Unedited Ramblings, Episode Three

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reduced echelon formI took linear algebra my last semester of college. On the day of the first exam, the professor entered the classroom and rhetorically asked us how we felt about the test, and the guy behind me said, “I can’t express my feelings in reduced echelon form.” He doesn’t know this, but I later quoted him on tumblr.  It was one of my most popular tumblr posts of all time, partly just because it sounds really deep and partly because someone who saw it came up with the brilliant idea of Echelon Poetry. I really wish that had caught on, even though I didn’t think much of the way some people interpreted that idea. Arranging words in a triangle is not the same as putting words in echelon form. I’m not sure how one would go about putting words in echelon form, but it fascinates me to imagine that there is a way.

But even if we are talking about actual matrices rather than poetry, I have an inclination to want to believe that it ought to be possible to describe emotions in reduced echelon form. There ought to be a way to notate feelings and then perform mathematical procedures to make sense of them.

My last attempt to do so lasted only two days, because there were just so many difficulties involved. How many different kinds of emotions are there? Is humor an emotion? What should the numerical scale be? One to ten? One to twelve? One to eight? One to six? Sixteen point twelve to thirty-nine and a half? Should the arithmetic be done in base ten or some other base? Should I use standard numerals or invent my own form of numerical notation specifically for the purpose of this exercise? Is there any logical reason to do so, or am I only considering that because I want my matrix to be nonsensical and enigmatic to everyone except myself? And perhaps most importantly, when I do the math and find an answer, what will that answer actually mean?

As it turns out, it’s a major inconvenience to carry around a notebook and commit to writing numbers in it every four hours. I could have changed my system so that I didn’t have to collect data so frequently, but I felt like that would compromise the accuracy. Between that inconvenience and the fact that I didn’t actually have any useful information to gain by proceeding with this plan, I ended it. But that doesn’t mean I won’t try again at some point.

Maybe I’m just strange, but I find it horribly frustrating to be incapable of quantifying feelings. There are so many things in life that can be accurately and thoroughly described by little numbers written in little boxes. Those numbers are knowledge and power and safety; not only do they convey information, but they allow you to assume that the thing being described by those numbers is subject to all the normal rules of mathematics. But if something can’t be described in numbers, then it’s unclear what the rules are.

Once, I spent several weeks keeping track of my feelings on a one-dimensional scale from one to ten, while simultaneously assigning a numerical value to every noteworthy event in order to determine how much of an impact it should have on my mood. The point was to determine whether or not my mood was a logical and objective response to the events of my life. As you can probably guess, this experiment also ended mainly because it was absurdly time-consuming. But in the meantime, I noticed that, interestingly enough, my actual feelings corresponded very closely to what they should have been if they were in fact an objective response. This trend quite surprised me even though it was what I had hoped to discover.

As far as I can tell, there are three possible explanations. One is that I took such a subjective approach to the whole project that even the numerical values I assigned to events was determined based upon how I felt about it at that particular time. That is admittedly very likely, but given the fact that I made sure to keep those values constant when an event re-occurred, it would seem that the effects of this bias would have decreased over time, which wasn’t what my numbers indicated. The second possibility is that it’s actually true that my feelings are a rational and quantifiable response to external events. I’d like to believe that, but it seems extremely far-fetched. The third possibility is the really fascinating one. Maybe, the act of trying to quantify feelings is therapeutic in the sense that it actually regulates emotions to the extent that they actually do begin to function in a completely logical way. Maybe, by quantifying one’s emotions, one can actually make them follow an algorithm.

Whether the second or third of those possibilities is the correct answer, that’s a good reason to work towards the goal of finding a way to quantify feelings. But the fact remains that it’s mathematically ridiculous to do so, at least not without somehow taking neurological factors into consideration, allowing for differences between different people, and using an extremely well-informed psychology-based rationale for every aspect of the method of quantification. In other words, such an undertaking is well beyond my capabilities. I am forced to live with the annoying and frustrating reality that I cannot quantify my feelings.

