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.

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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 23, 2019 at 12:27am

#959477

Book Learnin'

What is your learning style? Do you prefer to learn through reading, images, audio, discussion, hands on, etc.? What is something new you learned in the last 30 days?

Well, first of all, let's talk about the idea of learning styles.

https://www.theatlantic.com/science/archive/2018/04/the-myth-of-learning-styles/...

Now that we've debunked that, let's talk about my favorite subject: Me!

It's probably because I went to engineering school that I have a particular system that helps me learn and retain stuff: reading then doing. Like, when I was taking a calculus course, I'd read the chapter and work through the exercises at the end.

Or, at least, that's the theory. In practice, I'd go out and have beers and then take whatever test whilst hung over.

I had to repeat a couple of calculus classes.

And yet, I remember. I don't mean every single detail, of course - I'd have to look up how to take the integral of, say, (sin x² / log_e x) - but I remember the concepts.

Now, this was before the internet, and just after people stopped programming mainframe computers using punch cards in the Fortran language, so we didn't have a lot of the tools available to us today, like video courses.

It's possible that this is why I fucking hate learning from videos. Doesn't matter what subject: cooking, getting past certain boss monsters in video games, how to count cards at blackjack, how to not get caught counting cards at blackjack, History of Beer, programming in something other than Fortran, or whatever.

Every purported training video I've ever encountered either goes too fast or too slow. And yes, I am aware that there are tools colloquially known as "pause," "fast forward," "rewind," and whatever; those annoy me.

No, give me clear, written instructions, and I'll learn the shit out of something.

Which is not to say that videos are totally useless. I sometimes like to use them to get an overview of something. Like, when I'm at the gym, to keep my mind as active as my body, I've been watching educational lecture videos . Short on fancy graphics, these are mostly someone with a PhD in something or other standing there talking about their favorite subject. My current lecture series is about the quantum states of electrons. It's enough to give me a basic idea of how electrons work, but I could never actually calculate, say, the energy released when the outermost excited electron in a carbon atom drops down to its minimum-energy state. But I do know that this process occurs and has to emit a photon of a certain wavelength.

If I could be arsed to read a book about it, though, I could probably learn more. But most such books assume a knowledge that I just don't have, because the only actual course I took in quantum theory was at 8 am, and I don't do 8 am.

So, see, even though I find videos less than useless for most purposes, they work for me to some extent. A lot depends on the subject matter and my interest in it. Different approaches seem to work for different things. For example, in assembling furniture or a model rocket, I want text and diagrams. For learning more about spreadsheets, I want examples that I can plug into a workbook and play with.

These days, most of what I learn, I learn with the purpose in mind to somehow use it in my writing (the exception might be card-counting, if I actually studied that, which I will not admit to doing).

So - what is something new I learned in the last month? Well, it's a question that's been bugging me on an intellectual level and I'd never found a clear answer for: As you know, opposite charges attract. An electron has a negative charge; a proton has an equivalent positive charge. And yet, in an atom, the electron kind of hovers around the nucleus (which contains protons and neutrons, the latter of which are (wait for it) neutral in charge) without them getting together and having babiesannihilating their charges. So what is it that keeps the electron away from the proton? That was what was bugging me, and this course I'm watching explained it to my satisfaction.

The actual explanation is related to how white dwarf stars don't just collapse spontaneously into neutron stars or black holes.

Hey, what can I say? I write science fiction, and I want to at least try to get the science right.

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