you and on your men; but I will not have him go near the suitors, for they are very insolent, and are sure to ill treat him in a way that would greatly grieve me; no matter how valiant a man may be he can do nothing against numbers, for they will be too strong for him.”
Then Ulysses said, “Sir, it is right that I should say something myself. I am much shocked about what you have said about the insolent way in which the suitors are behaving in despite of such a man as you are. Tell me, do you submit to such treatment tamely, or has some god set your people against you? May you not complain of your brothers—for it is to these that a man may look for support, however great his quarrel may be? I wish I were as young as you are and in my present mind; if I were son to Ulysses, or, indeed, Ulysses himself, I would rather some one came and cut my head off, but I would go to the house and be the bane of every one of these men. 139 If they were too many for me—I being single-handed—I would rather die fighting in my own house than see such disgraceful sights day after day, strangers grossly maltreated, and men dragging the women servants about the house in an unseemly way, wine drawn recklessly, and bread wasted all to no purpose for an end that shall never be accomplished.”
And Telemachus answered, “I will tell you truly everything. There is no enmity between me and my people, nor can I complain of brothers, to whom a man may look for support however great his quarrel may be. Jove has made us a race of only sons. Laertes was the only son of Arceisius, and Ulysses only son of Laertes. I am myself the only son of Ulysses who left me behind him when he went away, so that I have never been of any use to him. Hence it comes that my house is in the hands of numberless marauders; for the chiefs from all the neighbouring islands, Dulichium, Same, Zacynthus, as also all the principal men of Ithaca itself, are eating up my house under the pretext of paying court to my mother, who will neither say point blank that she will not marry, nor yet bring matters to an end, so they are making havoc of my estate, and before long will do so with myself into the bargain. The issue, however, rests with heaven. But do you, old friend Eumaeus, go at once and tell Penelope that I am safe and have returned from Pylos. Tell it to herself alone, and then come back here without letting any one else know, for there are many who are plotting mischief against me.”
“I understand and heed you,” replied Eumaeus; “you need instruct me no further, only as I am going that way say whether I had not better let poor Laertes know that you are returned. He used to superintend the work on his farm in spite of his bitter sorrow about Ulysses, and he would eat and drink at will along with his servants; but they tell me that from the day on which you set out for Pylos he has neither eaten nor drunk as he ought to do, nor does he look after his farm, but sits weeping and wasting the flesh from off his bones.”
“More’s the pity,” answered Telemachus, “I am sorry for him, but we must leave him to himself just now. If people could have everything their own way, the first thing I should choose would be the return of my father; but go, and give your message; then make haste back again, and do not turn out of your way to tell Laertes. Tell my mother to send one of her women secretly with the news at once, and let him hear it from her.”
Thus did he urge the swineherd; Eumaeus, therefore, took his sandals, bound them to his feet, and started for the town. Minerva watched him well off the station, and then came up to it in the form of a woman—fair, stately, and wise. She stood against the side of the entry, and revealed herself to Ulysses, but Telemachus could not see her, and knew not that she was there, for the gods do not let themselves be seen by everybody. Ulysses saw her, and so did the dogs, for they did not bark, but went scared and whining off to the other side of the yards. She nodded her head and motioned to Ulysses with her eyebrows; whereon he left the hut and stood before her outside the main wall of the yards. Then she said to him:
“Ulysses, noble son of Laertes, it is now time for you to tell your son: do not keep him in the dark any longer, but lay your plans for the destruction of the suitors, and then make for the town. I will not be long in joining you, for I too am eager for the fray.”
As she spoke she touched him with her golden wand. First she threw a fair clean shirt and cloak about his shoulders; then she made him younger and of more imposing presence; she gave him back his colour, filled out his cheeks, and let his beard become dark again. Then she went away and Ulysses came back inside the hut. His son was astounded when he saw him, and turned his eyes away for fear he might be looking upon a god.
“Stranger,” said he, “how suddenly you have changed from what you were a moment or two ago. You are dressed differently and your colour is not the same. Are you some one or other of the gods that live in heaven? If so, be propitious to me till I can make you due sacrifice and offerings of wrought gold. Have mercy upon me.”
