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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 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 3 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.

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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 29, 2023 at 10:40am

#1050227

What You Wish For

Today in "You can get what you want and still not be happy":

The Unbearable Costs of Becoming a Writer
After years of hard work and low pay, the risks I took to work in publishing are finally paying off. But now, I wonder about the price my family paid, and whether it was too steep.

When I was younger, I was always told: "Be careful what you wish for; you might just get it." So I stopped wishing.

My parents didn’t understand my job.

No one's parents understand them.

There were stretches when I made so little money writing or editing that I couldn’t blame my parents for assuming they were hobbies.

Most people think it's easy and anyone can do it. And then they try, and they think they've done a good job because they get positive feedback from those who are afraid of crushing them with negative feedback. But in reality, they suck.

I'm aware I'm probably in that category, too. After all, I've never made money from writing (or editing), unless you count GPs for WDC newsletters. That's how we're measuring success now, right?

But, okay. I get it. Sometimes outsiders don't understand that it's work, whether you're getting paid for it or not. Often rewarding work, and requiring very little physical activity, but it's no less work than, say, website development.

I became an editor by volunteering for an Asian American magazine, a nonprofit mission-driven labor of love where no one drew a salary. Ten to fifteen hours of unpaid labor a week in exchange for the editorial experience I wanted was, to me, an acceptable trade—nearly all my labor then was unpaid.

And I get this. (Obviously; I do that sort of thing, myself.) But it's a symptom of a larger problem, which is that what creative-types do is largely undervalued. Hey, can your band do this gig? We can't pay you, but you'll get experience and exposure! Say, I like your art. Can you do a poster for me? I don't have a lot of money, but you can put your contact info on it.

Then one of my favorite indie websites hired me to edit on a part-time basis. The job started at thirteen dollars an hour, twenty hours a week, and after a couple of months I was brought on full-time and granted a salary in the mid-30s.

Don't mistake my above complaints for bitterness. Would it be nice to be paid for my writing? Sure. But if that did happen, I know it would come with a raft of requirements. Like, right now, I can write about anything I want to, say anything I want to, be honest about my opinions (or at least honestly lie about them). My only restrictions are self-imposed. But consider what might happen if, say, someone approached me to do a beer blog. I'd then be beholden to them, and two of my favorite hobbies (writing and drinking beer) would become, by definition, work, with deadlines and expectations.

Everything becomes promotion, then. Hell, even the article I'm highlighting today is basically an ad for the author's book. (I know I've said this before, but that never stops me, by itself, from featuring something; writing ad copy is also a writing skill, and this is a site about writing.)

Fortunately, I don't need the money. So I can pursue it as a hobby. (Still, free beer would be a nice perk.)

But there was also a reality I had to face: I was running out of time to help my family. My diabetic father’s health was in decline, and my parents were struggling to pay for his care and medications. My husband and I couldn’t pay for full-time childcare for both our kids, let alone provide the kind of support my parents needed. The little assistance I could offer wasn’t enough to make a dent, and as my father grew sicker, my guilt and anxiety intensified.

This is understandable, too, though I neatly sidestepped a lot of it by not having kids I knew I wouldn't be able to afford.

I continue to grapple with the instability of this industry and what kind of opportunities will be available to me in the years to come, as well as larger questions about whether my editorial work was valued.

Editorial work is, if anything, even less valued than writing. Don't believe me? Pick a few articles off the internet and see if they look like an editor saw them. Or, hell, this blog, which certainly doesn't have an editor. I do try to catch obvious typos and awkwardly-phrased passages, but I don't always succeed.

I think about who gets to be a writer or an editor, who can afford to wait for that livable salary or that higher advance. Who can choose to prioritize their creative goals, take potentially career-making risks, invest precious years in this work without the guarantee of financial stability.

And this is why I chose engineering way back when. Fortunately, for me, it wasn't an either-or situation; I'm equally bad at both tech and creative stuff.

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