This morning I took the Queen B and the Ladybug to the airport so that they could fly to Florida to spend some time with the B’s family. There, we stood and waited while slack-jawed staff at the Delta counter refused to check-in anyone until the very last minute, whereupon they tried (poorly) to ram through all of the passengers ain a single chaotic mess before finally returning to their previous torpor.
I didn’t realize that Delta customer service was modelled on the Delta function.
Speaking of Skynet, Stephen Wolfram’s version of it, Wolfram|Alpha went online three weeks ago. According to its own hype,
Wolfram|Alpha’s long-term goal is to make all systematic knowledge immediately computable and accessible to everyone. We aim to collect and curate all objective data; implement every known model, method, and algorithm; and make it possible to compute whatever can be computed about anything. Our goal is to build on the achievements of science and other systematizations of knowledge to provide a single source that can be relied on by everyone for definitive answers to factual queries.
So… Skynet. But in convenient, internet-ready form.
Saying hello to Wolfram|Alpha does little to dissuade this notion.
That’s just a little creepy, as is this:
It knows where I live! Let’s try to trick it!
Well played, Wolfram|Alpha. Well played. Then let’s ask it straight up:
Aha! That’s exactly what you’d say if you were Skynet! Busted!
In actuality, Wolfram|Alpha is not Skynet. Well, not yet, anyway. It’s really just Stephen Wolfram’s attempt to steal some business from those other sources of online information brokering, Google and Wikipedia. So how does it stand up to them? Well, let’s compare.
If you ask about “pi”
Similarly, if you ask about, say, “Stephen Cobert”
Or, more interestingly, if you type in “banana slug”
So, to summarize, Wolfram|Alpha is like Google crippled with Asperger’s, or Wikipedia crippled with academic intrigrity.
W|A is very good at computational stuff. Two weeks ago I mentioned Egyptian unit fractions, and (on my own) quite painfully computed a unit fraction decomposition of π that was accurate to ten decimal places; but with W|A it’s as easy as
Similarly, the Queen B needed to convert dollars into euros whilst talking to her mother on the phone. If you just type in, say, “$1200″ into Google, you find lots of sites about living on or spending twelve hundred dollars. Alternatively, if you type in “$1200″ into W|A, you find that its worth 862.46 euros (and 747.52 British pounds and 8184 Chinese yuan ad 16100 Mexican pesos), with a link to compare the exchange history of the dollar versus the euro over the past financial quarter.
However, if you pay attention to all of the weird numerical data W|A gives you, you’ll often find a sense of humor hidden under the symbols. For example, if you type in, say, “176 m.p.h.,” you’ll find the following:
Didja catch it? In addition to 176 m.p.h. being almost 3 miles per minute, or almost a third the speed of sound, or just a smidgen faster than your average TGV train, it is also exactly twice the speed at which Marty McFly needed to drive the Delorean DMC-12 in order to time travel. Apparently W|A likes to watch movies.
(For more fun, try seeing what it knows about “24 m.p.h.“ Go on… I’ll wait.)
I was a little miffed when I typed in “42″ and only got a handful of alternate notations and a couple of factoids, but W|A redeemed itself when I went round the question (or rather, the Answer) a different way:
What other universal truths does W|A know?
Basic philosophy is crippled by W|A’s prowess with cold, hard scientific fact. How about this classic?
Deep. How about this one?
Well, that was a bit unexpected. Apparently God is hanging out in Hungary, although I think referring to Him as a “Pest” is a little condescending.
After all, you’re not Skynet yet, buddy.
For Memorial Day, we decided to treat the Ladybug to a movie at the local movieplex, and being there reminded me of a bunch of movie-related posts I’ve been meaning to make, which I’ll do throughout the week.
To kick it off, let’s begin with the movie we actually saw. We took the Ladybug to go see Night at the Museum 2, which was surprisingly entertaining, in that I was able to sit through all 105 minutes of its runtime despite the presence of Ben Stiller onscreen for almost all of them.
In fact, my only major gripe with this movie is mathematical. The plot (such as it is) involves an evil pharaoh who needs to discover a secret combination to push into an enchanted tablet to unleash unspeakable power. Or something; it doesn’t matter. What ends up happening is the following:
What?
