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 10 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 12 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.

May 8, 2022 at 12:05am

#1032112

Sea What I Did There

And now, today's burning (or really, drowning) question:

Are the Great Lakes Really Inland Seas?
Well, yes. And no. Actually, it depends on where you stand, in more ways than one.

That's right up there with "Is a hot dog a sandwich?" in terms of categorization questions. (It's not, by the way. It's a taco.)

Also, I like how the subhead answers the question in the headline without actually shedding light on it, making it more likely that people will read on without being clickbait. (Source is Atlas Obscura, not a clickbait site.)

The water reared up and slammed onto the sand like an ambush predator.

Ooooh, someone's taken a creative writing class. (The first paragraph goes on like that, but it mostly just sets the tone.)

The Great Lakes of North America’s midsection—Superior, Michigan, Huron, Erie, and Ontario—together span nearly 100,000 square miles, with a combined coastline just shy of 10,000 miles.

It might be helpful here to have some comparisons. For example, that's about the size of Colorado or Oregon. Or New Zealand. As for coastline, well, that's notoriously tricky to measure, which is how fractal dimensions were discovered, but that's another topic.

They even have their own U.S. Coast Guard district, the only lakes with such a distinction. And the Guard’s rescue teams stay busy: Superior and its siblings are capable of storm surges, rip currents, tsunamis, rogue waves, unique extreme weather phenomena, and destructive surf.

I think we're all aware of how angry the Lakes can become.

They have claimed more than 6,000 ships, more than the Gulf of Mexico and the Black Sea combined, according to estimates.

Why is it a "gulf" and not a "sea?" And why is it a "sea" and not a "lake?" Well...

“The most accurate answer you’re going to get is, ‘I don’t know,’” says John Richard Saylor, author of the upcoming Lakes: Their Birth, Life, and Death. “I do think it comes down to semantics, what you want to call a ‘sea.’”

Yep. Is Pluto a planet? Depends on definition of "planet." Under current internationally accepted definition, no. Shut up about it already, sheesh.

For many, the Great Lakes are indeed greater than lakes. The United States Environmental Protection Agency, for example, describes them as “vast inland freshwater seas.” A seminal 2017 paper in Limnology and Oceanography, authored by some of the most influential researchers studying the lakes, also refers to them as ‘inland seas.’ But what makes a sea varies by source.

Virginia and three other US states are technically Commonwealths. We still refer to them as states.

You might, for example, associate seas with saltwater, but “Whether water is salty or fresh does not cleanly separate lakes from seas,” says Robert Sterner...

Well, not really, no. The Dead Sea is famously salty as hell, sure, but so is the Great Salt Lake, which is about 7 times bigger than the Dead Sea.

The article lists other examples of things that are called seas and lakes.

Meanwhile, a growing number of scientists believe the Red Sea may actually be a young ocean.

That's a technicality based on continental drift models.

While the languages of these earlier people are not known, over the last few millennia the Great Lakes have been home to several Native American and First Nations peoples, most of which belong, culturally and linguistically, to the Anishinaabe. The term covers a number of communities dispersed over a broad and varied geographical area, but there is continuity among them in how the Great Lakes are perceived.

While one could argue that the people who lived there the longest should get to define their geographical features, I don't think the natives would have had the global perspective to, say, compare their size to New Zealand.

So, the semantics remain imprecise across languages, but what about the science? Do the people studying the Great Lakes see them as inland seas? The answer is a resounding “sort of.”

Thanks for that clarification.

Despite their size, the lakes are beholden to what happens on the land that surrounds them in a way larger seas are not. For example, precipitation and runoff that drains into the lakes significantly affects their water levels, chemical composition, and other characteristics.

Not to mention the ever-present industrial and human pollution.

In fact, the strongest case for describing the Great Lakes as inland seas may be to remind the public of the potential threat that they pose.

And yes, the article does reference the Gordon Lightfoot song.

University of Minnesota’s Coker suggests that the Great Lakes belong “in a category of their own.” Perhaps it’s not the lakes that fail to fit our definitions, but rather our words failing to describe their unique nature.

And that, to me, is the crux of the issue: whatever they are, the bodies of water we call the Great Lakes are what they are. Pluto didn't suddenly disappear when astronomers changed the definition of "planet;" in fact, we sent a probe out to take pictures of it. It's kind of like the Rock of Gibraltar. You could call it a hill, or a mountain, but no, it's called a rock.

On that note, codifying hills and mountains is a similar issue. The definition has changed over time. Here in the US, people who live out west look at the Blue Ridge Mountains and scoff, "Those are just hills!" Going in the other direction, all I could say was "Holy fuck, these are some big-ass mountains."

At one time, though, around the time the dinosaurs bit it, the Rockies were at the bottom of an inland sea, and the Appalachians were much, much higher (and originally extended into Scotland). Before the last Ice Age, there weren't even any Great Lakes for people to argue about their definition. Or people to argue about their definition.

But it's the nature of humans to codify and classify things. This can aid in understanding, but it can also become a semantic trap. After all, it doesn't matter if your car gets crushed by a boulder, a big stone, a small mountain, or a large rock; your car is still crushed. All continents can be considered big islands, and there is really only one world-spanning ocean surrounding all of them.

So don't worry whether a hot dog is a sandwich, a sub, a taco, a gyro, or whatever. Just enjoy the hot dog.

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