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 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.

July 11, 2020 at 12:01am

#987788

Sounding the Alarm

Oh man. I'm bracing myself for a whole lot of anti-human sentiment today. I really hope I'm wrong, but that's how it works when you're a pessimist: either you're right, or you're pleasantly surprised, and I'll take either one.

PROMPT July 11th

Imagine what the world would be like today if humans had never discovered/invented _________ (fill in the blank).

alarm clocks

Wow, what a wonderful world that would be, eh? No worries about not getting enough sleep, everyone chill about people getting places whenever they feel like it. I think if more people got the proper amount of sleep -- as defined by one's own particular circadian rhythms -- there'd be a lot less anger and conflict, and a lower level of general grumpiness.

The coffee industry would be practically nonexistent, but I don't care.

I know there are good reasons to keep things on a schedule to maximize cooperation among humans, but really, what's the cost? Also, it's more tempting to go to bed later and not get enough sleep if you know for sure that some shrill ringing or beeping is going to disturb you out of a perfectly nice dream so that you can go do something you don't really want to do in the first place.

No, if we didn't have alarm clocks, I think more people would pay attention to going to bed at whatever time they need to in order to get enough sleep, rather than letting the alarm (and then caffeine) take care of things.

Of course, I say this as someone who was ruled by alarms for many years, and now only rarely has use for one. But that just means I can speak to the difference.

In reality, of course, if the alarm clock didn't exist, someone would have to invent one. And all of our inventions are interconnected; you can't just take one out of the picture and work around the hole that would create.

Except maybe those ugly-ass big-head figures. Funko Pops or whatever they're called. I'll make an exception for them; the world would unarguably be better off without them, and we'd never even notice the difference.

Of course, I had to think for a while before coming up with "alarm clocks." So many inventions, so many discoveries, and so few of them either clear benefits or obvious Bad Ideas. Most things are a mix of the two, just like most humans. So I hope we'll be around for a while to discover and invent more things, and none of them will be perfectly good or perfectly evil, but most of them will be perfectly interesting. I'm just sad I won't be around to see it.

8 Like • Comment • View 9 comments

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