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Saturday, Nov. 16, 2024

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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:

Write_Mikey_Write!

a day 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.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

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

3 Like

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.

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.

1 Like

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').

1 Like

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.

May 18, 2024 at 7:40am

#1071313

What's the Opposite of Eureka?

Now, here's a kid who's going to be subject to government scrutiny for the next few years.

13-year-old has eureka moment with science project that suggests Archimedes’ invention was plausible

After which they'll probably recruit him.

Often called the father of mathematics, Archimedes was one of the most famous inventors in ancient Greece, with some of his ideas and principles still in use today.

While there is no doubt that Archimedes made important contributions to mathematics, calling him "the father of mathematics" is rather insulting to earlier mathematicians, not to mention horrifically Eurocentric.

But one fabled device has left scientists speculating on its existence for hundreds of years — the death ray. Now, a middle schooler may have some answers.

Hey, when I was that age, I was fascinated by death rays, too. But this was a simpler time, and as far as I know, I wasn't subject to government surveillance for my hobbies.

Brenden Sener, 13, of London, Ontario, has won two gold medals and a London Public Library award for his minuscule version of the contraption — a supposed war weapon made up of a large array of mirrors designed to focus and aim sunlight on a target, such as a ship, and cause combustion — according to a paper published in the January issue of the Canadian Science Fair Journal.

Also, calling it a "death ray" is probably sensationalism. Most sources I've seen use "heat ray," but I think even that is misleading. Regardless; the point is, people have argued over whether Archie could have actually constructed the weapon using Greek technology of the time, or if perhaps it was speculation (similar to da Vinci's later speculation about flying machines or whatever). Hell, René Descartes dismissed the idea.

There is no archaeological evidence that the contraption existed, as Sener noted in his paper, but many have tried to recreate the mechanism to see if the ancient invention could be feasible.

Absence of evidence isn't evidence of absence, of course. Oh, and no one, as far as I know, is saying the idea is unworkable; only that it might not have been feasible using tech available in the 3rd century BCE.

At the same time, evidence that it could have been built and used isn't the same thing as evidence that it was built and used.

The article describes his setup, and then:

Writing in the paper, Sener said he found these results to be “quite remarkable as it suggests that light is going in all directions and that the shape of the concave mirror focuses the light waves onto a single point.”

Well... that's a bit disappointing. Was CNN clickbaiting us with that headline? While this is certainly what I'd call a good middle school science fair project, it's not like we didn't already know that focusing light rays intensifies the resulting temperature. Hell, all this is, is the reflection version of using a magnifying glass to burn ants.

While the experiment doesn’t offer “anything significantly new to the scientific literature … his findings were a nice confirmation of the first law of thermodynamics,” which states energy or heat can be transferred, Ho said.

Blink. Blink.

Look, I'm not trying to diminish the kid's accomplishment. Hell, I couldn't ever come up with an idea for a science project at that age. It's not that I wasn't interested; it's that I wasn't creative enough. Still not. But it proves nothing about Archie's heat ray.

2 Like • Comment

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