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Tag Archives: transcendental numbers

*Converging…*

Back in October I published two posts involving the ubiquitous **exponential function**. [see: *Circular Math* and *Fourier Geometry*] The posts were primarily about Fourier transforms, but the exponential function is a key aspect of how they work.

We write it as *e*^{x} or as *exp*(*x*) — those are equivalent forms. The latter has a formal definition that allows for the complex numbers necessary in physics. That definition is of a *series* that converges on an answer of increasing accuracy.

As a sidebar, I thought I’d illustrate that convergence. There’s an interesting non-linear aspect to it.

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9 Comments | tags: exponential function, transcendental numbers | posted in Math, Sideband

Five years ago today I posted, *Beautiful Math*, which is about **Euler’s Identity**. In the first part of that post I explored why the Identity is so exquisitely beautiful (to mathematicians, anyway). In the second part, I showed that the Identity is a special case of **Euler’s Formula**, which relates trigonometry to the complex plane.

Since then I’ve learned how naïve that post was! It wasn’t wrong, but the relationship expressed in Euler’s Formula is fundamental and ubiquitous in science and engineering. It’s particularly important in quantum physics with regard to the infamous Schrödinger equation, but it shows up in many wave-based contexts.

It all hinges on the **complex unit circle** and the **exp**(i×π×a) function.

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11 Comments | tags: 3Blue1Brown, complex numbers, complex plane, Euler's Formula, Euler's Identity, Fourier transform, numbers, transcendental numbers | 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

Last time we considered the possibility that human consciousness somehow supervenes on the physical brain, that it only emerges under specific physical conditions. Perhaps, like laser light and microwaves, it requires the right equipment.

We also touched on how Church-Turing implies that, if human consciousness *can* be implemented with software, then the mind is *necessarily* an algorithm — an abstract mathematical object. But the human mind is presumed to be a natural *physical* object (or at least to emerge from one).

This time we’ll consider the effect of transcendence on all this.

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54 Comments | tags: AI, algorithm, brain mind problem, computationalism, computer model, computer program, consciousness, human brain, human consciousness, human mind, mind, software model, Theory of Consciousness, transcendental numbers, Yin and Yang | posted in Computers

But my brain is full!

You may have noticed that, in a number of recent posts, the topic has been math. The good-bad news is that there’s more to come (sorry, but I love this stuff). The good-good news is that I’m done with math foundations. For now.

To wrap up the discussion of math’s universality and inevitability — and also of its fascination and beauty — today I just have some YouTube videos you can watch this Sunday afternoon. (Assuming you’re a geek like me.)

So get a coffee and get comfortable!

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6 Comments | tags: algebraic numbers, math and sex, math ontology, mathematics, numbers, Philosophy of Math, Theory of Mathematics, transcendental numbers | posted in Math, Opinion

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