Category Archives: Science
I just finished reading Beyond Weird: Why Everything You Thought You Knew About Quantum Physics Is Different (2018) by science writer Philip Ball. I like Ball a lot. He seems well grounded in physical reality, and I find his writing style generally transparent, clear, and precise.
As is often the case with physics books like these, the last chapter or three can get a bit speculative, even a bit vague, as the author looks forward to imagined future discoveries or, groundwork completed, now presents their own view. Which is fine with me so long as it’s well bracketed as speculation. I give Ball high marks all around.
The theme of the book is what Ball means by “beyond weird.”
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12 Comments | tags: Philip Ball, quantum mechanics, wave-function | posted in Books, Physics
Since I retired, I’ve been learning and exploring the mathematics and details of quantum mechanics. There is a point with quantum theory where language and intuition fail, and only the math expresses our understanding. The irony of quantum theory is that no one understands what the math means (but it works really well).
Recently I’ve felt comfortable enough with the math to start exploring a more challenging aspect of the mechanics: quantum computing. As with quantum anything, part of the challenge involves “impossible” ideas.
Like the square root of NOT.
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21 Comments | tags: Artur Ekert, Philip Ball, quantum computing | posted in Computers, Physics
With COVID-19 putting a damper on social activity, “the gang” doesn’t get together very often, but we still gather occasionally (and carefully). One of the times recently I got into how, even though we’re all sitting essentially motionless in a living room, we’re moving through time at the speed of light. I explained why that was, and they found it pretty cool.
Then I ran into someone online who just couldn’t wrap his head around it — just couldn’t accept it (despite explaining in detail and even providing links to some videos). Physics is sometimes challenging to our daily perceptions of reality!
In this case, though, it’s just a matter of some rather simple geometry.
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17 Comments | tags: frame of reference, light speed, Lorentz equation, spacetime, Special Relativity, velocity | posted in Physics
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.
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9 Comments | tags: exponential function, transcendental numbers | posted in Math, Sideband
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.
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10 Comments | tags: fun with numbers, math phobia, mathematics | posted in Math, Rant, Society
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.
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11 Comments | tags: calculus, David Berlinski, derivatives, fun with numbers, integrals, numbers, Steven Strogatz, The Joy of X | posted in Books, Math
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.
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8 Comments | tags: complex numbers, complex plane, exponential function, exponentiation, Fourier transform, Heisenberg Uncertainty | posted in Math, Physics
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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14 Comments | tags: 3Blue1Brown, complex numbers, complex plane, Euler's Formula, Euler's Identity, exponential function, Fourier transform | posted in Math
Sunday I breezed through Seven Brief Lessons On Physics (2014), by Carlo Rovelli. It’s a quick read of only 96 pages that still manages to touch on some of the key aspects of physics.
His much longer book, Reality Is Not What It Seems: The Journey to Quantum Gravity (2014), covers the same territory in greater detail (and greater length: 288 pages). After I finished what amounted to an appetizer, I tucked into the main course. I’m about 30% through it and am enjoying it quite a bit more than I have his work so far.
Both books, but especially the longer one, explore the theory of Loop Quantum Gravity (LQG), of which Rovelli is a co-founder.
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28 Comments | tags: Carlo Rovelli, Loop Quantum Gravity, quantum gravity | posted in Physics
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.
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14 Comments | tags: Cartesian product, multiplication | posted in Math
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