osmie: (Default)
(for [livejournal.com profile] moizissimo, who wondered why a cake would contain exactly seven servings)

Once upon a time there were seven hungry fairies who hated continued fractions and infinite series. It was their birthday (for all fairies have the same birthday), and they all wanted cake.

Mickey O'Flaherty, the neighbourhood drunken leprechaun, took a perfectly sharp knife -- one which left no kerf at all -- and sliced the perfectly cubical cake cleanly in two: once vertically from the north, once vertically from the east, once horizontally from the south. There were now eight pieces for seven fairies, so he placed one piece of cake on each plate and started to hand them out.

"Hey!" shouted the fairies. "You're just a drunken leprechaun! You don't get any cake; it's our birthday!"

So Mickey O'Flaherty took the last piece of cake, itself a perfect cube, and with his perfectly sharp knife sliced it cleanly in two: once vertically from the north, once vertically from the east, once horizontally from the south. There were now eight smaller pieces for seven fairies, so he placed one piece of cake on each plate, balancing it carefully atop the first piece, and started to hand them out.

"Hey!" shouted the fairies. "You're just a drunken leprechaun! You don't get any cake; it's our birthday!"

So Mickey O'Flaherty took the last piece of cake, itself a perfect cube, and hesitated a moment with his perfectly sharp knife in hand. "You know, we could just keep doing this forever, because there'll always be a piece left over."

"No!" screamed the fairies. "Not a continued fraction! We hate infinite series!"

"But don't you find it fascinating that Σ(1/8)i is equal to one seventh?"

"No!" screamed the fairies. "Not an infinite series! We hate continued fractions!"

"But don't you find it beautiful to imagine infinitely many cubes, each exactly one eighth the size of the last, stacked one on top of the other on your plates?"

"No!" screamed the fairies, and magicked Mickey O'Flaherty's perfectly sharp knife out of his hand. With their collective will, they easily sliced the last piece of cake six times -- snick, snack, sneak, snuck, snook, snick -- into seven parallel slabs of perfectly equal sevenths. And Mickey O'Flaherty took another swig of whiskey and grumbled off, belching, into the ferns.
osmie: (Default)
(for [livejournal.com profile] moizissimo, who wondered why a cake would contain exactly seven servings)

Once upon a time there were seven hungry fairies who hated continued fractions and infinite series. It was their birthday (for all fairies have the same birthday), and they all wanted cake.

Mickey O'Flaherty, the neighbourhood drunken leprechaun, took a perfectly sharp knife -- one which left no kerf at all -- and sliced the perfectly cubical cake cleanly in two: once vertically from the north, once vertically from the east, once horizontally from the south. There were now eight pieces for seven fairies, so he placed one piece of cake on each plate and started to hand them out.

"Hey!" shouted the fairies. "You're just a drunken leprechaun! You don't get any cake; it's our birthday!"

So Mickey O'Flaherty took the last piece of cake, itself a perfect cube, and with his perfectly sharp knife sliced it cleanly in two: once vertically from the north, once vertically from the east, once horizontally from the south. There were now eight smaller pieces for seven fairies, so he placed one piece of cake on each plate, balancing it carefully atop the first piece, and started to hand them out.

"Hey!" shouted the fairies. "You're just a drunken leprechaun! You don't get any cake; it's our birthday!"

So Mickey O'Flaherty took the last piece of cake, itself a perfect cube, and hesitated a moment with his perfectly sharp knife in hand. "You know, we could just keep doing this forever, because there'll always be a piece left over."

"No!" screamed the fairies. "Not a continued fraction! We hate infinite series!"

"But don't you find it fascinating that Σ(1/8)i is equal to one seventh?"

"No!" screamed the fairies. "Not an infinite series! We hate continued fractions!"

"But don't you find it beautiful to imagine infinitely many cubes, each exactly one eighth the size of the last, stacked one on top of the other on your plates?"

"No!" screamed the fairies, and magicked Mickey O'Flaherty's perfectly sharp knife out of his hand. With their collective will, they easily sliced the last piece of cake six times -- snick, snack, sneak, snuck, snook, snick -- into seven parallel slabs of perfectly equal sevenths. And Mickey O'Flaherty took another swig of whiskey and grumbled off, belching, into the ferns.

