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 13, 2022 at 12:01am

#1040597

Revisited: "Never ends"

Once again, taking a look at a past entry. This one's from way, way back in October of 2008: "Never ends"

It's hard for me to remember last week, let alone that far back. I used Wikipedia to remind me of stuff that happened around the time of the entry. At that point, George W. Bush was still President (look, I'm not getting political here, just stating facts), and we were still a month away from electing Obama. The Great Recession, as it came to be called, had just begun; Bush had, a few days earlier, bailed out some of the banks involved. I was still working, still married. None of that mattered to me on October 7-8 because, apparently, on those days, I was in severe pain.

This was, obviously, from a very different time in my life, so I'll try to take it step by step.

Up until a week and a half ago, I had that nasty, unrelenting back pain and sciatica, the only relief for which was lying down and taking lots of meds.

Oh, yeah. I do remember how bad my back pain could get back in the noughties. At some point, I got steroid shots for it, and it got better. You know, those great big needles that they insert into your actual spinal nerves? Yeah. They suck. But not as hard as back pain.

I was okay for a week, then. I mean, I still had twinges in my back and leg, but nothing major.

Memory of pain is weird. It all blends together for me. Back and neck pain was just part of my existence back then. But this particular episode stands out as being utterly incapacitating.

Then, as I reported here, I got sick on Sunday, cutting short our anniversary celebration. This continued through Monday.

Hm. Somehow I had it in my head that our anniversary was closer to the end of October. In my defense, once she dumped me, I could release that date from long-term memory storage, so I did. I couldn't find anything in the archives about an anniversary celebration, only about everyone in the house, including the cats, being sick.

Monday night, I slept for a few hours, then was wide awake for a few hours, until maybe 15 minutes before my alarm went off. When I woke up, my neck and shoulders were stiff. No big deal, except my stomach was still upset, so I went to work. I left work early afternoon, figuring what I needed was to lie down with some heat on my neck.

No, I couldn't call in sick. Hard to do that when you own the company and don't have employees (not at that time anyway). The effects of the Great Recession on the business hadn't taken hold yet. Can't recall what projects we were working on that month, only that we were still able to make money.

So I heated up a neck thing and went to lie down, and pain exploded between my shoulder blades such as made the worst pain I experienced with my lower back (not to mention appendicitis) seem like a pleasant day in the Caribbean.

Like I said, memory of pain is weird. If I were to rank my pain as I remember it now, that day would only be about #4, behind the appendicitis and my heart attack (which happened later) and that time I got stung by a whole nest of yellow jackets (which happened in the eighties).

I couldn't move. Oh, I could move my legs and, to some extent, my arms, but I couldn't sit up or roll over. I couldn't even play dead because I kept looking for a position that minimized the pain.

Ha! "play dead." I crack me up.

My mobile phone was not nearby, so I couldn't reach it to call anyone. Every time I tried, I felt like someone was pushing a knife into my upper spine.

I do remember this particular episode of pain. Until I found this blog post at random, though, I couldn't even have guessed at the year, only that it had to be sometime in the noughties because my wife was involved.

I think I dozed off for a while. My phone rang. I had no way to get to it. I could only hope that my wife would come home before she went out to dance practice.

These days, I'd be utterly boned. Someone would find my emaciated, cat-chewed corpse.

Fortunately, she did. Unfortunately, she had no way of moving me. Fortunately, one of our close friends is a chiropractor. Unfortunately, the chiropractor was still at work. Fortunately, we were able to leave a message. Unfortunately, the ditzbrain who took the message didn't give it to her. Fortunately, I called her mobile phone an hour later to see if she got the message. Unfortunately, she hadn't. Fortunately, she was still able to come over and fix it so I could at least stand up - albeit with intense pain.

Remember a week ago I said some of these entries made me cringe? This is one of them. It's a bit embarrassing to me now. The idea of going to a quack to crack my back wouldn't fly with me these days. Sometimes you have to learn these things the hard way, I guess. It's entirely possible that chiropractic was the actual cause of much of my back pain in that era, though obviously there was some short-term relief from it.

Once I stood up, holding my head straight and not twisting or raising an arm, I was okay. We got back to the chiropractic clinic and she worked on me some more on the table. Then she said I couldn't get on the computer, so I sat with ice on my upper back.

Like I said, short-term relief. I haven't been to a chiropractor in well over ten years, and I rarely have these bouts of pain anymore. The one time I remember since then was neck pain coinciding with my month-long trip to Maui in... 2017, maybe? Some February in the teens. Really cramped my style; it's hard to snorkel when you can't move your head around to see where you're going. Bouncing around on the roads wasn't pleasant, either. At least there was copious alcohol.

I can only imagine how antsy I was without being able to compute. I don't think I've gone a day since 1979 (with the possible exception of a couple of vacation trips) without using some computer, somewhere, for work or school, or the internet or gaming. Not even that day; I would have used one at work the day of the incident, and obviously I was using one to make the blog post about it the following day. The thought of going without a computer for so much as a day fills me with the dread of possible boredom.

And look, I'm not trying to come down hard against alternative medicine. And I'm certainly not dissing my friend (I still call her my friend even though we've barely seen each other in the last decade or more. People drift apart; it happens.) It's just that these days, I need more scientific evidence before trying a course of treatment. Chiropractic has been shown to work for several things, but it's also been reported that there's a risk involved with adjustments, especially neck adjustments. To me, right now, the risk isn't worth the benefit.

I might change my mind if I find my neck in that much pain again, though. People in general will go to great lengths to make pain go away, especially hedonists like me. I'd be like "Fine, even having a stroke would be better than enduring this much agony."

But it was around that time that I came up with this:

I used to say "My back is killing me!" Now I say, "My quack is billing me!"

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