And Ulysses said, “I am no god, why should you take me for one? I am your father, on whose account you grieve and suffer so much at the hands of lawless men.”
As he spoke he kissed his son, and a tear fell from his cheek on to the ground, for he had restrained all tears till now. But Telemachus could not yet believe that it was his father, and said:
“You are not my father, but some god is flattering me with vain hopes that I may grieve the more hereafter; no mortal man could of himself contrive to do as you have been doing, and make yourself old and young at a moment’s notice, unless a god were with him. A second ago you were old and all in rags, and now you are like some god come down from heaven.”
Ulysses answered, “Telemachus, you ought not to be so immeasurably astonished at my being really here. There is no other Ulysses who will come hereafter. Such as I am, it is I, who after long wandering and much hardship have got home in the twentieth year to my own country. What you wonder at is the work of the redoubtable goddess Minerva, who does with me whatever she will, for she can do what she pleases. At one moment she makes me like a beggar, and the next I am a young man with good clothes on my back; it is an easy matter for the gods who live in heaven to make any man look either rich or poor.”
As he spoke he sat down, and Telemachus threw his arms about his father and wept. They were both so much moved that they cried aloud like eagles or vultures with crooked talons that have been robbed of their half fledged young by peasants. Thus piteously did they weep, and the sun would have gone down upon their mourning if Telemachus had not suddenly said, “In what ship, my dear father, did your crew bring you to Ithaca? Of what nation did they declare themselves to be—for you cannot have come by land?”
“I will tell you the truth, my son,” replied Ulysses. “It was the Phaeacians who brought me here. They are great sailors, and are in the habit of giving escorts to any one who reaches their coasts. They took me over the sea while I was fast asleep, and landed me in Ithaca, after giving me many presents in bronze, gold, and raiment. These things by heaven’s mercy are lying concealed in a cave, and I am now come here on the suggestion of Minerva that we may consult about killing our enemies. First, therefore, give me a list of the suitors, with their number, that I may learn who, and how many, they are. I can then turn the matter over in my mind, and see whether we two can fight the whole body of them ourselves, or whether we must find others to help us.”
To this Telemachus answered, “Father, I have always heard of your renown both in the field and in council, but the task you talk of is a very great one: I am awed at the mere thought of it; two men cannot stand against many and brave ones. There are not ten suitors only, nor twice ten, but ten many times over; you shall learn their number at once. There are fifty-two chosen youths from Dulichium, and they have six servants; from Same there are twenty-four; twenty young Achaeans from Zacynthus, and twelve from Ithaca itself, all of them well born. They have with them a servant Medon, a bard, and two men who can carve at table. If we face such numbers as this, you may have bitter cause to rue your coming, and your revenge. See whether you cannot think of some one who would be willing to come and help us.”
“Listen to me,” replied Ulysses, “and think whether Minerva and her father Jove may seem sufficient, or whether I am to try and find some one else as well.”
“Those whom you have named,” answered Telemachus, “are a couple of good allies, for though they dwell high up among the clouds they have power over both gods and men.”
“These two,” continued Ulysses, “will not keep long out of the fray, when the suitors and we join fight in my house. Now, therefore, return home early to-morrow morning, and go about among the suitors as before. Later on the swineherd will bring me to the city disguised as a miserable old beggar. If you see them ill treating me, steel your heart against my sufferings; even though they drag me feet foremost out of the house, or throw things at me, look on and do nothing beyond gently trying to make them behave more reasonably; but they will not listen to you, for the day of their reckoning is at hand. Furthermore I say, and lay my saying to your heart; when Minerva shall put it in my mind, I will nod my head to you, and on seeing me do this you must collect all the armour that is in the house and hide it in the strong store room. Make some excuse when the suitors ask you why you are removing it; say that you have taken it to be out of the way of the smoke, inasmuch as it is no longer what it was when Ulysses went away, but has become soiled and begrimed with soot. Add to this more particularly that you are afraid Jove may set them on to quarrel over their wine, and that they may do each other some harm which may disgrace both banquet and wooing, for the sight of arms sometimes tempts people to use them. But leave a sword and a spear apiece for yourself and me, and a couple of oxhide shields so that we can snatch them up at any moment; Jove and Minerva will then soon quiet these people. There is also another matter; if you are indeed my son and my blood runs in your veins, let no one know that Ulysses is within the house—neither Laertes, nor yet the swineherd, nor any of the servants, nor even Penelope herself. Let you and me exploit the women alone, and let us also make trial of some other of the men servants, to see who is on our side and whose hand is against us.”
