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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Brandiwyn🎶

a day ago
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.

3 Like

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.

2 Like

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.

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 19, 2022 at 12:01am

#1039398

Horrendous Space Kablooie

Some names just have staying power, I suppose.

The Trouble With “The Big Bang”
A rash of recent articles illustrates a longstanding confusion over the famous term.

As Calvin noted, it should have been called the Horrendous Space Kablooie. Some cosmologists do call it the HSK. There aren't too many instances of science referencing comic strips; the only other one I can think of is the thagomizer.

(At that link are some others that I hadn't remembered.)

By Sabine Hossenfelder

I'm including the byline here, because I keep seeing her stuff all over the place. She has a whole YouTube video series, and I just finished a book she wrote. She's a very good science communicator.

Unfortunately, knowing she wrote this article means I read it in my mind with a soft German accent.

I can’t blame readers for being confused by recent news stories about the Big Bang. The article that kicked them off, “The Big Bang Didn’t Happen,” is bad enough. But some of the rebuttals also don’t get it right. The problem is that writers conflate ideas in astrophysics and use the term “Big Bang” incorrectly. Let me set the record straight.

Science writers love to do this shit, and it pisses me off. "Darwin overturned!" "Einstein was WRONG!" "Big Bang didn't happen!" "Schroedinger's cat found ALIVE!" Stop it. It's misleading and only adds fuel to the fires started by the willfully ignorant, anti-intellectual "my opinion is better than your book learnin'" crowd.

Let’s call Big Bang #1 the beginning of the universe. It’s what most people think the expression means. This Big Bang is what we find in the mathematics of Einstein’s general relativity if we extrapolate the current expansion of the universe back in time. The equations say that matter and energy in the universe becomes denser and hotter until, eventually, about 13.7 billion years in the past, both density and temperature become infinite.

Hossenfelder certainly knows more about this shit than I do, but I also don't think talking about infinite density or temperature is helpful, because no one can really grok the idea of infinity. It's just not possible. Georg Cantor tried, and had some success, but he died insane.

Besides, there are different kinds of infinity.

This Big Bang is sometimes more specifically called the Big Bang Singularity. This word has somewhat fallen out of favor among physicists, partly because it’s clumsy, but also because I don’t know anyone who thinks this singularity is physically real.

Probably because of the whole "infinity" thing. I mean, to me the whole thing sounds like a limit problem. Like, you know you can't divide by zero because that's mathematically undefined, right? You learn that pretty early on. But if you take a series of fractions where the denominator approaches zero, you can approach infinity. Mathematically, not practically.

Since physicists don’t believe the singularity is real, the phrase “Big Bang” has come to refer to whatever event might replace the singularity in the to-be-found theory of quantum gravity in this Planck time. Let’s call it just that—the Big Bang Event.

This is, as far as I understand it, not what would be called the Horrendous Space Kablooie.

The problem has long been that the term Big Bang is used to refer to the expansion of the universe in general, and not to the event of the creation of the universe in particular. These are, however, two separate scientific hypotheses.

That's the HSK.

Historically, the first evidence for the expansion of the universe was Edwin Hubble’s observation that the light from other galaxies is systematically shifted to the red, indicating that they all recede from us.

And I finally got to see the telescope he used for those observations. It's in the mountains near L.A. (Thanks, NaNoNette

)

While this may have been the first evidence, the decisive evidence for the expansion of the universe was the discovery of the cosmic microwave background that ruled out the competing hypothesis, the “steady state” universe.

I'm sure you've seen an image of that, but if not, here.

One of the most mind-blowing things about this, at least in my opinion, is that the (observable) universe can be thought of as being inside-out. That is, the further you look, the more back in time you go, until you can't see any further back; that's the CMB. It's roughly the same in all directions. So, in a sense, I really am the center of the universe. Unfortunately for my ego, so are you. So is Alpha Centauri. So is everything else in the universe.

This confusion between the expansion of the universe and the Big Bang Event becomes apparent, for example, just by looking at the Wikipedia entry for Big Bang. It starts out in the first paragraph referring to something called the “Big Bang theory,” and explains that this is the theory for the expansion of the universe. In the second paragraph, the Big Bang theory is distinguished from its extrapolation to the Big Bang singularity. But by the fourth paragraph the distinction has gotten lost, and we are informed, “A wide range of empirical evidence strongly favors the Big Bang, which is now essentially universally accepted.” The reader is misled to think that evidence for the expansion of the universe is evidence that the universe began with the Big Bang Event, which is incorrect.

I'm aware that Wikipedia has its shortcomings, but if you're a physicist who's also an effective science communicator, like Sabine here, you could, you know... edit that.

I have not seen or heard the term “Big Bang Theory” being used by a physicist in a seminar or paper for the expansion of the universe.

I'm sure that's at least partly because the show of that name has ruined the expression for all time.

Anyway, after a long discourse about these problems in definition, she finally gets to the point made back at the beginning:

In the attention-grabbing article, “The Big Bang Never Happened,” Eric Lerner questions that the universe expands in the first place. His article was published in August by the Institute of Art and Ideas, a British organization that, by my own experience, prioritizes debate over scientific rigor.

Buuuuurrrrrrrnnnnn.

Lerner argues against the “cosmological establishment [that] has circled the wagons to protect this failed [Big Bang] theory with censorship,” presumably because Lerner has faced some difficulties in getting his alternative theory published.

This is directly equivalent to "spherical establishment that has circled the wagons to protect this failed Round Earth theory with censorship."

It becomes clear, later in Lerner’s essay, that he is not attacking the Big Bang Event (which can reasonably be questioned) but the expansion of the universe. And because it is true that the Webb telescope has delivered data in tension with the concordance model, the reader (or editor) who does not know the difference, may get away finding Lerner’s piece reasonable.

And this is why we need more effective science communication.

It is, as Hossenfelder notes, sometimes very difficult to replace precise scientific or mathematical terminology with imprecise words. She goes on about it in the book I just read, too (which contains very nearly no math and lots of words). This problem is both pernicious and persistent, though: even some of the most basic words, like "theory," have a different meaning to scientists and the general public, thus leading to such utterly ignorant dismissals as "evolution is 'only' a theory."

And as much as I'd like to fix it, I'm not actually a scientist. It's like: you're not a football player, right? But you probably watch football on TV. And you probably have some knowledge of players and strategies, at least enough to occasionally shout "OH COME ON!" at the screen. Well, that's like me with science. So I'm not in a position to fix it. I'll just watch people like Hossenfelder do that.

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