Bad math

Jun. 8th, 2009 10:28 pm
osmie: (Default)
As I get older, my mathematical abilities are falling away from me. Sometimes, as when I'm actually trying to converse with a mathematician, this becomes acutely embarrassing: for example, the vector (a1b1a2b2a3b3) is not really interchangeable with the scalar dot product a1b1 + a2b2 + a3b3, and real mathematicians tend to cringe when you speak of them as if they were the same thing.

A year or two ago, I reached a new personal high in Bad Math by covering several pages with chicken-scratches before I proved that all positive integers have a unique partition within the set {1}. For non-mathematicians, this means not only that you can write any positive integer in the form 1+1+1+...+1, but also that there's only one way to do so! See, you probably already knew this. It wasn't really as profound a result as I wanted it to be.

Today at work I was creaking through an ordinary differential equation, and wanted to find an expression in k, ρ and dk/dρ for zeroes. And so I blithely set my entire expression equal to zero. After a good deal more math, I discovered that one of the following must be true:

  • k = 0
  • ρ = 0
  • ρ → ∞
  • k = Cρ for some arbitrary C

For non-mathematicians, this means that k and ρ can have any relationship they darned well please, as long as k is finite. Which really ought not to be surprising, considering that I'd just gone and set the whole ruddy expression to zero to begin with. Sigh.

At which point there came an epiphany over me.

I rocked at mathematics all the way through elementary and high school. I kicked so much ass I had to move on to donkeys and mules. I sailed through single-variable calculus at the age of 17, and linear algebra at 18. I didn't have to think too hard until I hit vector calculus, around the point where div and curl and Green's Theorem were introduced -- at which point it became clear to me that I didn't really know how to learn math. I tried hard, and came out of the course with a grade in the high 80s ... but it's long been my contention that any mathematics grade below 90% to 95% means you've missed some vitally important concept.

And indeed, this was the point where my grades started slipping. As that vitally important concept I missed -- whatever it was -- become elementary to more and more later coursework, I ploughed through more and more exercises with no clear idea why I was following these particular steps. I got through third-year math with C+ grades, and fourth-year with a bare pass. My degree says I'm educated with a specialization in the mathematical and physical sciences, but it's always been an exaggeration to say I really understood all my senior coursework.

Today I had no trouble deriving, articulating and visualizing a fairly complicated parametric expression, in which y is a known function f(φ), and φ follows a known distribution P(x,ρ), but f(φ) contains a constant term k which might actually be an unknown, differentiable function h(ρ). It's just single-variable calculus with repeated applications of the chain rule, leading to a differential equation in k and ρ. I whizzed through these steps.

At which point I had derived an ordinary differential equation of a type I knew how to solve, but not when to solve. I plugged in garbage boundary conditions and produced a garbage answer.

And I finally realized that I haven't actually forgotten any mathematics that I ever truly understood. I've only forgotten the pieces I never finished learning in the first place. The guesswork and cramming and rote problem-solving skills have fallen away, to mark a clear line of incoherence between the math I know cold and the math about which I start babbling. Although I can be sad to have learned so little mathematics during my years ostensibly studying it, I no longer need to be embarrassed at forgetting everything -- or worried what I might lose next.

Bad math

Jun. 8th, 2009 10:28 pm
osmie: (Default)
As I get older, my mathematical abilities are falling away from me. Sometimes, as when I'm actually trying to converse with a mathematician, this becomes acutely embarrassing: for example, the vector (a1b1a2b2a3b3) is not really interchangeable with the scalar dot product a1b1 + a2b2 + a3b3, and real mathematicians tend to cringe when you speak of them as if they were the same thing.

A year or two ago, I reached a new personal high in Bad Math by covering several pages with chicken-scratches before I proved that all positive integers have a unique partition within the set {1}. For non-mathematicians, this means not only that you can write any positive integer in the form 1+1+1+...+1, but also that there's only one way to do so! See, you probably already knew this. It wasn't really as profound a result as I wanted it to be.

