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Wednesday, Dec. 11, 2024

6:39pm EST

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Complex Numbers

by Waltz Invictus

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:

Wordsmitty ✍️

about 6 hours ago
In response to "On the Label"

"June 25, 2024 5:40 AM" = Need time zone.

"Note to self:" = self-driving taxis

Some people definitely can be labeled, but we don't say those in polite company.

2 Like

Annette

about 6 hours ago
In response to "On the Label"

Only one label matters. Cat person or dog person. If the answer is anything except "yes" the person's life choices defy all logic analysis.

I wonder what is the difference between psychotherapist, psychologist, and psychoanalysis. Or are they like po-ta-to, po-tuh-two?

3 Like

tj-Merry Mischief Maker

Dec 9, 2024 at 2:51pm
In response to "Analog Log"

As time progresses, it amazes me how we have moved from tide clocks/analog computers to people eating tide-pods.

1 Like

Aiva Raine

Dec 9, 2024 at 2:37pm
In response to "Story of the Century"

Fantastic entry. Have I mentioned how I love your snark? Thanks once again for the grins and your awesome sarcastic responses to the articles.

1 Like

Anni Pon

Dec 8, 2024 at 10:01am
In response to "Revisited: "Millennials Killed Millennials""

It's kind of funny to me that people still use "millennials" to refer to "those darn kids" when most of them are in their 30's now with kids of their own. As an elder millennial (a Xennial, if you will), I haven't really found the generational definitions too useful for myself. I'm right on the border of generations, so I have a lot in common with both Millennials and Gen X. Growing up in silicon valley with tech savvy parents, I had earlier access to computers and technology than a lot of other people my age. But in terms of cultural milestones, I lived through and am old enough to remember things like the 90's, Y2K and 9/11. The borders are sort of arbitrary, but on a population level I can see how sometimes it can be useful to group people into generations.

2 Like

Elisa: Snowman Stik

Dec 8, 2024 at 9:40am
In response to "Revisited: "Millennials Killed Millennials""

*rubs hands together* Time for me to dive right in!

2. In this system, I'm early Generation X, which hardly anyone ever talks about, opting instead to rag on "boomers" or "millennials" or "Gen-Z." Or praise them, depending on the author's beliefs and age.

Looks like I need to change my name to "hardly anyone". Also, there has been more and more chatter about Gen X in some circles, but curiously this seems to be happening in institutions that have dropped labels (which I'll touch on in a second here).

On top of which, you'd have to convince me that our 1983-born X-er has more in common with someone born in 1966 than with someone born in 1986.

Very, very, VERY few organizations that have ever in their history used generational labels have set the Gen-X border any later than 1980. While I'm still very salty that Pew stopped this practice (one of the few things they've done wrong, IMO), I still use their Gen-X demarcation of 1965-1980. That demarcation nicely strikes the median of most demarcations. SO in this case it's not a matter of you needing to be convinced. It's a matter of the author leaning on a REALLY bad source.

Much of what sets "generations" apart, in modern terms, has to do with technological advancements and societal changes...then other people had to think that the original wearer was cool enough to start a fashion trend.

And how that message has been conveyed has drastically changed among generations, to boot.

So you know what I want to see Millennials finally kill?

Generations.

As a Millennial, I can tell you that this is probably the last thing our generation will try to kill. It's something that's persisting even as we reach middle age. And that's another thing. We're aging very strangely, namely that physically we're aging more slowly than people in Gen Z. We could have killed the concept of aging a certain way, but Gen Z's circumstances have effectively torpedoed that revolution.

Also, if anyone wants to read the original article, you can certainly try throwing the URL in the Wayback Machine. It doesn't always work, but it's worth a try. There are some other methods as well, but I haven't tried them myself yet. (Speaking of which, has anyone here tried Zette? I've been meaning to sign up for that but haven't had a chance. It's one of my goals for 2025.)

2 Like

Wordsmitty ✍️

Dec 7, 2024 at 12:41pm
In response to "Story of the Century"

Maybe it helps if you can dance - Dick Van Dyke and Coldplay - although he's just 99. (I assume everyone has read about that)

4 Like

🌖 HuntersMoon

Dec 6, 2024 at 10:07am
In response to "No Space for Time"

I think you just added clarity to the Twilight Zone...

3 Like

tj-Merry Mischief Maker

Dec 5, 2024 at 4:05pm
In response to "How To Give A Fock"

How many photons does it take to make a torpedo? Who gives a Fock! Actually, wasn't Fock Spock's half brother?

3 Like

Jayne

Dec 5, 2024 at 10:23am
In response to "How To Give A Fock"

Versatile Fock States is the name of my next thrash album.

3 Like

December 9, 2024 at 11:19am

#1080964

Analog Log

This one's been hanging out in my queue for a long time, but it's not exactly time-sensitive. As they say, time and tide wait for no one.

Lord Kelvin and His Analog Computer
This tide-predicting machine was one of many advances he made to maritime tech

The source is a publication of the IEEE, the electrical engineering professional organization. But fear not; the article isn't very technical.

Civilizations recognized a relationship between the tides and the moon early on, but it wasn’t until 1687 that Isaac Newton explained how the gravitational forces of the sun and the moon caused them. Nine decades later, the French astronomer and mathematician Pierre-Simon Laplace suggested that the tides could be represented as harmonic oscillations. And a century after that, [William] Thomson used that concept to design the first machine for predicting them.

Thompson was Lord Kelvin and, yes, he's the one the temperature scale is named after. One wonders what it would have been called had Thompson not become a noble, because Thompson is a boring name for a unit of measure.

Thomson’s tide-predicting machine calculated the tide pattern for a given location based on 10 cyclic constituents associated with the periodic motions of the Earth, sun, and moon. (There are actually hundreds of periodic motions associated with these objects, but modern tidal analysis uses only the 37 of them that have the most significant effects.)

Translation: it's complicated.

The most notable one is the lunar semidiurnal, observable in areas that have two high tides and two low tides each day, due to the effects of the moon.

Which is what most of us think of when we think of tides, but it's not as simple as "it's high tide when the moon is directly overhead." There's a lag, and there are local conditions that affect the timing of tides (such as sea floor depth).

On Thomson’s tide-predicting machine, each of 10 components was associated with a specific tidal constituent and had its own gearing to set the amplitude.

Basically, it's a very complicated clock. Sure, the article calls it an analog computer, and I'm not going to argue with professionals (especially ones not in my field) but I think that's a categorization issue. At some point of increasing complexity, a clock stops being a clock and starts being an analog computer. But in my view, if it involves the timing of natural phenomena like the movement of solar system bodies, it's a clock.

The device marked each hour with a small horizontal mark, making a deeper notch each day at noon. Turning the wheel rapidly allowed the user to run a year’s worth of tide readings in about 4 hours.

But this bit, an output device, is probably what pushes it into the computer category. It also was designed for prediction, not for reading what the tide is right now.

As with many inventions, the tide predictor was simultaneously and independently developed elsewhere and continued to be improved by others, as did the science of tide prediction.

One thing I'm still unclear on when it comes to tidal prediction: the wind plays a role. And wind is way, way harder to predict than the future relative position of the sun and moon. I'd ask my dad, the sailor, but I seem to have misplaced my Ouija board.

In any case, mostly, I just liked the article and the history lesson, and I wanted to muse about the differences between clocks and analog computers.

1 Like • Comment • View 1 comment

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