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Tuesday, May. 7, 2024

12:19pm EDT

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

🌕 HuntersMoon

May 5, 2024 at 8:50am
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

May 3, 2024 at 11:58am
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

May 2, 2024 at 11:37am
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 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, Annette

)

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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