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Why Base Twelve Would Be Awesomer Than Base Ten

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If you think about it, it’s silly that people count in base ten. Yes, it’s convenient because we happen to have ten fingers, but it’s inconvenient in numerous other ways. For example, although 1/2 and1/4 and 1/5 and 1/10 can be easily expressed by decimals, other common fractions like 1/3 and 1/6 and 1/8 are just weird if you try to write them in any other form. To give a less abstract example, the amount of time that I spend at work is measured according to a decimal system. That means that each 0.01 hour of work is 36 seconds, which is a kind of random unit of time. On some level, the human race is clearly aware that units of twelve are logical. The year has twelve months, a foot has twelve inches, and many products are sold in groups of twelve. Yet we still insist upon counting in base ten.

handsI think it says something about the selfish nature of humanity that we just assume that numbers are meant to be used in base ten simply because we have ten fingers. The human hand, according to our subconscious thought process, is clearly the standard by which we are supposed to measure everything in existence. No source of authority and no rational point of view outranks the supremacy of the human hand. Or something like that. But, mathematically speaking, there are better ways to count.

The short and simple way to say this is just to insist that base twelve is better than base ten because twelve has more factors than ten. But I’m going to back up a couple steps and ramble about some other things first. In all fairness, I must acknowledge that there’s a book I’m currently reading (How Acceptance of a Duodecimal (12) Base Would Simplify Mathematics by F. Emerson Andrews, copyright 1935 and 1944) that basically says everything that I’m saying in this blog post, and I’m sort of drawing from that book in writing this. But I also would like to point out, just for the sake of being a know-it-all, that none of the information or ideas I’m repeating here were new to me. These were all things I had heard, read, and thought about a long time before I happened to notice that book on the library bookshelf and was drawn in by its awesomeness.

The first thing about which I want to ramble is that even the tally mark system is pretty cool. We couldn’t count very high if it wasn’t for the clever construct of splitting numbers into handy units. If you count on your fingers, you only have two sets of five at your disposal, and you’re going to lose count pretty quickly once you get past ten. And if you try to count by writing down one mark for every unit, that’s not going to improve matters much. But by sorting those individual units into groups of five and then counting fives, you can count an awful lot higher. There’s no particular reason that five has to be the base used or that the notation method has to be tally marks as we know them; it’s the system of individual units and larger group units that is so clever and useful. Even though we take that system completely for granted, it’s pretty awesome when you think about it.

Roman PIN numberEarly forms of number notation were basically always tally-mark-type systems. Even Roman numerals are really just a glorified form of tally marks. You’ve got the individual unit written as I, the group of five units written as V, the group of ten units written as X, and so forth and so on. As an added bonus, numbers could be written in a more concise way by putting a smaller numeral in front of the greater numeral to indicate that the smaller unit is to be subtracted from the bigger unit rather than added onto it. For example, nine isn’t IIIIIIIII or VIIII because that’s kind of hard to read. It would be easy to accidentally confuse VIIII with VIII. So nine is IX, which means I less than X. So the Roman numeral system definitely had its benefits, but it still is of the same caliber as tally mark systems, and it still is really bad for doing arithmetic. (Quick, what’s MCDXXIV plus XXVII?)

But then the world was revolutionized by the numeral zero, which is the awesomest thing ever invented by humanity with the possible exception of that time when some random person thought of the idea of grinding up coffee beans and filtering hot water through them. Of course, the concept of “none” had always existed and there were ways of expressing the quantity of “none” in words. But there was no numeral zero as we use it now, and so place value didn’t work. It’s difficult to attribute the origin of zero to a specific time or place, because various cultures had various different ways of mathematically denoting zero-ness. But the significant advancement was the use of place value that was made possible by the use of the numeral zero, and that came from India and then gradually became commonly used in Europe during the medieval period. It wasn’t until the 16th century that the current system for writing numbers finished becoming the norm.

I think we can all agree that the Hindu-Arabic number system is much easier to use than Roman numerals. It’s easier to look at 1040 and 203 and know right away that they add up to 1243 than to look at MXL and CCIII and know that they add up to MCCXLIII. And it isn’t hard to add 48 and 21 in your head and get 69, but adding XLVIII and XXI to get LXIX is a little messy. A numerical system that relies on place value is inherently simpler to use than a system that doesn’t.

