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Tag Archives: natural 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

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

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

In this post I’ll show how Set Theory allows us to define the natural numbers using sets. It’s admittedly a very abstract topic, but it’s about something very common in our experience: counting things. Seeing how numbers are defined also demonstrates (contrary to some false notions) that there is a huge difference between a *number* and how that number is “spelled” or represented.

**Note:** *I am not a mathematician!* This topic is right on the edge of my mathematical frontier. I wanted this addendum to the previous post, but be aware I may misstep. I welcome any feedback from Real Mathematicians!

But go on anyway… keep reading… I dare ya!

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6 Comments | tags: counting, counting numbers, natural numbers, numbers, set theory, successor function | 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

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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Comments Off on Smooth or Bumpy | tags: analog, digital, discrete, infinity, natural numbers, quantum gravity, rational numbers, real numbers, Yin and Yang | posted in Basics, Life, Science

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