#
Tag Archives: rational 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…”*

Continue reading

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

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.

Continue reading

16 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** are **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.

Continue reading

6 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.

Continue reading

6 Comments | tags: math theory, mathematics, natural numbers, nominalism, numbers, Plato, Platonic, Platonism, rational numbers, real numbers, Theory of Mathematics | posted in Math, Philosophy

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!

Continue reading

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*.

Continue reading

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.

Continue reading

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.

Continue reading

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!

Continue reading

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.

Continue reading

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