August 2018

Coin Theory of money

Toward a theory of Monee

You and your friends are members of an alien species that is fascinated by a dead planet known as E-rth and in particular with the extinct species called Huemon. In particular, you are interested in understanding a peculiar aspect of Huemon society, known as monee. It is thought by many that everything in the Huemon society is somehow made of monee. One of the problems that you face in understanding monee is your vision, which detects light only in the short wave radio range. As a result of this, you and members of your species can only resolve, that is, see, objects that are larger than about 7 cm in size. Therefore, while many of you suspect that there is fine structure to monee, no one has been able to discern what that structure is.
Among the things known about monee are:
  1. Collections of monee behave as a dense liquid. That is, when poured into a container they have a defined volume, but adopt the shape of the container.
  2. When poured, monee makes a characteristic "jingling" sound.

You are the planets leading expert in monee (since scientists are revered on your planet, that makes you kind of a big deal…kind of like basketball players are here). You have attained the status for two main reasons. First you have found a way to purify different types of monee based upon their optical properties. You have, to date, isolated three distinct types: type "A," is extremely shiny; type "B" is somewhat dull in appearance but similar in color to A; type "C" has two forms, a shiny one and a dull one, but both are darker in color than either A or B. The second reason for your success, is that you have discovered a Huemon device known as the "Kandimusheen." When specific combinations of monee are put into the Kandimusheen, various tasty treats come out (this is part of why your species thinks that everything on E-arth is made of monee). You have combined these two advantages to determine that specific combinations of each form of monee results specific treats.
Since you cannot handle money directly, you have a robot to carry out your experiments. In a typical experiment, you begin with a known mass of each type of monee in each of three cups. The robot then takes monee from each of two cups and combines them with the help of the Kandimusheen, yielding a particular substance. After that, you can determine how much of each type of money is used by determining mass remaining in each cup.
The results of typical experiments are shown below:


Mass of "A" usedMass of "B" usedMass of "C" usedYield
20g10.0g6.0g2 peesogum
45.0g10.0g0g2 chalklatkiss
610g10.0g0g2 Kookie
7**5.0g5.0g0gchalklatkiss AND 2.5g of A left over

The key observations you make are:

    Your task is to come up with a detailed theory of monee, and explain to me, an amiable but somewhat dim King, as much as you can about how monee works.
    John Dalton 1766-1844

    John Dalton lived in England, on planet Earth, where scientists are not revered nearly enough. Among nerds like myself, Dalton is famous for many things. He also trained a graduate student who went on to become famous, named James Prescott Joule, who distinguished himself working on energy. Dalton worked on matter, and is perhaps best known for his
    Atomic Theory of Matter.
    Dalton's notebooks show that he based it on data and observations such as this:
          This last set of observations led Dalton to what's known as: "the law of multiple proportions." the key observation here, is that you had to add exactly twice as much oxygen. As was the case for the two different types of money we were discussing, the best way to explain this is if there is a particle of a definite mass. Just as we had to combine two dimes with a nickel to get $.25, or one dime with a nickel to get $.15, but could not add some fraction of the dime to get an intermediate amount, Dalton proposed you had to add a whole atom of oxygen to a compound made of a definite number of each type of atom. In the example of the two gases, the deadly one is carbon monoxide, which has one carbon atom paired with one oxygen atom; whereas the less deadly version is carbon dioxide, which has two oxygen atoms paired with a single carbon atom.
          From this, Dalton came to very similar explanations as you did for monee.
            So, if we assume the deadly gas is one carbon atom and one oxygen atom, carbon monoxide, then the non deadly has one carbon atom and two oxygens (accounting for the mass of oxygen doubling). This would be carbon dioxide.
            The ratio of the masses of Carbon to Oxygen is 3:4. Hydrogen was found to be the lightest (it is given the weight of one Dalton), then oxygen is 16 times that mass, and therefore oxygen must be 16 daltons, and Carbon must be 12 (to maintain the 3:4 with Oxygen). Notice, you don't know the actual mass in grams. But, you know the ratios.

            Classifications of matter

            This covers elements, compounds and mixtures and the states in which they exist
            Read More…

            Energy Blog

            Here is a summary of some energy rules I want you to know. We will return to this many times, including much more quantitatively later in the year. The beginning part, up to "work work work," is just the discussion I presented about "capacity to cause pain." But, you should definitely read from that point (work work work). Read More…

            Practice spreadsheet

            You can pick up the practice excel spreadsheet here.