“Father,” replied Telemachus, “you will come to know me by and by, and when you do you will find that I can keep your counsel. I do not think, however, the plan you propose will turn out well for either of us. Think it over. It will take us a long time to go the round of the farms and exploit the men, and all the time the suitors will be wasting your estate with impunity and without compunction. Prove the women by all means, to see who are disloyal and who guiltless, but I am not in favour of going round and trying the men. We can attend to that later on, if you really have some sign from Jove that he will support you.”
Thus did they converse, and meanwhile the ship which had brought Telemachus and his crew from Pylos had reached the town of Ithaca. When they had come inside the harbour they drew the ship on to the land; their servants came and took their armour from them, and they left all the presents at the house of Clytius. Then they sent a servant to tell Penelope that Telemachus had gone into the country, but had sent the ship to the town to prevent her from being alarmed and made unhappy. This servant and Eumaeus happened to meet when they were both on the same errand of going to tell Penelope. When they reached the House, the servant stood up and said to the queen in the presence of the waiting women, “Your son, Madam, is now returned from Pylos”; but Eumaeus went close up to Penelope, and said privately all that her son had bidden him tell her. When he had given his message he left the house with its outbuildings and went back to his pigs again.
The suitors were surprised and angry at what had happened, so they went outside the great wall that ran round the outer court, and held a council near the main entrance. Eurymachus, son of Polybus, was the first to speak.
“My friends,” said he, “this voyage of Telemachus’s is a very serious matter; we had made sure that it would come to nothing. Now, however, let us draw a ship into the water, and get a crew together to send after the others and tell them to come back as fast as they can.”
He had hardly done speaking when Amphinomus turned in his place and saw the ship inside the harbour, with the crew lowering her sails, and putting by their oars; so he laughed, and said to the others, “We need not send them any message, for they are here. Some god must have told them, or else they saw the ship go by, and could not overtake her.”
On this they rose and went to the water side. The crew then drew the ship on shore; their servants took their armour from them, and they went up in a body to the place of assembly, but they would not let any one old or young sit along with them, and Antinous, son of Eupeithes, spoke first.
“Good heavens,” said he, “see how the gods have saved this man from destruction. We kept a succession of scouts upon the headlands all day long, and when the sun was down we never went on shore to sleep, but waited in the ship all night till morning in the hope of capturing and killing him; but some god has conveyed him home in spite of us. Let us consider how we can make an end of him. He must not escape us; our affair is never likely to come off while he is alive, for he is very shrewd, and public feeling is by no means all on our side. We must make haste before he can call the Achaeans in assembly; he will lose no time in doing so, for he will be furious with us, and will tell all the world how we plotted to kill him, but failed to take him. The people will not like this when they come to know of it; we must see that they do us no hurt, nor drive us from our own country into exile. Let us try and lay hold of him either on his farm away from the town, or on the road hither. Then we can divide up his property amongst us, and let his mother and the man who marries her have the house. If this does not please you, and you wish Telemachus to live on and hold his father’s property, then we must not gather here and eat up his goods in this way, but must make our offers to Penelope each from his own house, and she can marry the man who will give the most for her, and whose lot it is to win her.”
They all held their peace until Amphinomus rose to speak. He was the son of Nisus, who was son to king Aretias, and he was foremost among all the suitors from the wheat-growing and well grassed island of Dulichium; his conversation, moreover, was more agreeable to Penelope than that of any of the other suitors, for he was a man of good natural disposition. “My friends,” said he, speaking to them plainly and in all honestly, “I am not in favour of killing Telemachus. It is a heinous thing to kill one who is of noble blood. Let us first take counsel of the gods, and if the oracles of Jove advise it, I will both help to kill him myself, and will urge everyone else to do so; but if they dissuade us, I would have you hold your hands.”
