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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Turkey DrumStik

about 3 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

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

Nov 14, 2024 at 12:10pm
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').

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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

January 26, 2019 at 12:58am

#950466

The Final Countdown

For this final Creation Saturday, write about something that’s in its final countdown.

Okay, let's get this inevitability out of the way.

Yeah, couldn't resist. I'm betting I'm not the only one to post that, but if I have to have the earworm, so do you.

So... something that's in its final countdown...

Wouldn't that be... everything?

I mean, far be it from me to be a downer , but everything ends. Even the universe itself is slowly dying.

A countdown, though, is all about time. I've been seeing a lot of articles lately smugly asserting that time is an illusion.

This is, of course, bullshit. Oh, maybe it's an emergent property of other processes, but so is consciousness, and if that's an illusion, then everything is. And since it's perfectly clear that not everything is illusory, then to use the word "illusion" to describe it stretches the definition of "illusion" past the point of usefulness.

Now, sure, our perception of time can be modified. Time appears to pass more quickly, for instance, the older we get. Time spent doing something boring seems to go by less quickly than time spent doing something we enjoy. These two observations appear mutually incompatible, because, obviously, life is more boring as we get older - always fewer new experiences to absorb. So, yeah, but that's not an observation about time but about our perception of it. Disregarding relativistic effects that have only tiny effects on our daily lives, time can be measured to a high degree of precision.

And that's the polar opposite of an illusion: something that, regardless of our wishes or desires, acts in the same way in all commonly encountered situations.

A word about relativistic effects, though: Yes, someone traveling at close to the speed of light will have a different reading on their clock. This effect has been measured and observed, and is uncontroversial. But, again, it doesn't affect what time we have to get up in the morning. It has some bearing on things like GPS satellite clocks and ultra-precise scientific measurements, but if we agree to have lunch at noon, and you show up at 12:30 claiming time is an illusion, that's the last time we have lunch.

So why do these reports like to breathlessly call time an illusion? Well, because to a scientist, the only "real" effects are those not tied to a particular frame of reference. For example: centripetal force is considered real, while its buddy centrifugal force is dependent on reference frames so it's called imaginary. And yet, if you've ever been on a whirligig (or as I like to call it, a kiddy catapult), to you, the observer, centrifugal force is very real. Anyway, since the passage of time is dependent on frame of reference, it is, in a sense, not real. And a lot of science reporters seem to have been taken in by Buddhist philosophy.

Bottom line: time is real, everything is dying, and we're all in the final countdown.

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