            How do we choose units

            The important thing to remember here, is that units are usually chosen to be convenient. I made a couple of jokes about needing a unit for distance that is about the width of your thumb or a unit of volume that’s the right amount to order for a glass of beer (or orange juice, in your case). So, using the units you are more used to, a “cup” of orange juice or a “pint” of beer may seem about right. Measuring out gasoline in cups would be a little less convenient than gallons. Similarly, measuring distances between your home and downtown San Diego in inches would be inconvenient. Thus, we try to have units such that the amount we are measuring provides us with a number that makes sense in our feeble brains. (No offense intended, no humans are any good at comprehending really large or really small numbers.) The problem with the system that you grow up with is that there is no consistent relationship between units that are optimized for measuring out a cup of orange juice, a gallon of gasoline, or a barrel of oil, just as there is no consistent relationship between an inch, a foot, a yard and a mile. You have to remember all of the unit conversions. At this point you probably know most of them however, I have to say it is a bit of pain in the butt. We will be using the SI Units. That stands for the “international system” of units (or rather, it stands for the French version of that). The great advantage here is that in each case we have one base unit, say, the meter for distance, and then units derived from that as factors of 1,000 larger or smaller. In some cases, factors of 10 or 100 may be used if we have need for a unit in that range. A meter is about the length from the tip of your nose to the tip of your finger, that is, if you are an average male in the 17th century (actually, the definition of a meter has been updated many times over the last couple of hundred years). The common unit “centimeter” is one 100th of a meter. Normally, as I said, we would multiply or divide by factors of 1,000. So we have: a millimeter, 1/1000 of a meter; a micrometer, 1/1000 of a millimeter; a nanometer, 1/1000 of a micrometer, and so on. Whether we are speaking about meters, grams, liters, or volts, or any other basic unit, milli means 10
            -3, micro means 10-6 and nano means 10-9. The reason for the centimeter is that the length is just convenient for many things we measure. Sure, I could refer to the width of my index finger as about .01 m, or 10 mm, but one centimeter just seems easier (note that all of those values are the same). The same basic strategy applies when we go to larger and larger amounts. I could talk about my new hard drive and the fact that it can store up to 1 trillion bytes, but nobody really comprehends what one trillion means (just try talking about our national debt some time). However if I say 1000 GB, or 1 TB, those are numbers we can think about at least reasonably well. You have a sense from working with storage on your computer about how much a gigabyte really is. I could tell you that something has a mass of 10,000 grams; but, even if you have a sense of what a gram is, you cannot really conceive of what 10,000 of them would be. On the other hand, 10 Kg makes some sense, if you have an idea of what a Kg feels like. Below is a table of the prefixes that we will use commonly. Of the ones listed below, I expect you to know: centi; milli; micro; nano; Kilo and all the abbreviations. The others will rarely come up.

            Dimensional analysis

            The basic idea is this: the units must cancel out to give you the proper units for your answer. If you want your answer to be in grams, the other units in the calculation must cancel. The nice thing about this is that it shows you how to set of the calculation. Rather than try to remember whether two numbers should be multiplied, or one divided into the other, you set up the equation just so that the units cancel. You can set up a long running calculation with the numbers left out, just with units, so that they cancel properly. The you just fill in the numbers. So, here is an example: A car is traveling 60.0 miles/hour. Given that a mile=1,609.344 meters, how many seconds will it take to travel 10,000.0 meters. So, you want the answer in seconds, and the only starting number you have is in miles/hour. You are given the conversion factor for meters/mile, and the others you should know. I'm going to start by inverting miles/hour, because I want time to be on top in the final answer. You could save that step to the last, if you want. you see how miles, hours, meters and minutes cancel leaving seconds? Now, just plug in the numbers where they belong. By the way, am I Okay on the sig. figs? Let's see, the speed is given in 3 sig. my answer should not have any more than that. What about the others? The 60 minutes in an hour and 60 seconds in a minute are both exact numbers. That is, they are not measurements; they are definitions. The other numbers are more sig. figs. So, I should report it at only three sig. figs, or 373 seconds. Here is another one: A furlong is 220.0 yards and an inch is 2.54 cm exactly. A really fast (but believable) racehorse can run 10 furlongs in about 2 minutes. If a horse wins a 10 furlong race with a time of 2 minutes flat (2.00 minutes), how fast was he traveling in meters/second?

