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!

a day 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

a day 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.

August 19, 2019 at 6:00am

#964418

Missionary Position

http://nautil.us/blog/how-will-our-religions-handle-the-discovery-of-alien-life

How Will Our Religions Handle the Discovery of Alien Life?

Poorly, I'd imagine. Because religions are made up of people, and people will handle it poorly.

For the religious, knowing that life on Earth is not unique may demand radical new ways of thinking about ourselves: How special and sacred are we? Is Earth a privileged place? Do we have an obligation to care for beings on other planets? Should we convert ET to “my” religion? These questions point to a deeper issue about whether our religions can adapt to the idea that humans are not the only sentient beings in the universe capable of worshiping God.

And once again we have the conflation of "extraterrestrial life" with "intelligent space aliens." As I've noted before, those qualities we call "intelligence" aren't required to exist by some Law of Evolution. But... that's not the most egregious assumption made here, is it?

I approach the question of alien life from the standpoint of evolution, and science in general. This doesn't mean I'm right, of course, but I tend to ignore the religious / spiritual questions. This sometimes puts my own thoughts in sync with some of the religious thinking as described in the linked article. That is purely coincidental; when I say something like "intelligent life is probably rare in the galaxy," that does not mean the same thing as when a religious person says something like "humans are unique," even though the statements are roughly similar.

At the same time, I'm not going to conclude that humans are insignificant to be contrary, just because some religious people believe we're somehow special (as in chosen by God or whatever). We're not special, nor are we insignificant. We're just... us.

History has also shown us that many religions don’t hold back to invite or even force non-believers into the fold. The efforts by European colonists and Islamic armies to convert native peoples to Christianity and Islam have an oppressive, bloody history. A similar future could unfold in which one or more of our major religious groups attempts to convert aliens.

Just what we need - religious fanatic space aliens. Can we put some First Contact protocols in place to exclude proselytizers from meeting our new alien overlords? That would be great, thanks.

Is it possible that learning of alien life will encourage us to tolerate the religious beliefs of our fellow humans, too?

Ha ha ha ha ha! Oh, wait; you're serious. Let me laugh some more: Ha ha ha ha!

I note that this author didn't list certain religions in the link. Notably absent as far as major world religions go is Hinduism. I have no idea how Hindus might approach the idea of space aliens (unless, presumably, the aliens resemble cows.) (Is that a rude joke? I don't even know. I don't mean to be rude.)

Also absent from the list is Scientology. While I don't think we could call it a "major" religion in terms of number of adherents, the idea that there are space aliens is, as I understand it anyway, baked into the religion. I have to wonder what they'd think if sentient aliens showed up and have never heard of Xenu.

I'd also be interested in the LDS church's take on the idea.

Anyway, there are a number of problems with this kind of speculation. First one I can think of is that we have, by some measures of "intelligence," several intelligent species here on Earth. Sure, dolphins don't build spaceships, and octopodes don't have vast underwater cities (at least, none that we've found so far), but they and some other species communicate with each other with some sophistication - and yet, we humans can communicate in only the most rudimentary way with these other species. These are species with whom we share a common ancestor; how can we expect to be able to really communicate with alien life?

Another problem is about time. We haven't found evidence of intelligent ETs, or even simple life beyond our world. Finding so much as a microbe (or equivalent) would be a significant event. But will it happen next year, next century? How much longer before we encounter any flying-saucer-pilot type aliens? It's laughable to think that world religions would be the same in 1000 years, considering how different they were 1000 years ago.

Then there's the persistent idea floating around that the reason ETs haven't shown their face-equivalents around here is because we're somehow ideologically impure. Maybe they're waiting for us to outgrow this "religion" stuff, or at least the "killing in the name of" part of it. Or maybe they're waiting for all of us to convert to Scientology. I think the whole "waiting for humanity to grow up" crap is little more than self-flagellation, personally. Yeah, yeah, I get it - you think humanity sucks and you're projecting that onto the motivations of purely hypothetical life-forms the nature of which we have no way of knowing.

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