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 24 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

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.

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.

December 17, 2019 at 12:29am

#971586

I'm a Loser

Yes, I'm linking GQ today. Remember yesterday when I said most of the child-choice pieces I see are by and for women? Well, that's equally true for weight loss articles.

https://www.gq.com/story/what-i-learned-when-i-lost-50-pounds

Not that there's anything wrong with that. I mean, most of us, male, female or otherwise, could stand to lose a few pounds. In my case, it stretched the definition of "a few" past the point of tearing tendons. Still, it's come to my attention that male and female bodies are different - who knew? - so I seek perspectives more similar to my own.

What I Learned When I Lost 50 Pounds

Last summer, Rohan Nadkarni decided to get in shape. He expected it to be difficult; he didn’t expect a whole new set of anxieties.

Fifty pounds? Amateur.

The stars were aligning. From that day in early August through Christmas I had lost weight rapidly—at one point reaching the 50-pound mark.

So, (math) that's ten pounds a month. Seems a bit fast, but my own story isn't much slower: 80 pounds in 12 months, or 6-7 per month on average.

But when I stepped on the scale Christmas morning, and the display showed an arbitrary number I’d chosen for myself months ago, I felt as flustered as I felt accomplished: What the hell was I supposed to do now?

Yeah, see, that's the part that hooked me, not the earlier bits. What do I do ifwhen I reach my goal weight?

But what happens after you achieve these goals? Losing weight was always this huge mountain for me: I thought once I climbed it, I would be able to conquer anything. It hasn’t been that simple.

Fortunately, that isn't my problem. I know damn well I'll still have things to work on. I've probably said this before, but I fully expect to find that the problem isn't my weight, but my personality. I can't exercise that away.

I was lucky to have the fake (and flexible) job of being a “writer,” a steady income, and access to resources like groceries and a gym...

Oooooh, look at Mr. Makes-Money-From-Writing over here.

I made healthier choices when I could, and went to the gym/a spin class/for a long walk when I could.

My own journey is a bit more regimented, with short breaks (a week or less) every now and then because if I can't travel, what's the point of living? I eat 1300-1500 calories a day, get on an elliptical trainer (or walk) for 30 minutes a day, pump weight machines (not free weights) six days a week, and obsess over shit.

You won't catch me dead in a "spin class." Especially when I finally found out what one was: you get on an exercise bike with a bunch of other people and, presumably, listen to music and/or motivational shouts from some drill instructor-type trainer. No, thanks. I spend my time on the elliptical thingie watching science, math, philosophy, and/or writing lectures. Yes, just lectures; no fancy CGI. No one talks to me besides the guy or gal on the screen. I like it that way.

Point is, everyone has to find what's right for them. Spin classes worked for the author here? Great.

I haven’t eaten an added sugar in weeks: Is it okay for me to eat this brownie? Should I be calculating and tracking how many calories I’m about to consume, or can I scoop these cashews up by the handful since I’m high as hell right now?

Substitute "drunk" for "high," and that's closer to my own experience.

One disconcerting side effect of trying to be healthier is that people feel way more comfortable commenting on your body than they ever did in the past.

Body dysmorphia isn't limited to women. This is why I wanted a dude's perspective. It's somehow more socially acceptable to call fat guys fat. I don't think it hurts any less, but whatever.

I think that so many of the anxieties I’m describing are ten times worse for women, for whom the definition of the “ideal body”—as far as I can tell—has historically been much narrower than for men.

This may be true, but we're held up against skinny and/or muscular guys on TV, movies, magazines (*cough*GQ*cough*) all the fucking time.

What I wish someone had told me in August 2018 is: Be healthy, but also broaden your definition of what “health” is.

Here's the thing for me: I'm not necessarily chasing "health." Yeah, I want to avoid another heart attack and diabetes and maybe improve my sleep apnea (I think I've made progress in some of this), but you want to know how I keep from eating more than I should? I'll tell you anyway. Nicotine is a very effective appetite suppressant, and cigars take long enough to smoke that I can put off eating while I'm smoking one. Is it healthy? No. Does it work? Oh, hell yes.

Point is, it's not health I'm after; it's a 32-inch waist. By any means necessary.

Not every decision you make about food or your diet should be made in service of being thin. You should value your comfort and mental well-being just as much. Be physically healthier, for sure, but not at the cost of your sanity.

And that's the takeaway for me. This author wants to try "every bubble tea in Southern California." Me? I want to try every beer in existence. It'll never happen, but a guy has to have goals.

There's no point to living longer if you have to give up all the things that make life worth living. If someone told me I could die next month, or stop drinking beer, I'd make sure my will was updated while swigging a nice Imperial stout. You can do everything "right," and get hit by a meteor. (Technically it's a meteor until it hits the ground, at which point, when someone finds it next to my corpse, it'll be a meteorite.)

And if you're worried about what other people think - it seems the author of the linked article is, moreso than I am anyway - rest assured no one cares if you're "healthy."

They only care if you're thin.

In four days, I'll have been at this crap for one year, so consider this the update: 80 pounds lost, more to go, and I've surprised myself by actually sticking to the plan for that long - again, with a week off here and there so I could travel. Oddly enough, that seems to help in the long run even if it slows progress in the short term. I don't know why; maybe it resets metabolism or something? Like I'll come back from a trip and I'll have gained five pounds, but it comes back off fairly quickly, as does subsequent avoirdupois.

In August, I have plans for another cross-country road trip. By then, I'll have either reached my goal weight - or finally given up on the whole idea.

I'm aware I will fail at some point.

Just not today.

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