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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 14 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 16 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.

April 14, 2020 at 12:14am

#981056

The Electric Slide

Perhaps you've caught something going around this past month. No, not a virus; a meme in the original sense of the word.

https://contingentmagazine.org/2020/03/31/the-trouble-with-triscuits/

The Trouble with Triscuits

On March 25, Sage Boggs shocked the twittersphere with his revelation that the brand name Triscuit was a portmanteau of “electricity” and “biscuit.”

You know, in British English (and in French), the word "biscuit" refers to what we here in the US would call a "cookie" - a sweet baked product. In the US, a "biscuit" is a light, fluffy, savory, doughy thing (I think I got the adjectives in the right order there) that you eat for breakfast. In neither case is it a crunchy, lightly salted, whole-wheat square. That's a "cracker" in American English. No idea what the Brits call it. Or the French, for that matter. So the whole thing smacked of marketing to me.

The account even added a lightning bolt to its username and changed its bio to “elecTRIcity biSCUIT.”

Yep. Marketing.

Triscuit’s confirmation of the theory satisfies many that Boggs’s theory is correct, and it also helps that the theory is the sort of surprising factoid people love to share at parties.

What are these "parties" to which the author refers? Besides, even if we could have parties, by this time pretty much everyone knows this "sort of surprising factoid" via internet. I'm not even on Twitface and I've heard it.

In 1903, Triscuit ads proclaimed that Triscuits were the first commercial bread product to be “baked by electricity.” Some of the earliest designs even incorporated lightning bolts into the lettering of “Triscuit.”

You know why I want to believe this, though? Not because it's compelling, but because the early ad shows they were made in Niagara Falls, which, combined with the "electricity" thing, harks back to Tesla, and Tesla was awesome.

But did the Natural Food Company name their cracker “Triscuit” specifically to evoke the middle syllable of “electricity?” That’s a much harder question, and requires both a broader understanding of the culture of advertising in the early 20th century and also a deeper inquiry into how the “Triscuit” name operated at its inception.

Which, of course, the article delves into, and I'm not going to reproduce the argument here. Just read it.

Other theories about the name exist, some more plausible than others. Many have suggested the name derived from triticale, a late-19th-century hybrid of wheat and rye, but there is no evidence that this was used to make Triscuits.

To paraphrase James T. Kirk, "Who put the quadrotriticale in the Triscuits?"

A more compelling explanation is that the Triscuit is just a next-level Biscuit, perhaps referring to being thrice-baked, relying on the etymology of “biscuit” as meaning “twice-baked.”

And so we have yet another meaning of "biscuit." Language is confusing, especially since all the cookies and biscuits I've ever made have been baked exactly once.

Even if the electricity biscuit thesis is true—and that certainly remains a possibility—electrical baking as a selling point was obsolete within two years of the invention, when Pillsbury rolled out electrically baked biscuits in 1905.

I'm sure baking things via electricity back when electricity was young and vibrant was a huge selling point. I prefer gas for cooking; one of the first things I did for my house after I bought it was replace the electric oven/range with a gas-fired one. In neither case does it impart any particular flavor to the food, not like charcoal does, so my choice is for reasons of control and economy.

The sheer enthusiasm on display renders any skepticism a buzzkill.

I can hear people groaning inwardly whenever someone starts a sentence with "Well, actually..." Even when it's not the dreaded "mansplaining."

In any case, there’s far more to the Triscuit story than this etymological quest. The early days of the Triscuit take us to a world newly powered by electricity, a nation obsessed with scientific progress and pure food.

That same progress also brought us white bread, which is a pox on our existence.

Triscuits and Shredded Wheat were intended to be a central part of the daily diet, topped with everything from creamed peas to fricasee of oysters to Bromangelon, a pre-Jell-O gelatin dessert. It was a wild time.

Okay, ew. I'll take today, pandemic and all, rather than having to eat that stuff. Today is a much better time for a pandemic. After all, the internet, which certainly didn't exist 120 years ago, is the only thing keeping a lot of us semi-sane right now, by doing stuff like this. Even if it was on that blasted Twitface site.

I like Triscuits; I keep them in stock when I can (which, these days, is sporadically), but I generally only eat them topped with cheese.

Anything else is too much work.

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