But there’s still that whole thing about base ten. To say that we count in base ten means that ten is the number that we write as 10. 10 means one group of ten plus zero ones. 12 means one group of ten plus two ones.  176 means one group of ten times ten, seven groups of ten, and six ones. But if, for instance, we counted in base eight, then 10 would mean one group of eight and no ones, which is 8 in base ten. 12 would mean one group of eight and two ones, which is 10 in base ten. 176 would mean one group of eight times eight, seven groups of eight, and six ones, which is 126 in base ten. If that sounds complicated, it’s only because we’re so used to base ten. We instinctively read the number 10 as ten without even thinking about the fact that the 1 in front of the 0 could refer to a different number if we were counting in a different base.

I’m not really advocating for getting rid of base ten, because it would be impossible to change our whole system of counting overnight. It took centuries for Hindu-Arabic numerals to replace Roman numerals in Europe, and switching to a different base would be an even bigger overhaul. Base ten is a very familiar system and it would just be confusing for everyone to suddenly change it, not to mention the fact that everything with numbers on it would become outdated and mathematically incorrect. So I’m perfectly content to stick with base ten for the most part, but I still think it’s worth pointing out that base twelve would technically be better. And this brings me to my actual point, which is why exactly base twelve is the best of all possible bases.

It goes without saying that the only feasible bases are positive integers. But I’m saying it anyway just because I am entertained by the notion of trying to use a non-integer as a base. It is also readily apparent that large numbers don’t make good bases. Counting and one-digit arithmetic are basically learned by memorization, and the larger the base is, the more there is to memorize. But small numbers don’t make good bases, either, because it requires a lot of digits to write numbers. Take base three, for instance. Instead of calling this year 2013, we’d be calling it 2202120. (Disclaimer: it’s entirely possible that I made an error. That’s what happens when I use weird bases.) And it wouldn’t be a good idea to use a prime number as a base. Even though I happen to be fond of the number seven and have said before that the people on my imaginary planet count in base seven, I realize that’s weird. (That is, counting in base seven is weird. It’s completely normal that I have an imaginary planet that uses a different mathematical system.) In base ten, we have a convenient pattern; every number that ends in 5 or 0 is divisible by 5, and any number that doesn’t end in 5 or 0 is not divisible by 5. That pattern works because 5 is a factor of 10. Using a prime number as a base would complicate multiplication and division because we wouldn’t have useful patterns like that.

So the numbers that would work relatively well as bases are eight, nine, ten, and twelve, and maybe six, fourteen, fifteen, and sixteen, if we want to be a little more lenient about the ideal size range. Eight and sixteen win bonus points for being 23 and 24, which is nice and neat and pretty, and nine and sixteen win bonus points for being squares. (Squares are cool, y’all) But twelve is the real winner here, because its factors include all of the integers from one to four. That means that it’s easily divisible by three and four as well as by two, and a multiplication table in base twelve would have lots of handy little patterns. Every number ending in 3,6,9, or 0 would be divisible by 3; every number ending in 4, 8, or 0 would be divisible by 4; every number ending in 6 or 0 would be divisible by 6. All multiples of 8 would end in 4, 8, or 0, and all multiples of 9 would end in 3, 6, 9 or 0. As in base ten, all even numbers would end with an even digit and all odd numbers would end with an odd digit. And obviously, every number divisible by twelve would end in 0.Basically, base twelve has the most convenient patterns of any base in the feasible size range.

Base Twelve Multiplication TableTo prove its convenience, I made this multiplication table myself rather than copying the one in the aforementioned book. (For the record, X refers to ten, because the notation 10 now means twelve, not ten, and ε refers to eleven, because the notation 11 now refers to thirteen, not eleven. I got those additional digits from the book. Part of me wanted to make up new ones, but there was some logic to the way it was done in the book, so I decided to just go with that.) I did double check it against the book just to be sure, and I suppose I ought to confess that I made a couple errors in the 5 and 7 columns. 5 and 7 are a little problematic in base twelve in the same way that 3 and 4 and 6 and 7 and 8 are a little problematic in base ten. But this didn’t take me very long at all to do, and the columns for 2, 3, 4, 6, 8, 9, X, and ε were extremely easy. Since basic arithmetic isn’t exactly a great strength of mine, the fact that I found it easy to construct this multiplication table proves the mathematical ease of arithmetic using base twelve.

