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Tag Archives: irrational numbers

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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5 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

Well, it’s Pi Day once again (although this date becomes more and more inaccurate as the century proceeds). So, once again, I’ll opine that Tau Day is cooler. (see: *Happy Tau Day!*)

Last year, for extra-special Pi Day, I wrote a post that pretty much says all I have to say about Pi. (see: *Here Today; Pi Tomorrow*) That post was actually published the day before. I used the actual day to kick off last Spring’s series on Special Relativity.

So what remains to be said? Not much, really, but I’ve never let that stop me before, so why start now?

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10 Comments | tags: irrational numbers, normal number, pi, pi day, tau, tau day, transcendental numbers | posted in Math

Take a moment to gaze at *Euler’s Identity*:

It has been called *“exquisite”* and likened to a *“Shakespearean sonnet.”* It has earned the titles *“the most famous”* and *“the most beautiful”* formula in all of mathematics, and, in a mere seven symbols, symbolizes much of its foundation.

Today we’re going to graze on it!

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9 Comments | tags: complex numbers, discrete mathematics, Euler's Formula, Euler's Identity, geometry, irrational numbers, Leonhard Euler, natural numbers, numbers, rational numbers, real numbers, transcendental numbers, trigonometry, Yin and Yang | posted in Math, Opinion, Philosophy

It’s **pi** day! Be irrational!

Earlier this week I mentioned that *“this coming Saturday is a doubly special date (especially this year).”* One of the things that makes it special is that it is **pi day** — 3/14 (at least for those who put the month before the day). What makes it *extra-special* this year is that it’s 3/14/15— a **pi** day that comes around only once per century. (*Super-duper extra-special* **pi** day, which happens only once in a given calendar, happened way back on 3/14/1529.)

I’ve written before about the magical **pi**, and I’m not going to get into it, as such, today. I’m more of a **tau**-ist, anyway; **pi** is only half as interesting. (Unfortunately, extra-special **tau day** isn’t until 6/28/31, and the *super-duper extra-special* day isn’t until 6/28/3185!)

What I do want to talk about is a fascinating property of **pi**.

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138 Comments | tags: cake, Carl Sagan, Contact (book), Ellie Arroway, irrational numbers, Magnum Pi, normal sequence, numbers, pi, pi day, pie, real numbers, tau, tau day, transcendental numbers | posted in Math

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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4 Comments | tags: Cantor, Cantor's Diagonal, finite, Georg Cantor, infinity, integers, irrational numbers, natural numbers, numbers, rational numbers, real numbers | posted in Math, Sideband

You probably have some idea of what *infinity* means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!

As a simple example, a small circle is infinite in the sense that you can loop around and around the circle forever. At the same time, your entire path along the circle is *bounded* in the small area of the circle. Compare that to the straight line that extends to infinity. If you travel that line, you follow a path that goes forever in some direction.

What if we draw a larger circle outside the small circle. If there are an infinite number of points on the small circle and an infinite number of points on the large circle, does the larger circle have the same number of points as the small one? [The answer is *yes*.]

To understand all this, we have to first talk a bit about numbers.

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18 Comments | tags: countable, counting numbers, Georg Cantor, infinity, irrational numbers, natural numbers, pi, rational numbers, uncountable | posted in Math, Science