recommended.csv


      
        
          
            ["Entropy is a measure of disorder in a thermodynamic system. If you tell your science teacher that your room is messy, he may tell you that you are not standing in the way of entropy! It is used to predict whether processes are forbidden despite obeying the requirement of conservation of energy as expressed in the first law of thermodynamics. What does the second law say? The second law may be formulated by the observation that the entropy of isolated systems left to spontaneous evolution cannot decrease, as they always arrive at a state of thermodynamic equilibrium, where the entropy is highest at the given internal energy. An increase in entropy accounts for the irreversibility of natural processes, often referred to in the concept of the arrow of time. Historically, the second law was an empirical finding that was accepted as an axiom of thermodynamic theory. Statistical mechanics provides a microscopic explanation of the law in terms of probability distributions of the states of large assemblies of atoms or molecules. The second law has been expressed in many ways. Its first formulation, which preceded the proper definition of entropy and was based on caloric theory, is Carnot's theorem, credited to the French scientist Sadi Carnot, who in 1824 showed that the efficiency of conversion of heat to work in a heat engine has an upper limit. The first rigorous definition of the second law based on the concept of entropy came from German scientist Rudolph Clausius in the 1850s including his statement that heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.[END]"]
            ['When it comes to collecting statistical data, there are two main ways to go about it: surveys and experiments. For example, an opinion poll is one kind of survey - we pick a small number of people and ask them questions. Then, we use their answers as the data. The choice of which individuals to take for a survey or data collection is important, as it directly influences the statistics. When the statistics are done, it can no longer be determined which individuals are taken into account. Suppose we want to measure the water quality of a big lake. If we take samples next to the waste drain, we will get different results than if the samples are taken in a far-away and hard-to-reach spot of the lake. Sampling at different parts of the lake will give us different data - this is called a sampling error. There are two kinds of sampling errors that can occur: if there are many samples, they will likely be very close to what they are in the real population. If there are very few samples, however, they might be very different from what they are in the real population. This error is called a chance error. The individuals for the samples need to be chosen carefully. Usually, they will be chosen randomly. If this is not the case, the samples might be very different from what they really are in the total population. This is true even if a great number of samples are taken. This type of error is called bias.[END]']
            ["In calculations, the number of significant figures is important in order to maintain accuracy. This means that when dealing with numbers consisting of two digits, your final answer should only have two significant figures as well. However, many more digits might be displayed on your calculator depending on its mode setting. These extra digits are meaningless and should be disregarded. In this book, results are often rounded to match the least number of significant figures in the given data; however, sometimes an extra significant figure is retained if it doesn't change the answer significantly. When discarding leftmost digits 5 or more in a number, round up if the last remaining digit is less than 5; otherwise, retain it as is. For example, 11.3516 would be rounded to 11.4, and 11.3279 would become 11.3. When dealing with numbers like 3000 where all zeros might not appear to be significant, we assume that they all are instead. Just remember not to make this assumption elsewhere! Another thing to keep in mind is that significant figures and decimal places are two different concepts. For example, the lengths 35.6 mm, 3.56 m, and 0.00356 m all have three significant figures, but one, two, or five decimal places respectively.[END]"]
            ['If you’ve taken Statistics before, you may have heard of a median. The median is the middle item of the data - to find it, you sort the data from the smallest number to the largest number, and then choose the number in the middle. If there is an even number of data, there will not be a number right in the middle, so you choose the two middle ones and calculate their mean. In our example above (sample data: 23, 26, 49, 49, 57, 64, 66, 78, 82, 92), there are 10 items of data. The two middle ones are "57" and "64". So the median is (57+64)/2 = 60.5. As another example - like the income example presented for mean - consider a room with 10 people who have incomes of $10, $20, $20, $40, $50, $60, $90, $90, $100 and $1000000. Here, the median is $55 because that\'s the average of the two middle numbers ($50 +$60) /2 = 55. If you ignore the extreme value of $10000000 then the mean is 53. In this case, the median is close to the value obtained when the extreme value is thrown out. The median solves the problem of extreme values as described in the definition of mean above.[END]']
            ['If you\'ve taken geometry, you are probably familiar with proofs. A proof is a line of reasoning that will convince others that a statement is true. A proof is different from a good argument- a good argument can convince others that a statement is true without being stated as a proof. A good argument is one that is based on good evidence and good reasoning. Neat! Proof theory is critical in math. It is the area of math that studies the process of mathematical proof.  Proof theory is syntactic in nature. This means that it focuses on the form of a proof rather than what it actually proves. Model theory is semantic in nature- it focuses on the meaning of a proof. Imagine a criminal trial- the judge and jury will primarily be interested in whether or not the evidence presented is sound. They won\'t focus on what the conclusion actually is- only whether or not it is supported by the evidence. Model theory is similar in that it doesn\'t focus on what a proof actually proves- it focuses on whether or not it is supported by the premises. The judge and jury won\'t care if the conclusion they came to is actually true- they will only care if the evidence they used to come to that conclusion is sound. Model theory is interested in whether or not a proof is sound- it doesn\'t care if the conclusion is true or false. A model theorist would say "So what? It proves something false- it\'s still a bad proof!" This is similar to how a judge or jury would feel about a criminal case. They might say "It\'s definitely true that the killer did it- but since the evidence is bad, the case is still bad". A model theorist would say "It\'s definitely true that it proves something false- but since it\'s not supported by the premises, it\'s still a bad proof". Just like how a bad criminal case can\'t be saved by a truth testimony.[END]']