Thus did he speak, and his words pleased them well, so they rose forthwith and went to the house of Ulysses, where they took their accustomed seats.
Then Penelope resolved that she would show herself to the suitors. She knew of the plot against Telemachus, for the servant Medon had overheard their counsels and had told her; she went down therefore to the court attended by her maidens, and when she reached the suitors she stood by one of the bearing-posts supporting the roof of the cloister holding a veil before her face, and rebuked Antinous saying:
“Antinous, insolent and wicked schemer, they say you are the best speaker and counsellor of any man your own age in Ithaca, but you are nothing of the kind. Madman, why should you try to compass the death of Telemachus, and take no heed of suppliants, whose witness is Jove himself? It is not right for you to plot thus against one another. Do you not remember how your father fled to this house in fear of the people, who were enraged against him for having gone with some Taphian pirates and plundered the Thesprotians who were at peace with us?  Leave off doing so, and stop the others also.”
To this Eurymachus son of Polybus answered, “Take heart, Queen Penelope daughter of Icarius, and do not trouble yourself about these matters.  I say—and it shall surely be—that my spear shall be reddened with his blood; for many a time has Ulysses taken me on his knees, held wine up to my lips to drink, and put pieces of meat into my hands. Therefore Telemachus is much the dearest friend I have, and has nothing to fear from the hands of us suitors. Of course, if death comes to him from the gods, he cannot escape it.” He said this to quiet her, but in reality he was plotting against Telemachus.
Then Penelope went upstairs again and mourned her husband till Minerva shed sleep over her eyes. In the evening Eumaeus got back to Ulysses and his son, who had just sacrificed a young pig of a year old and were helping one another to get supper ready; Minerva therefore came up to Ulysses, turned him into an old man with a stroke of her wand, and clad him in his old clothes again, for fear that the swineherd might recognise him and not keep the secret, but go and tell Penelope.
Telemachus was the first to speak. “So you have got back, Eumaeus,” said he. “What is the news of the town? Have the suitors returned, or are they still waiting over yonder, to take me on my way home?”
“I did not think of asking about that,” replied Eumaeus, “when I was in the town. I thought I would give my message and come back as soon as I could. I met a man sent by those who had gone with you to Pylos, and he was the first to tell the news to your mother, but I can say what I saw with my own eyes; I had just got on to the crest of the hill of Mercury above the town when I saw a ship coming into harbour with a number of men in her. They had many shields and spears, and I thought it was the suitors, but I cannot be sure.”
On hearing this Telemachus smiled to his father, but so that Eumaeus could not see him.
Then, when they had finished their work and the meal was ready, they ate it, and every man had his full share so that all were satisfied.


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6.1: The Lymphatic and Immune System
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6.1.1: Introduction
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Acquired immune deficiency syndrome (AIDS) turned out to be a new disease caused by the previously unknown human immunodeficiency virus (HIV). Although nearly 100 percent fatal in those with active HIV infections in the early years, the development of anti-HIV drugs has transformed HIV infection into a chronic, manageable disease and not the certain death sentence it once was. One positive outcome resulting was that the public’s attention became focused on functional and healthy immune systems.
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6.1.2: Anatomy of the Lymphatic and Immune Systems
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The immune system is the complex collection of cells and organs that destroys or neutralizes pathogens that would otherwise cause disease or death. The lymphatic system, for most people, is associated with the immune system to such a degree that the two systems are virtually indistinguishable. The lymphatic system is the system of vessels, cells, and organs that carries excess fluids to the bloodstream and filters pathogens from the blood.
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6.1.3: Barrier Defenses and the Innate Immune Response
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The immune system can be divided into two overlapping mechanisms to destroy pathogens: the innate immune response, which is relatively rapid but nonspecific and thus not always effective, and the adaptive immune response, which is slower in its development during an initial infection with a pathogen, but is highly specific and effective at attacking a wide variety of pathogens.
