Tag Archives: rational numbers
I don’t know why I’m so fascinated that the rational numbers are countable even though they’re a dense subset of the uncountable real numbers. A rational number can be arbitrarily close to any real number, making you think they’d be infinite like the reals, but in fact, nearly all numbers are irrational (and an uncountable subset of the reals).
So, the rational numbers — good old p/q fractions — though still infinite are countably infinite (see this post for details).
More to the point here, a common way of enumerating the rational numbers, when graphed results in some pretty curves and illustrates some fun facts about the rational numbers.
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3 Comments | tags: charts, fractions, graphs, rational numbers | posted in Brain Bubble, Math
Zeno’s famous Paradoxes involve the impossibility of arriving somewhere as well as the impossibility of even starting to go somewhere. And that flying arrows have to be an illusion. [Time flies like an arrow, but fruit flies like a banana.]
If Zeno were alive today, he’d be over 2500 years old and would have seen his paradoxes explained in a variety of ways by a lot of very smart people. Yet at heart they still have some metaphysical oomph. And the thing is, at least in some contexts, Zeno was (sort of) right. There is something of a paradox here involving space and time.
Or at least something interesting to think about.
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7 Comments | tags: rational numbers, real numbers, Zeno of Ela, Zeno's paradoxes | posted in Brain Bubble, Math, Philosophy
It’s been a few minutes since my last post. Lately, the effort of writing hasn’t seemed worth the almost non-existent return. I find I’ve lost faith in humanity, and the phrase that seems most resonant is: “Really, when you come right down to it, what’s the point of it all?” I think, at least in our case, the Fermi Paradox seems resolved.
Perhaps more crucially, this damned dark cloud over me seems all I can write about. Everything else seems ephemeral. If we can’t solve our most basic human problems (education, race, gender, poverty, pollution) then the rest of it really is fiddling while Rome burns.
It makes me angry. Humanity can do better than this. I think.
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16 Comments | tags: maze generator, mazes, pi, rational numbers | posted in Friday Notes
Multiplying by i…
Recently I did a series of posts about how the complex numbers arise from a natural progression of math realizations. I’ve done posts in the past about how the natural numbers lead through the integers and rationals to the real numbers. (And I’ve done posts about how weird the real numbers are, but that’s another topic.)
I recently came across another way a progression of obvious natural questions directly leads to the necessity of a new type of number, and this progression takes us all the way from the naturals to the complex numbers.
All by asking, “What do you get when you…”
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6 Comments | tags: complex numbers, complex plane, group theory, groups, integers, irrational numbers, natural numbers, rational numbers, real numbers, sets, town barber paradox | posted in Math
Yes, this is a math post, but don’t run off too quickly. I’ll keep it as simple as possible (but no simpler), and I’ll do all the actual math so you can just ride along and watch. What I’m about here is laying the groundwork to explain a fun fact about the Mandelbrot.
This post is kind of an origin story. It seeks to explain why something rather mind-bending — the so-called “imaginary numbers” — are actually vital members of the mathematical family despite being based on what seems an impossibility.
The truth is, math would be a bit stuck without them.
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17 Comments | tags: complex numbers, complex plane, fun with numbers, imaginary unit, integers, Leopold Kronecker, natural numbers, numbers, rational numbers, real numbers | posted in Math
In the Rational vs Real post I mentioned that real numbers were each “an infinitely tiny island separated from direct contact with all other numbers.” The metaphor of each real number as an island comes from how, given any real number, it’s not possible to name the next (or previous) real number.
It’s easy enough to name a particular real number. For instance 1.0 and 3.14159… real numbers. There are infinitely many more we can name, but given any one of them, there is no way to get to any other number other than by explicitly naming it, too.
This applies to a variety of numeric spaces.
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8 Comments | tags: complex numbers, continuum, integers, math theory, mathematics, natural numbers, numbers, rational numbers, real numbers, Theory of Mathematics | posted in Math
One of the great philosophical conundrums involves the origin of numbers and mathematics. I first learned of it as Platonic vs Aristotelian views, but these days it’s generally called Platonism vs Nominalism. I usually think of it as the question of whether numbers are invented or discovered.
Whatever it’s called, there is something transcendental about numbers and math. It’s hard not to discover (or invent) the natural numbers. Even from a theory standpoint, the natural numbers are very simply defined. Yet they directly invoke infinity — which doesn’t exist in the physical world.
There is also the “unreasonable effectiveness” of numbers in describing our world.
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8 Comments | tags: math theory, mathematics, natural numbers, nominalism, numbers, Plato, Platonic, Platonism, rational numbers, real numbers, Theory of Mathematics | posted in Math, Philosophy
Oh, no! Not math again!
Among those who try to imagine alien first contact, many believe that mathematics will be the basis of initial communication. This is based on the perceived universality and inevitability of mathematics. They see math as so fundamental any intelligence must not only discover it, but must discover the same things we’ve discovered.
There is even a belief that math is more real than the physical universe, that it may be the actual basis of reality. The other end of that spectrum is a belief that mathematics is an invented game of symbol manipulation with no deep meaning.
So today: the idea that math is universal and inevitable.
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41 Comments | tags: alien math, cardinal numbers, cardinality, counting, counting numbers, first contact, Leopold Kronecker, math origins, math theory, mathematics, natural numbers, numbers, Philosophy of Math, rational numbers, real numbers, Theory of Mathematics | posted in Math, Opinion
We’re still motoring through numeric waters, but hang in there; the shore is just ahead. This is the last math theory post… for now. I do have one more up my sleeve, but that one is more of an overly long (and very technical) comment in reply to a post I read years ago. If I do write that one, it’ll be mainly to record the effort of trying to figure out the right answer.
This post picks up where I left off last time and talks more about the difference between numeric values and how we represent those values. Some of the groundwork for this discussion I’ve already written about in the L26 post and its followup L27 Details post. I’ll skip fairly lightly over that ground here.
Essentially, this post is about how we “spell” numbers.
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1 Comment | tags: base 10, base 2, base 8, Frederik Pohl, Heechee, irrational numbers, Leopold Kronecker, natural numbers, number bases, number names, numbers, pi, prime numbers, rational numbers | posted in Math, Sideband
Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the Theory of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: countable and uncountable. It also connects with the Smooth or Bumpy post, which considered differences between the discrete and the continuous.
This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including Star Trek). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)
So, let’s consider Cantor.
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7 Comments | tags: Cantor, Cantor's Diagonal, finite, Georg Cantor, infinity, integers, irrational numbers, natural numbers, numbers, rational numbers, real numbers | posted in Math, Sideband
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