            Accuracy and Precision/Sig Figs1

            This combines a couple of older blogs. There may be some redundancy. It considers accuracy, precision and how to report your uncertainty. Read More…

            Facts, Laws and Theories

            Facts, Laws and Theories

            How do we know things? Fundamentally, what is the basis of knowledge? This is a completely separate class. However, it is essential that we address it a bit. Science is not about certainty, but about testability. It is a system of knowledge that is based on testing one's ideas against observation. The only untestable assumptions we allow are: that there is something we call reality; and that we can interact with that reality in some meaningful way. That is the core of what an experiment is: an experience of reality from which we extract meaning.
            One could fairly point out that science is therefore based ultimately on untestable assumptions. However, it is also fair to say that anyone interacting with the world at all—walking, talking to others, any interaction at all—makes these same two untestable assumptions. In science, these two untestable assumptions are the only two we make.

            Testable Assumptions:

            Every other assumption can be tested by comparing the predictions of your assumptions to the reality you experience. There will be mistakes and dead ends, that's true. However, by continuing to check our predictions against reality, we can assess how well our explanations fit. You may notice that in this system, there are no "absolutes" accept for rare instances where they arise as part of a definition (as in “absolute zero” in temperature). Otherwise, we merely define and limit our uncertainty. The very idea of certainty is somewhat out of place in science. A few years back at the dedication of our Science building, Nobel Laureate David Baltimore put it like this: “in science, we move from uncertainty to less uncertainty.”

