Tag Archives: transcendental numbers

Sideband #70: The exp Function

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 ex 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.

Happy Pi Day!

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?

Transcendental Territory

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.

Moar Math!

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!

Here Today; Pi Tomorrow

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.