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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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.

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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').

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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, 2019 at 12:08am

#958501

In Space, No One Can Watch You Drink

In the future, if space travel became possible, would you want to go? What would most influence your decision whether or not to leave Earth?

If you'd asked me that when I was 15, my answer would have been, "Hell yeah, sign me up."

At that time, I had every reason to believe it was not only possible, but inevitable. For fuck's sake, we'd just lofted people to the goddamn moon. Certainly, Mars would be next, then maybe a generation ship to the Alpha Centauri system or some such science fiction trope. And I desperately wanted to be involved.

What took me by surprise was that we spent all the time since then never leaving Low Earth Orbit (LEO). Oh, I know there are reasons for this: political and financial costs, greater awareness of the effects it would have on a human body to leave the safety of our natural radiation shield, whatever. And I guess I should be glad we didn't abandon even that, but we could have done so much more, especially as technology advanced.

We went to the moon using little more than a slide rule. It would be another ten years before you could stuff a real computer into a spaceship. Whatever you're using to read this, laptop or desktop or tablet or mobile phone, has orders of magnitude more computing power than Collins, Aldrin and Armstrong had access to.

We've done other stuff, of course. We populated Mars with robots, and sent probes throughout the solar system and beyond, and that's all very cool. Nothing like boots on the ground, though - or floating in a capsule.

But I digress. I tend to do that when we're talking about space exploration. I'm convinced that Western civilization hit its peak on or about July 20, 1969, and we've been on a downhill slide ever since. But that could be age talking; I don't know.

I've heard all the arguments against space exploration, and I summarily reject all of them.

Now? Well, now I realize that there's no beer in space. Oh, I'm fully aware that astronomers have identified entire clouds of ethanol out there. Not the same thing, though. You'd also have to find barley, hops, and yeast, and there are plenty of those things right here on Earth. To go the rest of my life without beer? You couldn't pay me enough.

Happy to sit back and watch other people endure the trips, though. Go for it.

7 Like • Comment • View 8 comments

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