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6.1.4: The Adaptive Immune Response - T Lymphocytes and Their Functional Types
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Innate immune responses (and early induced responses) are in many cases ineffective at completely controlling pathogen growth. However, they slow pathogen growth and allow time for the adaptive immune response to strengthen and either control or eliminate the pathogen. The innate immune system also sends signals to the cells of the adaptive immune system, guiding them in how to attack the pathogen. Thus, these are the two important arms of the immune response.
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6.1.5: The Adaptive Immune Response - B Lymphocytes and Antibodies
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Antibodies were the first component of the adaptive immune response to be characterized by scientists working on the immune system. It was already known that individuals who survived a bacterial infection were immune to re-infection with the same pathogen. Early microbiologists took serum from an immune patient and mixed it with a fresh culture of the same type of bacteria, then observed the bacteria under a microscope.
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6.1.6: The Immune Response Against Pathogens
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When following antibody responses in patients with a particular disease such as a virus, this clearance is referred to as seroconversion (sero- = “serum”). Seroconversion is the reciprocal relationship between virus levels in the blood and antibody levels. As the antibody levels rise, the virus levels decline, and this is a sign that the immune response is being at least partially effective (partially, because in many diseases, seroconversion does not necessarily mean a patient is getting well).
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6.1.7: Diseases Associated with Depressed or Overactive Immune Responses
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This section is about how the immune system goes wrong. When it goes haywire, and becomes too weak or too strong, it leads to a state of disease. The factors that maintain immunological homeostasis are complex and incompletely understood.
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6.1.8: Transplantation and Cancer Immunology
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The immune responses to transplanted organs and to cancer cells are both important medical issues. With the use of tissue typing and anti-rejection drugs, transplantation of organs and the control of the anti-transplant immune response have made huge strides in the past 50 years. Today, these procedures are commonplace. Tissue typing is the determination of MHC molecules in the tissue to be transplanted to better match the donor to the recipient.


9.3: Terminating or Repeating?
You’ve seen that when you write a fraction as a decimal, sometimes the decimal terminates , like:
\[\frac{1}{2} = 0.5 \quad \text{and} \quad \frac{33}{100} = 0.033 \ldotp \nonumber \]
However, some fractions have decimal representations that go on forever in a repeating pattern, like:
\[\frac{1}{3} = 0.33333 \ldots \quad \text{and} \quad \frac{6}{7} = 0.857142857142857142857142 \ldots \nonumber \]
It’s not totally obvious, but it is true: Those are the only two things that can happen when you write a fraction as a decimal.
Of course, you can imagine (but never write down) a decimal that goes on forever but doesn’t repeat itself, for example:
\[0.1010010001000010000001 \ldots \quad \text{and} \quad \pi = 3.14159265358979 \ldots \nonumber \]
But these numbers can never be written as a nice fraction \(\frac{a}{b}\) where and are whole numbers. They are called irrational numbers . The reason for this name: Fractions like \(\frac{a}{b}\) are also called ratios . Irrational numbers cannot be expressed as a ratio of two whole numbers.
For now, we’ll think about the question: Which fractions have decimal representations that terminate, and which fractions have decimal representations that repeat forever? We’ll focus just on unit fractions .
A unit fraction is a fraction that has 1 in the numerator. It looks like \(\frac{1}{n}\) for some whole number .
- Which of the following fractions have infinitely long decimal representations and which do not? $\frac{1}{2} \quad \frac{1}{3} \quad \frac{1}{4} \quad \frac{1}{5} \quad \frac{1}{6} \quad \frac{1}{7} \quad \frac{1}{8} \quad \frac{1}{9} \quad \frac{1}{10} \ldotp$
- Try some more examples on your own. Do you have a conjecture?