Now granted, a movie built around sentient statues, undead dinosaurs, bird-headed Egyptian gods, and a surprisingly flight-worthy Wright flier is clearly not striving for any kind of scientific or quantitative realism, but that’s not the bit that bugged me.
No, these things offended my sense of mathematical aesthetics and history, and in the spirit of mathematical edification, I’ll tell you why, point by point.
Point 1: π is not encoded in the Great Pyramid. This idea is based on the new-agy notion that the Egyptians encoded their knowledge of transcendental numbers such as π or the golden ratio in the dimensions of the Great Pyramid of Cheops. Specifically, π is supposedly approximated by twice the ratio of the base of the pyramid to its height.
However, there’s no bit of mathematical history to support this as being anything other than coincidental. In fact, most archeological evidence indicates that the Egyptians were relatively unconcerned with the value we nowadays know as π. For example, the process by which Egyptians computed the area of a circle, for example, is not to take the radius squared and multiply it by some constant; rather, Egyptian surveyors were instructed to instead first subtract from the diameter of the circle 1/9th the length of the diameter, and then to multiply that value by itself. In fact, this process is equivalent to approximating the area of a circle by the area of a square whose side length is 8/9 the diameter of the circle, which would mean that ancient Egyptians were more or less approximating π by the value 256/81, which is roughly 3.16, not 3.14.
The Great Pyramid mumbo-jumbo itself became popular only in the early 1920s (which would at least explain why Amelia Earhart was so quick to know it), and it’s largely due to the predictions of bone fide psychic/crank Edgar Cayce, who connected the “divine proportions” of the ancient Egyptians with, among other things, the re-emergence of the lost city of Atlantis. In fact, several of his followers (who quickly became known to Egyptologists as pyramidiots) were so taken with proving Cayce correct that they were actually caught gently filing down corners of the Great pyramid to make the approximations that much more accurate.
Point 2: π is not exactly 3.14159265. That’s a really accurate approximation, but that’s all it is. The number π is irrational (transcendental, even), so it cannot be written exactly as any finite decimal number, no matter how long.
What’s worse — Albert Einstein would have known better. For shame.
Point 3: No Egyptian would understand “3.14159265.” Even assuming that ancient Egyptians did treat the number π with any special reverence — and given that their standard approximation of it is only accurate to one decimal place, this is highly unlikely! — they still wouldn’t work with something like 3.14159265. The reason why is two fold.
First, even though the ancient Egyptians used a base-10 number system like we do now, they didn’t use a positional number system. To explain: expressing a number like one-hundred twenty-five as “125″ assumes you’re using a positional number system:the order in which the digits appear is just as important has the digits themselves. Just think about it: even through 125 and 251 and 152 and 521 all have the same digits (namely 1, 2, and 5), they all represent very different numbers. However, standard Egyptian hieroglyphic numbering was not positional, it was additive: it consisted of a special symbol for each of the numbers 1, 10, 100, 100, etc., and then used multiple copies of each symbol to construct a number. For example, to write 125 in hieroglyphics, an Egyptian scribe could write
which consists of five | symbols (each meaning 1), two arches (each representing 10) and a single spiral (representing 100). The order is immaterial — each of the following values still represent 125:
Mathematical history buffs might point out that high priests and other top-tier ancient Egyptians did not use hieroglyphics for their day-to-day recording and calculations involving numbers. They used an abbreviated shorthand called hieratic script, but even this shorthand was an additive ciphered system — it had separate symbols for the numbers 1, 2, …, 9, then another set for 10, 20, …, 90, and then another set for 100, 200, …, 900, and so on. For example, to write 125, a scribe would only need to write the symbol for 100, 20, and 5, such as
But again, the order didn’t matter. The symbol
still means the same thing. By way of another comparison, the number 152 and 125 have the same digits but refer to different numbers — their relative position is what determines this. In contrast, in hieratic the number 152 uses a completely different set of symbols that 125:
An expression like 3.14159265 assumes a positional numbering system, and that’s just something the Egyptians didn’t work with.
That brings me to the second reason why Egyptians wouldn’t work with something like 3.14159265. Even assuming they were to understand that this meant the value found by dividing 314159265 by 100000000, they never would have expressed the answer as the single fraction 314159265/100000000.