Today at work I was creaking through an ordinary differential equation, and wanted to find an expression in k, ρ and dk/dρ for zeroes. And so I blithely set my entire expression equal to zero. After a good deal more math, I discovered that one of the following must be true:

  • k = 0
  • ρ = 0
  • ρ → ∞
  • k = Cρ for some arbitrary C

For non-mathematicians, this means that k and ρ can have any relationship they darned well please, as long as k is finite. Which really ought not to be surprising, considering that I'd just gone and set the whole ruddy expression to zero to begin with. Sigh.

At which point there came an epiphany over me.

I rocked at mathematics all the way through elementary and high school. I kicked so much ass I had to move on to donkeys and mules. I sailed through single-variable calculus at the age of 17, and linear algebra at 18. I didn't have to think too hard until I hit vector calculus, around the point where div and curl and Green's Theorem were introduced -- at which point it became clear to me that I didn't really know how to learn math. I tried hard, and came out of the course with a grade in the high 80s ... but it's long been my contention that any mathematics grade below 90% to 95% means you've missed some vitally important concept.

And indeed, this was the point where my grades started slipping. As that vitally important concept I missed -- whatever it was -- become elementary to more and more later coursework, I ploughed through more and more exercises with no clear idea why I was following these particular steps. I got through third-year math with C+ grades, and fourth-year with a bare pass. My degree says I'm educated with a specialization in the mathematical and physical sciences, but it's always been an exaggeration to say I really understood all my senior coursework.

Today I had no trouble deriving, articulating and visualizing a fairly complicated parametric expression, in which y is a known function f(φ), and φ follows a known distribution P(x,ρ), but f(φ) contains a constant term k which might actually be an unknown, differentiable function h(ρ). It's just single-variable calculus with repeated applications of the chain rule, leading to a differential equation in k and ρ. I whizzed through these steps.

At which point I had derived an ordinary differential equation of a type I knew how to solve, but not when to solve. I plugged in garbage boundary conditions and produced a garbage answer.

And I finally realized that I haven't actually forgotten any mathematics that I ever truly understood. I've only forgotten the pieces I never finished learning in the first place. The guesswork and cramming and rote problem-solving skills have fallen away, to mark a clear line of incoherence between the math I know cold and the math about which I start babbling. Although I can be sad to have learned so little mathematics during my years ostensibly studying it, I no longer need to be embarrassed at forgetting everything -- or worried what I might lose next.
osmie: (Default)
The more I think about the equal-tempered scale, the more convinced I become that √2 -- the augmented fourth, or F# -- is its only pure interval. Which leads me to several other speculations:
  • I wonder what kind of music results from rational tunings within ℝ/√2 instead of simply ℝ.
  • I wonder what the interval √3 sounds like.
  • I wonder whether the aural beats of an irrational (or even a tempered) tuning affect the mind similarly to the visual beats of fluorescent lights.
  • I wonder whether the insistent low-level cacophony of music that's almost in a rational tuning, along with the increasing number of venues by which such music is delivered to our ears, has any correlation to the prevalence of depression in Western society.


Edit: All the stylized Rs in that post should have been stylized Qs. I must have misread the Unicode font table, and now I'm too lazy to go try again.
osmie: (Default)
The more I think about the equal-tempered scale, the more convinced I become that √2 -- the augmented fourth, or F# -- is its only pure interval. Which leads me to several other speculations:
  • I wonder what kind of music results from rational tunings within ℝ/√2 instead of simply ℝ.
  • I wonder what the interval √3 sounds like.
  • I wonder whether the aural beats of an irrational (or even a tempered) tuning affect the mind similarly to the visual beats of fluorescent lights.
  • I wonder whether the insistent low-level cacophony of music that's almost in a rational tuning, along with the increasing number of venues by which such music is delivered to our ears, has any correlation to the prevalence of depression in Western society.


Edit: All the stylized Rs in that post should have been stylized Qs. I must have misread the Unicode font table, and now I'm too lazy to go try again.
osmie: (Default)
So I've been happily working my way through Dummit & Foote's Abstract Algebra, whose manuscript once served as the textbook to the only math course I ever failed.

Now, it's been my opinion since at least high school that if you get less than 90% in a math class, you didn't get it. Stupid mistakes can pull you down to 90%, but anything lower means you're missing at least one basic concept ... and the moment you proceed to a higher grade level, you won't understand any of the three concepts which build on the one you're missing. And the problem only gets worse (one might even say it "exponentiates"): two years down the road, you'll be confused by nine concepts, and start to suspect that you're no good at math. The third year, if you're even paying attention anymore, you'll have to scramble to eke out a pass. No: better to make someone repeat the first class they didn't understand, than a course so advanced that they don't even understand its terminology.