A fraction \(\frac{1}{b}\) has an infinitely long decimal expansion if:
________________________________.
Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 2. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)
Try even more examples until you can make a conjecture: What is the decimal representation of the unit fraction \(\frac{1}{2^{n}}\)?
| Fraction | Denominator | Decimal |
|---|---|---|
| \(\frac{1}{2}\) | \(2\) | \(0.5\) |
| \(\frac{1}{4}\) | \(2^{2}\) | \(0.25\) |
| \(\frac{1}{8}\) | \(2^{3}\) | \(0.125\) |
| \(\frac{1}{16}\) | ||
| \(\frac{1}{32}\) | ||
| \(\frac{1}{64}\) | ||
| \(\frac{1}{128}\) | ||
| \(\frac{1}{256}\) |
Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 5. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)
Try even more examples until you can make a conjecture: What is the decimal representation of the unit fraction \(\frac{1}{5^{n}}\)?
| Fraction | Denominator | Decimal |
|---|---|---|
| \(\frac{1}{5}\) | \(5\) | \(0.2\) |
| \(\frac{1}{25}\) | \(5^{2}\) | \(0.04\) |
| \(\frac{1}{125}\) | \(5^{3}\) | |
| \(\frac{1}{625}\) | ||
| \(\frac{1}{3125}\) | ||
| \(\frac{1}{15625}\) |
Marcus noticed a pattern in the table from Problem 7, but was having trouble explaining exactly what he noticed. Here’s what he said to his group:
I remembered that when we wrote fractions as decimals before, we tried to make the denominator into a power of ten. So we can do this: $\begin{split} \frac{1}{2} &= \frac{1}{2} \cdot \frac{5}{5} = \frac{5}{10} = 0.5 \ldotp \\ \frac{1}{4} &= \frac{1}{4} \cdot \frac{25}{25} = \frac{25}{100} = 0.25 \ldotp \\ \frac{1}{8} &= \frac{1}{8} \cdot \frac{125}{125} = \frac{125}{1000} = 0.125 \ldotp \end{split}$When we only have 2’s, we can always turn them into 10’s by adding enough 5’s.
- Write out several more examples of what Marcus discovered.
- If Marcus had the unit fraction \(\frac{1}{2^{n}}\), what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?
- Now think about unit fractions with powers of 5 in the denominator. If Marcus had the unit fraction \(\frac{1}{5^{n}}\), what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?
Marcus had a really good insight, but he didn’t explain it very well. He doesn’t really mean that we “turn 2’s into 10’s.” And he’s not doing any addition, so talking about “adding enough 5’s” is pretty confusing.
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Complete the statement below by filling in the numerator of the fraction.
The unit fraction \(\frac{1}{2^{n}}\) has a decimal representation that terminates. The representation will have n decimal digits, and will be equivalent to the fraction \(\frac{?}{10^{n}} \ldotp\)
- Write a better version of Marcus’s explanation to justify why this fact is true.
Write a statement about the decimal representations of unit fractions \(\frac{?}{5^{n}}\) and justify that your statement is correct. (Use the statement in Problem 9 as a model.)
Each of the fractions listed below has a terminating decimal representation. Explain how you could know this for sure, without actually calculating the decimal representation. \[\frac{1}{10} \quad \frac{1}{20} \quad \frac{1}{50} \quad \frac{1}{200} \quad \frac{1}{500} \quad \frac{1}{4000} \ldotp \nonumber \]
The Period of a Repeating Decimal
If the denominator of a fraction can be factored into just 2’s and 5’s, you can always form an equivalent fraction where the denominator is a power of ten.
For example, if we start with the fraction
\[\frac{1}{2^{a} 5^{b}}, \nonumber \]
we can form an equivalent fraction
\[\frac{1}{2^{a} 5^{b}} = \frac{1}{2^{a} 5^{b}} \cdot \frac{2^{b} 5^{a}}{2^{b} 5^{a}} = \frac{2^{b} 5^{a}}{2^{a+b} 5^{a+b}} = \frac{2^{b} 5^{a}}{10^{a+b}} \ldotp \nonumber \]
The denominator of this fraction is a power of ten, so the decimal expansion is finite with (at most) \(a+b\) places.
