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.
And the total is…?
Oh the irony of it all. Two days ago I post about two math books, at least one of which (if not both) I think everyone should read. This morning, reading my newsfeed, I see one of those “People Are Confused By This Math Problem” articles that pop up from time to time.
Often those are expressions without parentheses, so they require knowledge of operator precedence. (I think such “problems” are dumb. Precedence isn’t set in stone; always use parentheses.)
Some math problems do have a legitimately confusing aspect, but my mind is bit blown that anyone gets this one wrong.
There are many science-minded authors and working physicists who write popular science books. While there aren’t as many math-minded authors or working mathematicians writing popular math books, it’s not a null set. I’ve explored two such authors recently: mathematician Steven Strogatz and author David Berlinski.
Strogatz wrote The Joy of X (2012), which was based on his New York Times columns popularizing mathematics. I would call that a must-read for anyone with a general interest in mathematics. I just finished his most recent, Infinite Powers (2019), and liked it even more.
Berlinski, on the other hand, I wouldn’t grant space on my bookshelf.
Last time I opened with basic exponentiation and raised it to the idea of complex exponents (which may, or may not, have been surprising to you). I also began exploring the ubiquitous exp function, which enables the complex math needed to deal with such exponents.
The exp(x) function, which is the same as ex, appears widely throughout physics. The complex version, exp(ix), is especially common in wave-based physics (such as optics, sound, and quantum mechanics). It’s instrumental in the Fourier transform.
Which in turn is as instrumental to mathematicians and physicists as a hammer is to carpenters and pianos.
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 naive 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.
At the beginning of the month I posted about a neat Japanese visual method for multiplying smallish numbers. Besides its sheer visual attractiveness, it’s interesting in allowing one to multiply numbers without reference to multiplication tables (which, let’s face it, typically require rote memorization).
As I mentioned last time, my interest in multiplication is linked to my interest in generating Mandelbrot plots, which is a multiplication-intensive process. But for those learning math, digging into basic multiplication has some instructive value.
With that in mind, here are some other multiplication tricks.
123 × 321 = 39,483
My interest in number multiplication goes back to exploring algorithms for generating Mandelbrot plots, which can require billions of multiplication operations on arbitrary precision numbers (numbers with lots and lots of digits).
Multiplying two numbers — calculating their product — is computationally intense because of the intermediate Cartesian product. Multiplying two 12-digit numbers creates a 24-digit result (12+12), but it also has an intermediate stage involving 144 (12×12) single digit multiplications.
Recently I learned an intriguing Japanese visual multiplication method.
For a little Friday Fun I have a logic puzzle for you. I’ll give you the puzzle at the beginning of the post, detour to some unrelated topics (to act as a spoiler barrier), and then explain the puzzle in the latter part of the post. I would encourage you to stop reading and think about the puzzle first — it’s quite a challenge. (I couldn’t solve it.)
The puzzle involves an island with a population of 100 blue-eyed people, 100 brown-eyed people, and a very strange social practice. The logic involved is downright nefarious, and even after reading the explanation, I had to think about it for a bit to really see it. (I still think it’s twisted.)
To be honest, I’m kinda writing this to make sure I understand it!
Four years ago I started pondering the tesseract and four-dimensional space. I first learned about them back in grade school in a science fiction short story I’d read. (A large fraction of my very early science education came from SF books.)
Greg Egan touched on tesseracts in his novel Diaspora, which got me thinking about them and inspired the post Hunting Tesseracti. That led to a general exploration of multi-dimensional spaces and rotation within those spaces, but I continued to focus on trying to truly understand the tesseract.
Today we’re going to visit the 4D space inside a tesseract.
I just finished Humble Pi (2019), by Matt Parker, and I absolutely loved it. Parker, a former high school maths teacher, now a maths popularizer, has an easy breezy style dotted with wry jokes and good humor. I read three-quarters of the book in one sitting because I couldn’t stop (just one more chapter, then I’ll go to bed).
It’s a book about mathematical mistakes, some funny, some literally deadly. It’s also about how we need to be better at numbers and careful how we use them. Most importantly, it’s about how mathematics is so deeply embedded in modern life.
It’s my third maths book in a month and the only one I thoroughly enjoyed.