Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Write_Mikey_Write!

about an hour ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

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Adherennium Plotting something

about 3 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

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Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 14, 2023 at 9:27am

#1049580

Revisited: "Something about Nothing"

Today, we're going all the way back to June of 2020 for an article about quantum mechanics: "Something about Nothing"

. Now, don't freak out; it's not a technical treatise in any way.

The article referenced there (really a book promotion masquerading as an interview masquerading as an article) is a year older than that, but it's still up and can be found here,

with the misleading title of "Making Sense of Quantum Mechanics."

As I've noted, the main reasons I do these retrospectives are to see if there have been any updates to the material covered, and to elucidate how my own views on the subject may have evolved. I don't follow science news all that rigorously; that is, I'll read an article or a book, or watch a video, here and there, but it's not like I delve into much depth. But the one thing that comes to mind is that, recently, there was a lot of buzz about a Nobel Prize given to some scientists for their work on quantum entanglement.

Same as with everything else related to quantum theory, people got that stuff wrong, too. I don't mean the prize-winning scientists, who presumably had really tight results, but the way it was reported on made it look like the old "Einstein was Wrong" crowing, with an emphasis on how quantum entanglement means something travels faster than light.

It does not. The light speed barrier should more properly be termed the information speed barrier, and quantum entanglement does not, at least with our current understanding, imply the transmission of information faster than light. We can't use it to send instantaneous messages to Pluto, for instance. Mostly, from what I can tell, the usefulness is limited to the arcane workings of quantum computers. Perhaps there are other uses, or will be in the future, but mostly the prize was about experimental confirmation of a theory.

None of which really negates anything in the article I featured, as far as I can tell.

In that entry, I said:

I've always been of the opinion that anyone who claims to have figured out quantum mechanics is lying, at the very least to themself.

Nothing about that has changed.

But I do want to go back to that original article to note something I apparently missed the first time around:

Horgan: Good choice. What is the point, in our scientific age, of philosophy?

Albert: I'm not sure I see the connection. It's like asking, “What is the point, in our scientific age, of ice cream?" People like it. People - or some people - want to understand how things, in the broadest possible sense of the term, hang together. And it happens (moreover) that the business of trying to figure that out has had obvious and enormous and innumerable consequences for the history of the world. And if the thought is that what goes on in university science departments has lately somehow taken that function over, then that's just clearly and wildly wrong - and the fact that it's wrong, as I explained in my answer to your previous question, was part of the reason why I moved from a science department to a philosophy department.

Here, I think both people missed the mark.

Ice cream has nothing to do with science (except in the sense that everything does and that, reportedly, Einstein was a big fan). But—and I might have mentioned this before, but I don't remember—philosophy guides science, and science informs philosophy.

"Science" isn't a body of knowledge; it's a method. The scientific method is, at base, philosophy. It's philosophy that works, in that it's been shown to get useful results, unlike a lot of the mental self-pleasuring some philosophers do. But philosophy also has at least one other function in science, and that's to limit the lengths to which we'll go to investigate some hypothesis.

To note a basic example, in biology, animal testing is a thing. What limits animal testing isn't science itself, but ethics, which is a branch of philosophy. You can argue that the restrictions go too far, or, conversely, that they don't go far enough and maybe we shouldn't be doing animal testing at all. But by doing so, you're not doing science, you're doing philosophy.

As for "science informs philosophy," well, the thing about philosophy is that you can build entire logical edifices on the foundation of a false premise. One need look no further than the convolutions of a flat-earther to see what I'm talking about here, but, in general, if you're going to draw conclusions, it's best to start with a solid and well-tested premise, such as "the earth is basically round" or "gravity is an attractive force between masses."

Sometimes, when you do that, you might find a contradiction, or a paradox. That might lead to a revised premise, and that's okay.

My point is that the universe doesn't support our penchant for putting everything into neat little boxes. There's no sharp dividing line between, say, biology and chemistry. The boundary between our selves and our environment can get murky, too, and does so every time we take a breath, or eat, or shit.

So it is with science and philosophy. Though we were doing philosophy way before the beginnings of science as a discipline (physics was originally termed "natural philosophy"), it often led to some really wrong conclusions. Still does, of course.

Okay, enough of that. I guess I just had to defend why I bang on about both those things in here, when I'm not making dick jokes. And sometimes when I am.

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