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Tag Archives: real numbers

Folded into the mixed baklava of my 2018, was a special mathematical bit of honey. With the help of some excellent YouTube videos, the light bulb finally went on for me, and I could see **quaternions**. Judging by online comments I’ve read, I wasn’t alone in the dark.

There does seem a conceptual stumbling block (I tripped, anyway), but once that’s cleared up, quaternions turn out to be pretty easy to use. Which is cool, because they are very useful if you want to rotate some points in 3D space (a need I’m sure many of have experienced over the years).

The stumbling block has to do with quaternions having not one, not two, but three *distinct* “imaginary” numbers.

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16 Comments | tags: complex numbers, math, math theory, mathematics, natural numbers, number systems, numbers, octonion, quaternion, real numbers | posted in Math, Sideband

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

In the recent post *Inevitable Math* I explored the idea that mathematics was both universal and inevitable. The argument is that the foundations of mathematics are so woven into the fabric of reality (if not actually *being* the fabric of reality) that any intelligence must discover them.

Which is not to say they would think about or express their mathematics in ways immediately recognizable to us. There could be fundamental differences, not just in their notation, but in how they *conceive* of numbers.

To explore that a little, here are a couple of twists on numbers:

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7 Comments | tags: alien math, Frederik Pohl, Heechee, math, math origins, math theory, more math, prime numbers, real numbers, surreal number | posted in Math, Opinion, Sideband

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, math origins, math theory, mathematics, natural numbers, numbers, Philosophy of Math, rational numbers, real numbers, Theory of Mathematics | posted in Math, Opinion

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

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

Last time I wrote about analog recording and how it represents a physical chain of proportionate forces directly connecting the listener to the source of the sounds. In contrast, a digital recording is just numbers that *encode* the sounds in an abstract form. While it’s true that digital recordings can be more accurate, the numeric abstraction effectively *disconnects* listeners from the original sounds.

In the first month of this blog I wrote about analog and digital and mentioned they were mutually exclusive Yin and Yang pairs (a topic I wrote about even earlier — it was my seventh post).

Today I want to dig a little deeper into the idea of analog vs. digital!

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Leave a comment | tags: analog, digital, discrete, infinity, natural numbers, quantum gravity, rational numbers, real numbers, Yin and Yang | posted in Basics, Life, Science