What about fractions where the denominator has other prime factors besides 2’s and 5’s? Certainly we can’t turn the denominator into a power of 10, because powers of 10 have just 2’s and 5’s as their prime factors. So in this case the decimal expansion will go on forever. But why will it have a repeating pattern ? And is there anything else interesting we can say in this case?
The period of a repeating decimal is the smallest number of digits that repeat.
For example, we saw that
\[\frac{1}{3} = 0.33333 \cdots = 0. \bar{3} \ldotp \nonumber \]
The repeating part is just the single digit 3, so the period of this repeating decimal is one.
Similarly, we know that
\[\frac{6}{7} = 0.857142857142857142857142 \ldots = 0. \overline{857142} \ldotp \nonumber \]
The smallest repeating part is the digits 857142, so the period of this repeating decimal is 6.
You can think of it this way: the period is the length of the string of digits under the vinculum (the horizontal bar that indicates the repeating digits).
Complete the table below which shows the decimal expansion of unit fractions where the denominator has prime factors besides 2 and 5. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)
Try even more examples until you can make a conjecture: What can you say about the period of the fraction \(\frac{1}{n}\) when n has prime factors besides 2 and 5?
| Fraction | Denominator | Decimal |
|---|---|---|
| \(\frac{1}{3}\) | \(0.1 \bar{6}\) | \(1\) |
| \(\frac{1}{6}\) | \(0. \overline{142857}\) | \(6\) |
| \(\frac{1}{7}\) | ||
| \(\frac{1}{9}\) | ||
| \(\frac{1}{11}\) | ||
| \(\frac{1}{12}\) | ||
| \(\frac{1}{13}\) | ||
| \(\frac{1}{14}\) |
Imagine you are doing the “Dots & Boxes” division to compute the decimal representation of a unit fraction like \(\frac{1}{6}\). You start with a single dot in the ones box:
To find the decimal expansion, you “unexplode” dots, form groups of six, see how many dots are left, and repeat.
Draw your own pictures to follow along this explanation:
Picture 1: When you unexplode the first dot, you get 10 dots in the \(\frac{1}{10}\) box, which gives one group of six with remainder of 4.
Picture 2: When you unexplode those four dots, you get 40 dots in the \(\frac{1}{100}\) box, which gives six group of six with remainder of 4.
Picture 3: Unexplode those 4 dots to get 40 in the next box to the right.
Picture 4: Make six groups of 6 dots with remainder 4.
Since the remainder repeated (we got a remainder of 4 again), we can see that the process will now repeat forever:
- unexplode 4 dots to get 40 in the next box to the right,
- make six groups of 6 dots with remainder 4,
- unexplode 4 dots to get 40 in the next box to the right,
- make six groups of 6 dots with remainder 4,
- and so on forever…
On Your Own
Work on the following exercises on your own or with a partner.
- Use “Dots & Boxes” division to compute the decimal representation of \(\frac{1}{11}\). Explain how you know for sure the process will repeat forever.
- Use “Dots & Boxes” division to compute the decimal representation of \(\frac{1}{12}\). Explain how you know for sure the process will repeat forever.
- What are the possible remainders you can get when you use division to compute the fraction \(\frac{1}{7}\)? How can you be sure the process will eventually repeat?
- What are the possible remainders you can get when you use division to compute the fraction \(\frac{1}{9}\)? How can you be sure the process will eventually repeat?
Suppose that is a whole number, and it has some prime factors besides 2’s and 5’s. Write a convincing argument that:
- The decimal representation of \(\frac{1}{n}\) will go on forever (it will not terminate).
- The decimal representation of \(\frac{1}{n}\) will be an infinite repeating decimal.
- The period of the decimal representation of \(\frac{1}{n}\) will be less than n .
- Find the “decimal” expansion for \(\frac{1}{2}\) in the following bases. Be sure to show your work: $two, \quad three, \quad four, \quad five, \quad six, \quad seven \ldotp$
- Make a conjecture: If I write the decimal expansion of \(\frac{1}{2}\) in base b , when will that expansion be finite and when will it be an infinite repeating decimal expansion?
- Can you prove your conjecture is true?