The corollary to this opinion is that math classes should be two weeks long -- or maybe a month, tops -- because there's no point in repeating an entire semester, let alone a year, just to pick up that one missing concept. My ideal math curriculum would be pretty much entirely Montessori-based, so that each module contains just one or two concepts, and comes clearly marked with its own prerequisites: self-directed students could pick up any module for which they qualify; one-on-one tutors could quickly diagnose which concepts need more explanation; teachers could cover a different variety of modules each year, always lecturing to a class of all the students (across multiple grades) who qualify for that particular module. It's completely foreign to the conventional mathematics curriculum, and I'd love to be involved someday in helping to design it.

I failed Math 422 because I didn't pay attention in Math 322, and coasted to a 65% grade. I should have accepted this as a warning sign and repeated Math 322 ... but because I truly love algebra, I enrolled in Math 422 anyway. Guess what? I didn't even understand its basic terminology. "Plummet!" [livejournal.com profile] topdrop memorably shouted into my answering machine that year. "Plummet into the Atlantic!" I think he was just being surreal, but it described my comprehension level perfectly.

So.

The time has come to learn algebra properly: on my own, working from the textbook, doing all the exercises, and for no other reason than because I love it. But in the absence of a professor on whose door to knock when I get stymied, I turn to you, gentle readers. For I have a problem. I cannot seem to prove the normality of the commutator subgroup. )

Can anybody help my poor algebraically strapped brain? EDIT: Yes: [livejournal.com profile] grouchychris can! Yay [livejournal.com profile] grouchychris!
osmie: (Default)
So I've been happily working my way through Dummit & Foote's Abstract Algebra, whose manuscript once served as the textbook to the only math course I ever failed.

Now, it's been my opinion since at least high school that if you get less than 90% in a math class, you didn't get it. Stupid mistakes can pull you down to 90%, but anything lower means you're missing at least one basic concept ... and the moment you proceed to a higher grade level, you won't understand any of the three concepts which build on the one you're missing. And the problem only gets worse (one might even say it "exponentiates"): two years down the road, you'll be confused by nine concepts, and start to suspect that you're no good at math. The third year, if you're even paying attention anymore, you'll have to scramble to eke out a pass. No: better to make someone repeat the first class they didn't understand, than a course so advanced that they don't even understand its terminology.

The corollary to this opinion is that math classes should be two weeks long -- or maybe a month, tops -- because there's no point in repeating an entire semester, let alone a year, just to pick up that one missing concept. My ideal math curriculum would be pretty much entirely Montessori-based, so that each module contains just one or two concepts, and comes clearly marked with its own prerequisites: self-directed students could pick up any module for which they qualify; one-on-one tutors could quickly diagnose which concepts need more explanation; teachers could cover a different variety of modules each year, always lecturing to a class of all the students (across multiple grades) who qualify for that particular module. It's completely foreign to the conventional mathematics curriculum, and I'd love to be involved someday in helping to design it.

I failed Math 422 because I didn't pay attention in Math 322, and coasted to a 65% grade. I should have accepted this as a warning sign and repeated Math 322 ... but because I truly love algebra, I enrolled in Math 422 anyway. Guess what? I didn't even understand its basic terminology. "Plummet!" [livejournal.com profile] topdrop memorably shouted into my answering machine that year. "Plummet into the Atlantic!" I think he was just being surreal, but it described my comprehension level perfectly.

So.

The time has come to learn algebra properly: on my own, working from the textbook, doing all the exercises, and for no other reason than because I love it. But in the absence of a professor on whose door to knock when I get stymied, I turn to you, gentle readers. For I have a problem. I cannot seem to prove the normality of the commutator subgroup. )

Can anybody help my poor algebraically strapped brain? EDIT: Yes: [livejournal.com profile] grouchychris can! Yay [livejournal.com profile] grouchychris!
osmie: (Default)
Suppose I have a countably infinite union of sets.

A good example would be the union of all sets Q, where Qn is the set of rational numbers strictly between (1/n) and (n+1/n), and n is a positive integer. (This seems like a good example because given any two such sets, you can find a number that's in either set but not both.)

