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 26, 2020 at 12:03am

#999138

Gone Native

Oh good. I was afraid I was going to have to rant about "gratitude" today. More like "grrr-attitude" for me.

PROMPT November 26th

I am writing this prompt today from the current and ancestral home of the Anishinaabe people. In your entry today, research and write about the indigenous and native peoples who lived on the land where you currently reside. If you are in the United States, take time during your Thanksgiving meal to thank your ancestors and the indigenous people of your area.

I'm going to use the word "Indian" in today's entry. I know this is controversial, but the Indians I know call themselves that. No disrespect is intended.

I live in Monacan country. The Monacans got state recognition a while back, and now Federal recognition as well. Historically, as I'm sure everyone is aware, European settlement in the Americas was... not kind to the natives.

Before that happened, though, there were distinct groups of Indians in Virginia, with great variation in language and culture; while the Monacans controlled the Piedmont region, the Tidewater area was the domain of the Powhatan confederacy -- which is far better known, because, you know... Pocahontas.

I know the prompt is to talk about where I live now, but I have a stronger connection to the Powhatans, so I'm going to be talking more about them.

When I was a kid, we couldn't kick a clod of dirt without dislodging some sort of stone arrowhead or spear point. Turns out the place where I lived had been a major village of a tribe affiliated with the Powhatans, called the Patowomack.

If that name sounds somewhat familiar, it should; it was this tribe that gave its name to the Potomac River.

I didn't think much about this when I was a kid. When you're young, you tend to think that the way you're living is normal, nothing to get all excited about. So I lived on a farm with woodlands and wetlands, next to a wide body of water off the Potomac, and we kept finding arrowheads (and also Civil and Revolutionary War relics such as shot, buttons, and even sometimes swords). This was just ordinary life.

You get older and, hopefully, you get some perspective. Maybe even some wisdom; jury's still out on that. Definitely aches and pains, but hopefully more knowledge, at least.

I don't want to go too much into detail because I don't want to identify the exact spot; the last thing I need is random people crawling around my land looking for relics. Yes, I still own the land; I inherited it and have rebuffed all efforts to develop it. But the general outline is as follows:

I moved to Monacan country in the mid-80s, and ended up graduating from university and sticking around because I got a job here. Eventually, my dad began to decline, and a friend of the family moved in with him to assist, because I'd had trouble finding a job back there. This friend, who I'll call M, had an interest in history and spent his free time finding all of these relics, as well as researching a great amount. He found some old documents that revealed the presence of the Patowomack village, for instance. He also spent time cleaning and mounting a lot of the artifacts.

There is some evidence that Pocahontas herself spent time there, in the place where I spent my childhood.

In the early noughties, my father got to the point where he had to receive more professional care, and I had to make the difficult decision to place him in an Alzheimer's facility. M wanted to keep living there; it was the least I could do since he'd been good to my dad. During this period, the Patowomack received state recognition as a tribe and organized in my home county. M knew some of them, and they talked, because it was part of his research.

According to the law, anything found on a piece of land belongs to the owner of the land: my dad, and then after he died, me. But when M came to me and proposed that we give what he'd found to the tribe, I readily agreed. As I put it at one of their tribal meetings once, when they asked what would motivate me to do such a thing: "Well, I didn't put them there."

Now, this is Virginia; it's difficult to find what you might call pure Indians here. Over the generations they intermarried and lost a lot of their history and culture. Before M and I gave them the artifacts, they had nothing but family histories: no artifacts, no tribal memory. We changed that.

And so that is how I became an honorary member of an actual Indian tribe.

I consider this the greatest contribution of my life, and true to form for me, it took almost zero effort on my part; M did all the actual work. But it just didn't feel right to hoard the collection for myself, or even to take a tax break or whatever. All I got in exchange was the honorary membership, but I treasure that far more than anything else I've ever received, including my college degree or my professional engineering license.

It probably sounds like I'm tooting my own horn here, boasting or bragging. That's not my intention, though. It's nothing to do with me at all; the tides of history sweep over eastern Virginia like... well, like actual tides, and take me with it. Those people were on the land before it was mine, and they'll be around long after I'm gone. I've had archeologists come out to the place, too, and though it's been farmed for decades, I'm hoping they can discover something new. Well, something old that's new to us, that is.

I inherited the land, but I also inherited a political system that, for generations, tried to erase the people who were here before us. I feel no guilt about that, because I had nothing to do with it, but the least I can do -- literally -- is acknowledge their history. It's not one I directly share, but I'm glad to have contributed in some small way to the history of the people of the land where I have roots.

One final side note: one of my best friends, a guy who grew up here in the Piedmont, showed me genealogical research that his brother had done, and that research traced his family back to Pocahontas. I have no idea if it's true or not, but either way, it drove home to me that in a very deep way, we are all connected.

I've pretty much quit celebrating Thanksgiving, myself. There's just too much cultural baggage involved, and fortunately, I don't have family that insists upon it (yes, last year I went to New York because my cousin invited me for Thanksgiving, but it was a low-key affair and mostly just an excuse for me to visit breweries in the Northeast). The holiday is just too tied up in religion and colonialism, not to mention consumer culture.

But I did buy my traditional Thanksgiving bottle of Beaujolais Nouveau (a red wine from France that comes out in November and, fortuitously, tastes goddamned awesome with turkey or by itself), which I intend to polish off later today while I'm sitting around pointedly not doing anything else. Maybe I'll pour some out for the ancestors.

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