So, yeah, base twelve is cool and stuff.

More Numbers

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I just finished the second phase of the experiment which I began in early June and wrote about again here after finishing the first phase. (Making links to my earlier blog posts is really fun. I have no idea why, but it is.) For the last twenty days, I have been attempting to memorize a string of twenty digits in a minute twice each day, thereby attaining forty data points. For twenty of these, I put my hand on my face while I was trying to remember the numbers, and for the other twenty, I kept my hands away from my face. The point of this experiment was to discover whether or not I think better when my hands are on my face, and I was really hoping that the answer would be yes. It would be very convenient if I could instantly make myself smarter just by putting my hand on my face.

Actually, the real reason for this experiment was to give me a reason to tape stuff like this to my walls. The important thing here is interior design. My room looks very elegant decorated like this.

Unfortunately, just like in the first phase of this experiment, my results were inconclusive. Although there did seem to be trends in the results, the actual calculations revealed that the trends weren’t large enough to actually prove anything. The differences could conceivably have been due to random chance. I think it’s worth noting, though, that this time the results were almost good enough to be statistically significant. The trends probably mean something, even though I don’t have good enough evidence to insist that they definitely do. And what the trends show is that I remember stuff better when my hands are on my face. I definitely intend to make use of this information when I’m taking the GRE in two weeks.

Now I just have to decide what the next phase of this experiment is, because I don’t want to give up this delightfully fun game.

Statistical Stuff

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Three weeks ago today, I wrote this blog post in which I described an experiment that I was about to start. The first phase of the experiment was to take twenty days, and it has recently been completed. In case you didn’t see the first post about this experiment and don’t feel like reading it now, I will briefly summarize. Over the course of twenty days, I have collected sixty data points measuring how well I was able to memorize a string of twenty digits, acquired from an online random number generator. I collected these data samples three times a day (early morning, late morning, and late afternoon) and memorized the numbers using three different methods. (sitting at a desk without my hands on my face, sitting at a desk with my hands on my face, and pacing in a circle around my room)To reduce the effects of confounding variables, the time of day and the method used did not correspond. I used the online random number generator to determine the order in which I would use the three methods each day.

My scores ranged from two to twenty, but most of them were in between eleven and fifteen, and the later ones seemed to be higher on average. Apparently, my number-memorizing skills are improving. I have just now determined that the average was 13.60 and the standard deviation was 4.25. (I am very curious to know if that’s a ‘good’ score or not, but since nobody else has done an identical experiment, as far as I know, there’s no standard for comparison) For most of the last twenty days, I have been coming to the disappointing conclusion that there is no statistically significant difference between the three different methods, but I was still curious about what I would find when I added up the scores and did the math.

At first, it looked promising. The averages for the three different methods were 12.68, 14.10, and 13.24, which seems to be different enough to actually mean something. Out of curiosity, I also calculated the individual averages for the three different times of day and got 14.95, 14.48, and 11.23 respectively, which verifies my assumption that I tend to kind of be a morning person. But since I already knew that, the data that I was really looking at was the difference between the three memorization methods.  At that point in my statistical analysis, I thought I had actually discovered something interesting.

Unfortunately, the thing with the standard deviation messed that up. 4.25 is a pretty large standard deviation for something with a 20-point scale. The definition of ‘statistically significant’ depends upon where you want to set the margin of error, but 4.25 is a pretty reasonable number to use for that, and it makes the math really easy if margin of error is equal to one standard deviation. That means that the differences in my data are not significant unless they fall outside of the range from 9.35 to 17.85, which they do not.

Now, I suppose I could calculate the within standard deviations rather than using the overall standard deviation, but I’m pretty sure that my results wouldn’t be any different. Each method had approximately the same range, and since I already know they had similar averages, their standard deviations are probably going to be pretty close also, and I’ll have to reach the same anticlimactic conclusion, which is that there wasn’t a significant difference.

In case anyone is still interested in that slight little insignificant difference, I can inform you that the best score came from memorizing the numbers while sitting at the desk with my hands on my face, and the worst score came from memorizing the numbers while sitting at the desk without my hands on my face. I’m not entirely convinced that there isn’t something to my hypothesis that having one’s hands on one’s face somehow improves cognitive ability.

This calls for some further tests.