Suppose further that I have an arbitrary element q of this union, Q.

My question is: Can I assume that q is an element of one of the finitely numbered sets Qn? Or is it possible that q might be a limit point of Q, which in this example would be 0? (It's clear that Q needn't be closed; in this example, 2 is certainly not an element of Q, although (2-ε) is for rational ε.)

In other words, if I can find a sequence qn of elements of Qn which converges to q, then is q an element of the infinite union of Q? On one hand this seems to make sense, but on the other, it's possible to construct such a sequence qn which converges to any real number r between 0 and 1, and surely even the infinite union of rational sets doesn't contain any irrational numbers!

So ... is there any circumstance in which Q can contain an element not present in any of the finitely numbered sets Qn? If yes, how can this be codified without using a convergent sequence -- or is it in fact true that the infinite union of Q[0,1] with itself equals R[0,1]?

Thanks...

Edit: It occurs to me, a few hours later, that I may be confusing two separate concepts here. (Or maybe not. You tell me.)
  • Concept #1: the union of a countably infinite number of sets, A = (union) An over n from 1 through ∞
  • Concept #2: the limit of a union of a finite series of sets, as the number of sets goes to infinity, B = limn->∞ (union) Bn.

    Is concept #2 just the formalization of concept #1? Or are they actually distinct ideas, in which case it might make sense for B to include the closure (under the limit) of all sets Bn, but for A to include only those values which actually appear in at least one finitely numerable set An?


Later edit: Thanks to [livejournal.com profile] grouchychris (who gave me the right answer), [livejournal.com profile] topdrop (who proved it was the right answer) and [livejournal.com profile] dabrota (who showed me where I went wrong)! Concept #1 is valid. Concept #2 is not well defined, because the notion of a convergent series of bn requires that all Bn (beyond a finite N) be subsets of a topology with a well-defined metric -- which is far more structure than I wanted to assume. It's certainly more structure than the operation of a "union" implies.
osmie: (Default)
Suppose I have a countably infinite union of sets.

A good example would be the union of all sets Q, where Qn is the set of rational numbers strictly between (1/n) and (n+1/n), and n is a positive integer. (This seems like a good example because given any two such sets, you can find a number that's in either set but not both.)

Suppose further that I have an arbitrary element q of this union, Q.

My question is: Can I assume that q is an element of one of the finitely numbered sets Qn? Or is it possible that q might be a limit point of Q, which in this example would be 0? (It's clear that Q needn't be closed; in this example, 2 is certainly not an element of Q, although (2-ε) is for rational ε.)

In other words, if I can find a sequence qn of elements of Qn which converges to q, then is q an element of the infinite union of Q? On one hand this seems to make sense, but on the other, it's possible to construct such a sequence qn which converges to any real number r between 0 and 1, and surely even the infinite union of rational sets doesn't contain any irrational numbers!

So ... is there any circumstance in which Q can contain an element not present in any of the finitely numbered sets Qn? If yes, how can this be codified without using a convergent sequence -- or is it in fact true that the infinite union of Q[0,1] with itself equals R[0,1]?

Thanks...

Edit: It occurs to me, a few hours later, that I may be confusing two separate concepts here. (Or maybe not. You tell me.)
  • Concept #1: the union of a countably infinite number of sets, A = (union) An over n from 1 through ∞
  • Concept #2: the limit of a union of a finite series of sets, as the number of sets goes to infinity, B = limn->∞ (union) Bn.

    Is concept #2 just the formalization of concept #1? Or are they actually distinct ideas, in which case it might make sense for B to include the closure (under the limit) of all sets Bn, but for A to include only those values which actually appear in at least one finitely numerable set An?


Later edit: Thanks to [livejournal.com profile] grouchychris (who gave me the right answer), [livejournal.com profile] topdrop (who proved it was the right answer) and [livejournal.com profile] dabrota (who showed me where I went wrong)! Concept #1 is valid. Concept #2 is not well defined, because the notion of a convergent series of bn requires that all Bn (beyond a finite N) be subsets of a topology with a well-defined metric -- which is far more structure than I wanted to assume. It's certainly more structure than the operation of a "union" implies.

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