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 9 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 11 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.

June 19, 2021 at 12:03am

#1012126

Pompeii Attention Now

Yesterday, I finally got around to seeing Wednesday's episode of Marvel's Loki (and, later, a movie -- see below). So for anyone who's happened to see that episode, today's article is especially fortuitous.

Reconstructing the Menu of a Pub in Ancient Pompeii
Eat like a first-century Roman, using recent archaeological discoveries as your guide.

In the second century, Pliny the Younger wrote a letter to the Roman historian Tacitus, recounting the early stages of the eruption of Mount Vesuvius in 79 AD.

As this article is about a pub menu, it is even more fortuitous that there is a rare and highly-sought-after beer called Pliny the Younger. It's a triple IPA from Russian River in California, and I've never had it because I generally dislike West Coast IPAs.

A day after Pliny observed that dark cloud, a small tavern in a northeastern section of Pompeii collapsed, along with the rest of the town, under the weight of pumice and ash. This was later followed by a fast-moving pyroclastic surge of hot gas, volcanic debris, and ash that signaled the volcano’s final devastating blow: Those who stayed behind in Pompeii and Herculaneum were killed instantly by this infernal wave of heat, estimated to have been as high as 900° Fahrenheit. The barkeep of this tavern was one of these poor souls. He didn’t make it out of the establishment in time and perished in the cot where he slept, along with a dog and a man who had taken refuge inside the tavern with them.

Honestly, I can think of worse ways (and places) to perish.

In December 2020, archaeologists at the Archaeological Park of Pompeii announced that they had found the remains of these two men and the dog as they were excavating this ancient food establishment, known as a thermopolium.

I mean, sure, everyone (I hope) knows the story of Pompeii. I think it's awesome that they're still making discoveries there. Well. Rediscoveries, technically.

As a classical archaeologist whose research centers on food and food preparation in the Roman Mediterranean, I am overjoyed by finds like these, as the information obtained from them shines a bright light on the daily lives of classes of Roman society that are poorly represented in ancient literary sources: slaves and average, working Romans.

Wait, you mean they didn't have McDonald'ses? What a horrible time to have lived. Vesuvius would have represented a sweet release.

To recreate a meal that comes close to what may have been served here, one can begin by looking at the archaeological remains found inside.

The rest of the article goes into the findings and, as a bonus, provides a recipe. Spoiler: not even pizza!

Does all of this evidence suggest that meat boiled in broth at the corner popina is the Roman version of pub grub?

No burgers, no fish & chips, no Buffalo wings (which I consider to be a bonus; those things suck), and not even French fries.

The recipe, if you're not inclined to visit the site (you should; Atlas Obscura is my third favorite website), is for braised duck. Now, I have a personal rule against eating anything smarter than I am; this includes cuttlefish, cat, and duck. I do make an exception for bacon, as it's bacon. But I try not to impose my personal preferences on anyone else.

So there you have it. You can call it Volcano Duck, maybe. Or just read the article and marvel at what we've rediscovered through the wonders of archaeology.

One-Sentence Movie Review: The Hitman's Wife's Bodyguard

This film has almost everything I look for in a movie: Samuel L. Jackson, Ryan Reynolds, Salma Hayek, other big stars, gunfights, chases, explosions, great stunt work, snappy dialogue, and marvelous dark humor; in fact, the only thing it doesn't have is anything more than a basic plot -- the lack of which barely diminished my enjoyment of the spectacle.

Rating: 4.5/5

5 Like • Comment • View 2 comments

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