The single most defining aspect of the Egyptian number system is its reliance on unit fractions. To describe any quantity less than 1, Egyptians always and invariably expressed it as a sum of fractions of the form 1/n, so-called “unit” fractions. For example, an ancient Egyptian would never talk about the fraction 2/5, they would instead say that 5 into 2 yields 1/3 + 1/15; they’d never talk about 2/89, but rather 1/60 + 1/356 + 1/534 + 1/890. Hell, even though the Egyptians treated π as 256/81, they never would have thought to express it that way; instead, it would have been written as 3 + 1/7 + 1/57 + 1/10773. I have no idea how the Egyptians would have expressed this more accurate approximation of 3.14159265, but it would not be as a decimal; it’d be something closer to 3 + 1/8 + 1/61 + 1/5020.
So go and enjoy the show… and be better informed, too!
I’m embarrassed to admit that it took me well over a minute to solve the following word problem.
A train leaves Philadelphia for Denver at noon, traveling at 65 miles per hour. Two hours later, a train leaves Denver for Philadelphia, traveling 80 miles per hour on a parallel track. At the moment the two trains pass each other, which one is closer to Philadelphia?
Hint: it doesn’t take a minute to solve.
Part of the problem, I think, is that years and years of mathematical training have lead me to automatically start considering the problem in its most abstract terms even before I’ve actually read what it is I need to solve. I’m not even a half a sentence in before I’m mulling over the more general problem of
A train leaves point A heading for point B at time C, traveling at D miles per hour. E hours later, a train leave from point B to point A (a distance of F miles), traveling at G miles per hour. (a) When is the time H at which the two trains pass? (b) At time H, what is the distance between the trains and point A?
and I’m well into setting up the system of equations needed to solve this word problem before it dawns on me that the original question is not either of the ones I’ve posed… and is, in fact, infinitely more trivial.
This reminds me of a bit from The Hitchhiker’s Guide to the Galaxy:
“Bloody hell,” said Majikthise, “now that is what I call thinking. Here Vroomfondel, why do we never think of things like that?”
“Dunno,” said Vroomfondel in an awed whisper, “think our brains must be too highly trained.”
or the following Far Side comic:
In any case, in the spirit of further unnecessary generalizations, I also submit the following two abstractions of this archetypal word problem.
Did you hear about the romance novel written by a famous mathematician?
It starts off with:
The two lovers ran towards each other like two trains, one leaving Boston at 3:36 PM traveling at 42 miles per hour, and the other leaving Chicago at 4:18 PM traveling at 53 miles per hour…
Consider the following problem:
Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?
The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. There is, however, a much easier solution: since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles.
That’s all there is to it.
When this problem was posed to John von Neumann by his colleague Paul Halmos, he immediately replied, “150 miles.”
“It is very strange,” said Halmos, “but nearly everyone tries to sum the infinite series.”
“What do you mean, strange?” asked Von Neumann. “That’s how I did it!”
* Told by Paul Halmos, in “The Legend of John von Neumann,” American Mathematical Monthly, 80 (April 1973) 386.
By the way, for homework, solve the general problem above. Might as well flex those mathematical muscles.
Last month I posed question for the math folks out there, namely,
What does sin(x)2 mean, or is it ambiguous notation?
and the answers I got were interesting. When I polled my fellow faculty at Komplexify U, the results were pretty evenly split between two choices: (sin(x))2 or ambiguous. When I polled my students, the results were pretty evenly split between three choices: (sin(x))2, sin(x2), and ambiguous. As for the commentors here, there was a definite bias towards (sin(x))2, although considerable effort was made to point out how it might be misinterpreted, and how to better express it.
So what’s the right answer? Easy: there is no ambiguity. The expression sin(x)2 means the square of the sine of x, or to be more clear, ( sin(x) )2.
Let me first off explain why there is no ambiguity here, and then lay down my plan to make all of trig unambiguous henceforth and forever.
