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 15, 2019 at 12:17am

#958987

It's About Time

Share an instance when something blew your mind.

My mind isn't that easy to blow, these days. Call it age, experience, wisdom, cynicism... oh, wait, those are all the same thing.

Most of my experiences along those lines come to me when I'm driving alone on near-deserted roads and letting my mind wander. That's one reason I like to take road trips. (The other is beer.) Most, but not all. Some external things can still blow my mind. My first thought when I saw the prompt was the eclipse of August 2017, and that certainly qualifies, but it's not like I wasn't expecting and anticipating what I saw - it was just really, really cool that I actually saw it.

I don't know if anyone else does this, but I like to learn about physics and shit for recreation. I tell myself it's because I write science fiction, and that's true enough, but a deeper truth is that it simply intrigues me. To do this, sometimes you have to separate the wheat from the chaff, and boy is there a lot of chaff out there. No, you can't use quantum mechanics to create your own reality. No, seriously, you cannot. First of all, you don't understand quantum mechanics. I don't understand quantum mechanics. In the last century, since it became a thing, you could count on your fingers the number of people who did fully understand it - and one of them was confined to a wheelchair. Now, you can argue all you like that Stephen Hawking wanted to be in a wheelchair, but you'd have to back that shit up with some evidence. Point is, even the (arguably) most agile mind in the field couldn't shape his own reality just by thinking about it - so all the "quantum consciousness" bullshit is bullshit.

Combine that with the unfortunate tendency of science reporters to sensationalize things or simply get things wrong, and you get something close to a pure application of Sturgeon's Law (90% of everything is crap).

So I try to avoid the crap, or at least learn to identify it when I see it.

Therefore, I'll try to render this revelation faithfully, without adding to the misinformation that's out there. On the other extreme from quantum mechanics is general relativity - the thing Einstein is famous for. And you probably already know that, Star Trek notwithstanding, no physical object or information can exceed the speed of light - and that as something accelerates toward the speed of light, time seems to slow down for that thing. This is called "time dilation," and it's mind-boggling by itself. Someone who zips around the world at speeds approaching that of light would age notably less than someone who doesn't; that sort of thing.

In my search for a better understanding of this cosmic speed limit (though I'm still not sure we're looking at it correctly - the question may not be "why can't anything exceed light speed" but rather more like "why does anything not move at light speed") and time dilation, I came across a rather simplified explanation that... you guessed it... blew my mind.

Basically, it works like this. Say you're in a car that can only go 100 kph. It can't go slower or faster. And you've got an enormous parking lot to travel in. You can go 100 kph in the "north" direction; you can go 100 kph in the "east" direction. You can go 100 kph in some angle in between, and when you do that, you have a "north" component to your velocity and an "east" component, both of which are (you're familiar with vectors? Or at least the Pythagorean Theorem?) something less than 100 kph, but the resultant velocity is, and has to be, 100 kph.

Now, further say that there's an observer on the south side of the parking lot, measuring your speed. That observer doesn't have a radar gun and can only measure your speed relative to the east-west axis. If you're traveling entirely North, she measures your speed as 0. If you're only traveling East, she measures your speed as 100. If you're going northeast, though, she measures your speed as (using the Pythagorean Theorem) roughly 70.71 kph. Unless I screwed up the math, which is always possible.

Here's the thing that blew my mind, though: Now pretend that the North axis is actually Time, and the East axis is actually Space. Based on General Relativity, space and time have to be considered as a single entity, spacetime. And just as the car in the example can only travel at one speed, everything in the universe also can only travel at one speed - that of light - but it's traveling through spacetime. Consequently, the less you move through space, the more you move through time. As you move more through space, approaching the speed of light, you're moving less through time. But you're always moving at the same speed through the continuum.

Now, it's not a precise analogy - the actual equations are not circular but hyperbolic - but it conveys the situation fairly clearly, I think. And looking at it that way was, to me, mind-blowing.

(Incidentally, don't bother commenting with the whole "time is an illusion" nonsense. That's part of the chaff I described above. Time is as real as space, as it's all part of the same thing. Different observers experience it differently, but that doesn't make it any less "real." And when you start questioning whether something that we experience is "real" or not, you go down the rabbit-hole of thinking that, well, everything must be an illusion, which makes the words "real" and "illusion" semantically invalid. So, stop it.)

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