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 8 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 10 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.

March 13, 2024 at 10:52am

#1066209

It's "Desert" Not "Dessert"

I pronounce it "collection of endorrheic basins," but apparently, that's wrong.

Why Do Nevadans Pronounce Their State’s Name So Strangely?
How you say it certainly says something about who you are.

Look, it's an occupational hazard when you make a career out of hydrology to see the world in terms of drainage areas.

Seriously, though, I, too, pronounced Nevada in the non-Nevadan way until I spent some time as a guest of someone who lives there. Then, I learned the One True Pronunciation, ensuring that never again would I be caught by that particular shibboleth.

How exactly is the name of this state pronounced? Nevadans say “neh-VAD-uh.” Non-Nevadans typically say “neh-VAH-duh.”

To get it out of the way, there isn’t really a “correct” or “incorrect” way to pronounce Nevada in any objective sense; both “neh-VAD-uh” and “neh-VAH-duh” are perfectly understandable to all English speakers, which is really the only thing that matters.

Yeah, not really. Watch what happens if you mispronounce quinoa, for example.

It is not unusual for the residents of a state to have their own pronunciation of their state’s name; regional accents and dialects can affect all kinds of words.

Which, I suppose, is how we get Mississippi when the people who live in that state call it Misipi. And don't get me started on Ar-Kansas.

Even worse, few countries pronounce their names the same way that foreigners do. I'm sure that the way we say "France" here in the US grates on the very last nerve of the French.

Proponents of “neh-VAH-duh” will often say, look, nevada is a Spanish word (meaning “snowy” or “snow-capped,” and the state’s name is probably derived from the Sierra Nevada mountain range, part of which lies within its borders.

And the area, like most of California, was once claimed by Spain and Mexico, which ex-Spains all the Spanish names in the area. Want to piss off some Southern Californians? Pronounce La Jolla like you're English, not Spanish.

The problem is that Spanish, being a much more sensible language than English, has five vowels, and five vowels only. (Well, if we’re not counting dipthongs. Or tripthongs. All languages are complicated but stay with me here.)

I think there are more problems here than that.

There are a couple of possible explanations for this phenomenon. One is that Nevada, being a fairly new and historically largely unpopulated state, traditionally did not have much to differentiate it. If you’re proud to be a Nevadan, what could you do to present that to the rest of the world? Until the creation of the Vegas Golden Knights NHL hockey team in 2017, the state had no major professional sports team, which is often a way to signify geographical pride.

You know who else doesn't have a major professional sports team? Virginia. And I don't hear too much crap about pronouncing my state wrong, except for the hillbillies out west who insist on Virginny.

In any case, the article (from AO) is like candy to me because it involves history, linguistics, sociology, and philosophy. So there's no hard science; nothing's perfect. In other words, everyone can learn something there, if only the "correct" pronunciation of Nevada.

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