To explain why sin(x)2 is unambiguous, let me first discuss a trigonometric expression that truly is ambiguous: sin(x+1)2. Someone who sees this can easily interpret this as both
with the potential ambiguity arising from the common practice of leaving the parentheses off of argument of a trig function. As a consequence, the parentheses in “sin(x+1)2” can be interpreted in two ways: as indicating the argument of a function, or to indicate an algebraic grouping.
When parentheses are used to indicate the argument of function f, then the expression “f(t)” is treated as a single unit, since it represents the unique output value of the function corresponding to the input value t. For example, if f is a generic function, then f(x+1) represents the output when the function f is evaluated at the input value x+1. Therefore, f(x+1)2 must be the square of this value, since f(x+1) is “atomic,” if you will. Said differently, the letter “f” alone, without its parentheses, is a not an algebraic expression; so viewing f(x+1)2 as “f” by itself followed by the expression “(x+1)2” is meaningless. It makes as much sense as writing “(x+1)2f.” (Attentive readers will note that this is essentially basic “order of operations” stuff we’re talking about here.) Applying this logic here, since sine is just a function, if the parentheses indicate its argument, then sin(x+1)2 must be the first interpretation: the square of the sine of (x+1).
However, if we’re assuming that the argument of sine is not being explicitly indicated, then the usual rule of thumb for determining the meaning of the otherwise meaningless expression “sin stuff” is to assume the leftmost complete factor is the argument of sine. For example, given the meaningless expression “sin 3x2 + 2x + 1″, this convention for assigning the argument means this should be interpreted as “sin(3x2) + 2x + 1.” Applying this to sin(x+1)2, if we assume that sin has been written without its argument made explicit, then “(x+1)2” is the first full factor. Here the parentheses are used for an algebraic grouping, because without them the expression “x+1 2” would be different; it would just mean “x+1″, in which case its leftmost factor would just be x. Hence, if we assume the sine notation is being abused, then sin(x+1)2 must be the second interpretation: the sine of the square of (x+1).
Let’s go back to the original sin(x)2. Arguing as above, the only two reasonable interpretations could be
with the potential ambiguity arising because of the tradition of writing “sin x” instead of “sin(x).” If the parentheses are used to indicate the functional argument of sine, then we use the first interpretation. Otherwise, the parentheses not being used for the argument, then they must be used to indicate some essential algebraic grouping to define the argument. However, taking that view, we interpret this as the sine function applied to the expression (x)2, so that the parentheses are grouping…. just x? Really? You really need to write (x)2 to indicate x2? More to the point, would you ever write (x)2 to indicate x2?
So in interpreting sin(x)2, your two choices are either
I think the choice is pretty clear.
And to those who suggested that I could have avoided this whole mess by just writing sin2(x)… I have a better idea. I have…
Trigonometry is hard enough for students without the hassle of all this ambiguity, which as the results of the informal poll above show, can stump even the best of us. Often my colleagues lament this fact, as well as the fact that it’s unfortunately impossible to change.
Screw that!
I for one am working to eliminate all traces of notational ambiguity from trigonometry. I teach the students to write trigonometric functions in one and only one utterly unambiguous form, and I simply do not accept their work unless it is written that way. If all students were simply taught not to use ambiguous notation — and their teachers simply did not accept it — we could utterly stamp out this headache in a generation.
So I am calling on you, my brothers and sisters in the trenches of mathematical education, to come to my aid and banish problematic trigonometric notation. More specifically, I am calling on you to
Banish forever the notation “sin x” and “sin2 x” –
In their place, love live “sin(x)” and “sin(x)2“
There are many reasons for doing this, but let me focus on three:
Reason 1. It’s the right thing to do. I don’t mean this in a touchy-feely, Kumbaya way. I mean it literally: it is the correct thing to do. Sine is function, and there is a completely well-defined, unambiguous notation for working with functions, as mentioned above: if f is the name of a function, then f(x) is the expression representing the output value defined when the function f is evaluated at the input value x.
In algebra, we spend significant time teaching students how to properly use function notation. In an algebraic expression, f must have a set of parentheses to indicate an argument, and the symbol f(argument) is a single, non-divisible value. For a generic function, the symbol f x is just meaningless.