            A fact is something observed reproducibly in the physical world. Facts are our measurements. As absolute as the term "fact" may seem to you, the reliability of facts depends on the precision of the measurement. Thus, our appreciation of facts can change. You may say it is a fact that the desktop is solid matter. However, it is, like all objects you see, mostly space. Small particles pass through the desktop as though it isn't there. If your level of detection (resolution) is that of your eye, or your finger, objects do not pass through the desktop. If your level of resolution is considerably smaller (the size of a proton or electron), the is more "space" for particles to pass through than there is matter for them to hit. Anything outside your limits of resolution cannot be detected. That does not mean your measurements are meaningless. It just means that it is extremely important to know what your limit of resolution is. In a very meaningful way, the desktop is solid. Your conclusions from your interaction with the desktop were not wrong. We have just refined our understanding of what "solid" means.
            Laws: So, if there are no absolutes in science, then what about "laws?" The scientific concept of "law" is a hard one for students, sometimes. Laws amount to codification of our observations. They can be conceptual or mathematical. For example: "an object in motion will stay in motion unless acted upon by a force" is part of Newton's First Law. Force=mass times acceleration is also a law. Laws are subject to change as we understand things better. Newton's Law of Universal Gravitation is a darn good rule. But, after over 200 years of supremacy, Newton's Law had to be re-tooled when Einstein spotted a fundamental flaw in it. Einstein based this on another Law: nothing can travel faster than the speed of light. Could we learn that this too is an approximation? Sure. But, so far, it is holding up well. Moreover, Einstein's theory of gravity has an advantage over Newton's Law because it has a theoretical basis (yes, I am implying that a "theory" has distinct advantages over a "law." I will expand on that below). But, the General Theory of Relativity, which is Einstein’s attempt to resolve the problem of gravity, has it’s own shortcomings. Another example are the laws of "conservation of matter" and "conservation of energy." These laws say that neither matter nor energy can be created or destroyed. Once again, that annoying Albert Einstein found that there was a shortcoming in our understanding. While the laws are still fundamentally considered correct, Einstein and later deBroglie and Schroedinger forced us to realize that matter and energy were really aspects of the same stuff. Thus, we now allow that matter can be converted to energy and energy to matter (given by the relationship e=mc
            2). The two laws have become one, and we now say that the sum of all matter/energy in the universe is constant.
            Hypothesis/model: I use these two words more or less interchangeably. These are tentative explanations of principles underlying the facts we observe. A model also may be conceptual or mathematical. Any scientific model must have two attributes: it must fit with data already observed (known facts—it may modify our interpretation of those facts);
            and it must make predictions about future experiments. It is this second thing that is most important, because it allows us to test whether our tentative explanation has merit. Hypotheses are often wrong. Testable hypotheses that turn out to be wrong are still useful. Untestable explanations are worthless in science. Theories: Believe it or not, theories represent the height of scientific understanding. There is a tendency in the non-science world to think of a theory as some sort of weak fact, something we are less sure of than facts. This is not the case in science. Theories are as good as we get. Theories do not get "proved." They get tested, like hypotheses. A theory is sort of a grander hypothesis. A good theory offers a single explanation that links many different, often seemingly unrelated, facts into one cogent model that can be used to make predictions. It often subsumes many hypotheses. As with hypotheses, the power of a theory comes more from its predictions than from its explanations. It is through predictions that the theory obtains its usefulness. Often, the best predictions of a theory are unexpected: things that no one knew, or even suspected, before the theory. If those predictions turn out to be correct, they represent dramatic support for the theory. The Special Theory of Relativity is a great example: based on an attempt to explain some of the shortcomings of Newton’s Laws, Einstein came up with his famous equation linking mass and energy:. This predicts that a tiny amount of mass can be converted into an extraordinarily large amount of energy (“c,” the speed of light, is a huge number). Nuclear power plants confirm this prediction continuously. That represents a critical test of the theory. No one could have discovered nuclear power without the theory to predict it. When a new theory emerges, sometimes it forces us to reevaluate how we interpreted facts. Thus, a new theory can change things dramatically. We may have many facts that seem unrelated. These facts may be used in various hypotheses. Then, rarely, someone comes along and says: "all those things are actually related via this overarching theory." The old laws and theories usually are subsumed by the new one. For example, Newton’s laws now are considered a special case of Einstein’s Theories. They work fine under most conditions. Relativity tells you the conditions under which Newton’s laws will fail you. If there is a word in common use that corresponds to what we mean by theory, it is "explanation." This could be contrasted with a law, which is really just a "description" of what happens. The description (law) may be very useful. You may be able to predict not only that things will fall to Earth but also at what rate they will accelerate. Einstein’s theory of gravity does that, but it also makes a surprising prediction: space itself will be curved around massive objects (like a star) so light must follow a curved path near such an object. This prediction was tested and that’s exactly what happens.
            Can a theory be said to be "True" or "False?" Theories that are tested and are consistently good in predicting outcomes of experiments do not become absolute truths. However, we assume we are on the right track. Also, certain predictions or claims of a theory can be said to be true. If demonstrated reliably, these claims can become facts, subject to the same limitations above. For example, the central requirement of plate tectonic theory, that there are large areas of the Earth's crust that move relative to each other, has been demonstrated. Another example comes from the Theory of Evolution. It is a fact that organisms evolve and that one species can evolve into another related one. That has been observed. But, the basic structure of a theory remains just that: theoretical.
            A theory can be disproved. However, when a theory that has proven extremely useful fails to work in some context, we usually don't talk about disproving it, but rather finding its limits. For example, Dalton's Atomic Theory, which we will discuss soon, is right in many ways. However, one of Dalton's claims is not correct: that atoms are indivisible. So, we modify the theory a little and say that dividing an atom makes it a fundamentally different thing. An atom is not the smallest unit of matter (as Dalton would have said) but the smallest unit of an element that still has the properties of that element.
            So, What do we know?! It may seem that science at once claims to have some authoritative knowledge and yet to know nothing for certain. So, what are we supposed to make of it? It is one of the amusing paradoxes of human existence that you cannot learn something until you admit that you don't know it. I would not want to give the impression that there are no petty scientists who guard their positions and thwart new ideas. But, we are raised in science to think that it is wrong. We all are told "fables" by our mentors that teach us to be wary of getting too set in our ways. We are taught to get used to saying "I don't know." The important thing for you to take from this discussion is that gaps in our understanding are not seen as flaws, so much as opportunities for further study. “I don’t know” is not an admission of failure; it merely defines the next question. There will likely always be gaps; we will always have uncertainty. However, in the few hundred years that this method has been in use, there have been no reversals in our understanding (that’s a bit of an opinion on my part. There are philosophers who would argue that). We continually challenge what we know. We are surprised by new directions, but we haven't, in any cases that I know of, grossly lost our way. We refine our view. We find limits to our existing ideas, and find new ideas to cover the areas we didn't even know were there. Galileo said that the Sun was the center of the universe. It isn’t, but it was the center of the universe as he could measure it. The (currently known) universe is bigger than he knew. As we refine our measurements, we appreciate new levels of detail. But, we did not, for example, discover that he was wrong and that the Earth was the center of the universe (as the Catholic Church said it knew as a matter of certainty to be true). We did not overturn our understanding. We came to see a larger picture.