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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 1, 2022 at 12:02am

#1040090

Snip

I don't know why I keep finding Outside articles. But this one, at least, isn't directly about the (shudder) outside.

Speaking of which, I'll be venturing (ugh) outside (sort of) for the next few days. It's been a while since I did a road trip, and now that I have a car I can do one. Remember a couple of months ago, when I linked an Gastro Obscura article about non-grape wines? No? Well, here it is: "Keep Wining"

. There, I noted that one of the featured wineries is in New Jersey, which isn't that far away from me, considering that I have, several times, driven all the way across the country and back.

I described it as "might be a good overnight trip," but then I found out that it's actually half an hour away from Atlantic City (hookers and blackjack), and that there are other wineries in the area, as well as breweries. So I'm going until Friday. Blog posts may happen at hours unusual for me, but they will happen (barring catastrophe, of course).

That's got nothing to do with today's article.

I Got a Vasectomy Because of Climate Change
Getting one was, by far, the most powerful personal action I could take for our planet

I'd like to say that my decision to remain childfree was primarily motivated by concern for the ecosystem. I'd like to, but I can't. Oh, sure, it factored into it—I almost never make decisions based on just one thing, and absolutely never with something of that magnitude. But mostly, it was the other way around: I couldn't ethically justify bringing a child into a declining civilization.

The article itself is from the Before Time, almost three years ago.

I’ve always struggled to combine the idea of personal responsibility with the overwhelming need for human society to address the threat posed by climate change.

That's because it's our problem, but it's not our individual responsibility.

Since at least the 1970s, the massive energy corporations responsible for the vast majority of our carbon emissions have known about, and done nothing to mitigate, the harm they cause. Because they own politicians worldwide, there doesn’t appear to be any will to take government-level action. But I’m supposed to turn off a light? What possible impact could that ever have? And why is all this on my shoulders and not theirs?

Which is what I've been saying.

Bad example, though. Turning off a light has a direct impact on you: you pay a lower power bill. A better example might be, oh, I don't know, not taking road trips that waste gas just so you can try exotic New Jersey wines.

Incidentally, several years ago, I replaced all the lightbulbs in my house (except for oven and refrigerator) with low-consumption LEDs. Again, I'd like to say this is because I'm concerned with the environment, and that was certainly a part of it. Another part was spending less on electricity. But mostly it's because I'm too lazy to replace lightbulbs, and these LEDs are supposed to last 20 years.

When I got engaged, my fiancée, Virginia, and I started planning for the future. It wasn’t just my dog Wiley and me against the world anymore. All of a sudden, I started thinking ten to 20 or more years ahead.

Huh. I didn't need to get engaged to plan for the future.

Children are an obvious thing to plan. With a sudden focus on responsible decision-making, it no longer made sense to leave hypothetical future offspring up to chance. When should we have them? What did our careers look like on that timeline? Who’d be responsible for staying home and raising them? Couldn’t we just have one of the dogs do that?

To be fair, they'd do a better job than a lot of humans.

Is this a world we want to bring kids into? Is this a world it’s responsible to bring kids into?

No.

Of course, that's my opinion. Others have different opinions. That's okay. I would no more want everyone to decide to have all the offspring they can than I would want everyone to suddenly decide to stop breeding.

It looks like the pace of climate change is speeding ahead of science’s ability to understand or forecast it.

Lately I've been seeing more hopeful articles about climate change forecasts. By "more hopeful," I mean "we're still boned, but not as much as our worst fears." I suppose that's something, but it's still going to suck.

I can't help but think that the impetus behind such articles is to keep people from spiraling into despair. Despair isn't good for the economy. Gotta keep the economy going.

The future might be worse than any of us currently fear.

Again, older article. But even the "optimistic" predictions are pretty bad.

Then Virginia and I started talking about something we could do—for ourselves and to make a meaningful impact on the bigger problem. We could just forego the whole kid thing altogether.

Such a decision would certainly do more than, say, using only reusable bags, or foregoing plastic straws.

The image of personal climate change action doesn’t really match the reality. If I gave up my 15 mpg pickup truck—basically the mascot for climate inaction—and rode my bicycle everywhere, I’d save the planet 2.4 tons of carbon emissions a year. That’d be a massive sacrifice, but it’s nowhere near the carbon emissions I’ll save by skipping becoming a daddy, which comes in at around 58 tons annually, per kid. Any other action we could take, even all the actions we could ever possibly add up together, pale in comparison.

One can quibble about the math, but a) do you really need that pickup truck and b) the basic idea is sound: if your carbon footprint ends with you, you've done a lot.

That’s because there are simply too many humans on this planet.

Arguable. I mean, I happen to agree, but every time I say so, someone chimes in with "so why don't you kill yourself?" as if that's equivalent to not reproducing. There's some basic disconnect there in peoples' brains that I just don't understand and it may be that I never will. Urging someone to kill themselves is otherwise frowned upon, unless it's in that context. Makes no sense to me.

So, we’re not having kids. I found a colleague’s brother here in Bozeman who performs vasectomies and made an appointment. I was afraid of getting my scrotum operated on, but the procedure ended up being quicker and less invasive than most dental appointments.

It is true that, like most men, I can count the number of people I want holding a scalpel near... there... on the fingers of one foot. But, as with my eye surgery last year, sometimes you just gotta overcome your fears.

Once the anesthetic wore off, it felt like someone had kicked me in the balls pretty good, a feeling that dissipated over the next seven days. I took a Valium before the surgery and a few handfuls of ibuprofen afterward but otherwise didn’t need painkillers or even an ice pack.

Dude. Frozen peas. Come on. Also, it's not nearly as bad as getting kicked in the nuggets, unless you get it done by your sister's friend's cousin's coworker's uncle in some back alley.

The worst part was taking a week off from the gym; I’d been making good progress.

Where's my tiny violin? I left it around here somewhere.

Anyway, it's certain that a lot of people had, or will have, a visceral hatred of this article. The suicide comment above makes me sure of it. I want to emphasize again that I'm not judging people who do have kids. It's a choice that should be neither mandatory nor forbidden. Nor should we cave to social pressure either way.

There are, as I said, other reasons to get the procedure done. It could be argued that, well, why would someone without kids even care about the environment at all? Oh, I don't know, maybe we have a broader view of humanity than just our own immediate family?

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