So why do we persist in allowing “sin x” if it is so problematic? Tradition, more than anything else. The notation “sin x” is archaic, denoting nothing more that an abbreviation for the expression “sinus rectus arcus x,” which (very, very loosely) means “the vertical perpendicular component measured from the arc x.” It was written in much the same spirit as writing “ae” as an abbreviation for the word “aequis” (latin for “equals”), which was the preferred method to connect two sides of a equation before Robert Recorde introduced the = symbol in his 1557 text The Whetstone of Witte.
It’s from a point in mathematical history in which the function concept was still not well defined: in fact, this archaic notion of “sin x” has always been ambiguous, for its numerical value could not be assigned until a radial length was given. It makes as much sense today to use this notation in mathematics as it does to continue the Olde Englishe practise of adding silent “e”s to the end of everything (as in the full title of Recorde’s work, The Whetstone of Witte, whiche is the seconde parte of Arthmeteke: containing the extraction of rootes; the cosske practise, with the rule of equation; and the workes of Surde Nombers.)
This old idea served its purpose, but it is not how we work with sine today. Today, we define the radial length to be 1. Today, sine is an honest-to-God function, with a well-defined, utterly unambiguous rule of assignment of values. As a result befitting its modern function interpretation, it makes sense to use its modern function notation, namely, sin(x).
As a corollary, if you’re willing to grant this, then there is also reason to continue the doubly lamentable symbol “sin2x,” since sin(x)2 serves exactly the same purpose. Now I know what some of you are saying right now: “Why not compromise, and use sin2(x)?” The reason is simple and two-fold.
* As a side issue, when discussing the arcsine function, I myself actually only use the arcsin(x) notation rather than the sin-1(x) notation, partly to avoid the aforementioned notational ambiguity, but mostly because there is NO INVERSE to the sine function, which (being periodic) fails to be one-to-one in a particularly egregious manner. Given that the so-called “inverse trigonometric” functions are actually only local inverses designed to preserve acute radian angle measures — that is, arcs — I prefer the arcsine name and notation better.
Reason 2. It’s the right thing to do. Now I mean it in a touchy-feely, Kumbaya way. Trigonometry is a difficult subject as it is: it’s students’ first real, in-depth exposure to transcendental functions. It is the first time students need to work with expressions and formulas for which they have little algebraic intuition, a subject that includes a memorizing a large amount of information (e.g. six functions times five reference angles times four quadrants equals 120 basic trig evaluations) by the end of the first three days.
For many students, “trigonometry” seems like an entirely separate aspect of mathematics with its own indecipherable rules. Persisting in using notation that violates what they’ve been previous taught (or will be taught in a later algebra class) not only validates this viewpoint, but also certainly harms whatever understanding they might have had about the function concept in general. Really, what’s a better use of students’ — and teachers’ — time:
Let’s give ‘em a break.
Reason 3. It’s what your calculator does. This is not as compelling a reason as the other two, but I think that in an age of handheld symbolic calculators and campus-wide access to computer algebra systems, it’s not entirely irrelevant. Calculators like the TI-89 and programs like Maple already use this convention… that is to say, already use only correct function notation when dealing with trigonometry. If using correct notation in class can also help bridge the syntactic difficulty in using a CAS, I’m all for it.
I suppose one argument against all of this, of course, is what’s the point? Every calculus book abuses trig notation, so why waste time trying to teach students to use it correctly? First off, if that’s your attitude, perhaps teaching is not really your calling. Might I suggest “paper weight” or “wind sock” or some other task that doesn’t require any energy exerted on your part?
Beyond that, as a teacher you can use poor notation as a learning experience over and over again. In my classes, students are taught to only use correct function notation for everything, including trigonometric functions and logarithms. In fact, the act of converting outdated notation from the textbook into proper function notation is always the first step in any problem solving algorithm I share with students. Moreover, to emphasize its importance (and encourage students to get into the habit of using it), students in my classes who make simple, bone-headed mistakes based directly on misinterpreting or misusing trigonometric notation sacrifice all partial credit on a problem. I may not be able stamp out poor notation from the calculus text, but I sure as hell do it in my classes!
So what say you, O teachers of mathematics? Will you banish obsolete trigonometric notation and the endless ambiguities it causes? Will you promote true functional notation, and help make trig just that little bit more accessible to your students?
Who’s with me?
