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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 hours 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

about 4 hours 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.

August 26, 2022 at 11:57am

#1036955

Woolly Bully

As this entry is later than usual—as I vaguely remember mentioning in a note yesterday, I might have consumed an excessive amount of ethanol—I'm not going to comment too much. But it's an interesting bit of history, illustrating some of the best and worst of humanity.

The Story of Charles Willson Peale’s Massive Mastodon
When a European intellectual snubbed the U.S., the well-known artist excavated the fierce fossil as evidence of the new Republic’s strength and power

In the 18th century, French naturalist George-Louis Leclerc, Comte du Buffon (1706-1778), published a multivolume work on natural history, Histoire naturelle, générale et particuliére. This massive treatise, which eventually grew to 44 quarto volumes, became an essential reference work for anyone interested in the study of nature...

The Comte de Buffon advanced a claim in his ninth volume, published in 1797, that greatly irked American naturalists. He argued that America was devoid of large, powerful creatures and that its human inhabitants were “feeble” by comparison to their European counterparts.

Obviously, Buffoon wasn't familiar with Sasquatch. As for "feeble," well, those "European counterparts" were busy systematically destroying the inhabitants by means of more advanced technology.

The claim infuriated Thomas Jefferson, who spent much time and effort trying to refute it—even sending Buffon a large bull moose procured at considerable cost from Vermont.

You know, this is about when, normally, I'd stop reading. Why? Because according to this article, the Comte died in 1778. The Wikipedia page claims he died in 1788 (I know there was a major calendar switch in the 18th century, but not that major). The ninth volume was, again according to this article, published in 1797, either 19 or nine years after his death. And yet Jefferson sent him a moose? To what, his mausoleum?

So, okay, something's really wonky about the dates here, and that definitely needs resolved (especially as Leclerc was a noble and the French revolution was mostly a 1790s thing).

In 1739, a French military expedition found the bones and teeth of an enormous creature along the Ohio River at Big Bone Lick in what would become the Commonwealth of Kentucky.

I'm mostly just including this quote so that those of you unfamiliar with Kentucky can have a sensible 12-year-old chuckle at "Big Bone Lick."

Of course, the local Shawnee people had long known about the presence of large bones and teeth at Big Bone Lick.

It's right there in the name, folks. What? You didn't actually think the other definition applied?

For millennia, bison, deer and elk congregated there to lick up the salt, and the indigenous people collected the salt as well. The Shawnee considered the large bones the remains of mighty great buffalos that had been killed by lightning.

This is completely tangential to the article, but I've had this working hypothesis for a while now that the reason so many cultures have dragon myths is because they'd occasionally find dinosaur bones. Having no concept of deep time, they had to make up stories about how such enormous skeletons got to be part of the landscape, and those stories became dragon legends.

I have no real support for this, but it tracks with what I know about humans.

Anyway. Not much else to say, except that the article calls out my hometown, which I always think is cool (unless it's to recall the events of 2017). The rest of the story details the process of figuring out what those bones were (spoiler: mastodon), and, like I said, is an interesting look into the history of scientific discovery.

Oh, but before I go, don't give much credence to those stories you keep finding about people trying to Jurassic-Park the mastodon back into existence. Most of them are sensationalist.

1 Like • Comment

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