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 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 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.

February 24, 2022 at 12:01am

#1027320

Running for Precedent

Please note: I will be traveling for a week, starting Friday (for fun this time). Consequently, blog entries might appear at different times than usual. That's if World War 3 doesn't start in the meantime, of course.

Meanwhile, let's wrap up this month's "Journalistic Intentions"

[18+] with a quote, selected at random, that is nevertheless relevant to the world situation right now:

"It is clear now that we live in precedented times."

Another fun thing about English (see entry from two days ago) is that there are words that sound like they should have similar antonyms, but don't. Such as "nonplussed." It means to be perplexed, bemused, floored, flabbergasted, etc. (Yet another fun thing is how some ideas have so many synonyms). But the only time you hear about someone being "plussed" is when someone is commenting on how weird it is that the word "plussed" isn't actually a thing.

Apparently, that one comes from the Latin "non plus," which translates to "no more," so I guess the sense is that you have no more fucks to give to react to whatever some idiot is doing or saying. The opposite of plus in English is minus, but no one's ever nonminused, minused, or plussed. One can give one's undivided attention, but can't be nondivided, and I won't even get into multiplication or exponentiation.

Similarly, we're never gruntled or chalant.

This is distinct from some words that seem to form antonyms but don't, such as flammable and inflammable which, to the consternation of kids and ESL students everywhere, actually mean the same damn thing. (This actually makes sense because "inflammable" is kind of similar to "inflammatory.")

Which leads me to "unprecedented."

No one actually uses it like the author of the above quote, but it's clear that they were trying to riff off of "unprecedented." And it's weird that it's not used, but no weirder than the exclusion of "chalant" from the dictionary.

You want to make a point and have it be remembered? Use a thing that should be a word, but isn't, and that can be instantly understood by the reader.

This is a lot harder than it sounds.

I think it was Mark Twain who said that history doesn't repeat itself, but it rhymes. I mean, that sounds like something Twain would have said. All due respect to the great man (if indeed it was he who said it, which is unclear), but I say it doesn't repeat itself or rhyme, but it echoes.

Some people take the philosophical viewpoint that time is a circle; that, to quote Battlestar:Galactica, "all of this has happened before; all of this will happen again." Well, it's pretty clear to anyone who is actually paying attention that time is linear, no matter how many cycles we try to impose upon it, but there are certain things that reverberate through the ages: wars, uprisings, invasions, economic recessions, philosophy itself, and so on.

All very different from each other, but all with echoes of the ones that came before.

As they say in the Fallout games: War. War never changes.

The Book of Ecclesiastes asserts that "there is nothing new under the sun." Arrant nonsense, of course; you're not going to convince me that smartphones, for instance, aren't "new" (as compared to the technology of, say, 50 years ago). But when you break down what we use smartphones for, well, at base it's about communication. And communication has been going on for at least as long as there have been humans; only the means have changed.

But to think that nothing changes at all? No. The details are always different. And yet we find commonalities, because that's what we do.

There's precedent for that, too.

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