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 22 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 23 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.

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.

January 2, 2020 at 12:06am

#972434

Livin' In The Future

PROMPT January 2nd

Yesterday, we looked to the past. Today, we’ll write about the future! (No resolutions, I promise, Robert

)

2030 sounds so futuristic, but it’s only ten years away! What do you think the world will be like ten years from now? What inventions do you think we will have by then? What are some changes you think will happen in the world in the next ten years?

I discovered science fiction young.

Between the TV shows like Star Trek, short stories (which were still a published thing at the time), novels and even comic books, I got to see what I thought were different authors' visions of The Future™.

Later, I came to understand that it is not the job of a science fiction author to foretell the future; it is his or her job to prevent it.

I don't mean the post-apocalyptic wasteland subgenre, or not just that. I mean that science fiction is, essentially, a philosophical form of writing that explores the repercussions of change on humans and their societies. "But not all science fiction is like that," one might object. "Sometimes it's just space empires and adventure stories." This is not the place to get into the fine details of what is and is not science fiction; besides, any such argument starts to wiggle perilously close to the No True Scotsman logical fallacy. Yes, I understand the objection. In rebuttal, I'll just post a link to Sturgeon's Law.

Notably, Theodore Sturgeon was a science fiction author.

Point being, every once in a while you come across a "news" story or fluff piece that explores what this or that science fiction author "got right" and what they "got wrong." While sometimes intriguing, they miss the point; again, SF authors aren't here to foretell the future. On those occasions when they do explicitly foretell the future, the only reason they might have a slightly better track record is because they are, pretty much by definition, interested in the trends of science and technology, and anyone with such interests can prognosticate some advancements to some degree of accuracy.

On the other hand, those are some of my interests as well, and yet I have no idea what things are going to look like in 10 years because it's not just about science and inventions; it's also about human interaction with them.

Let me provide a quick example. It was fairly common, in the 60s and 70s, to predict the advent of videophones. As I recall, the idea was you have the existing telephone line technology, only instead of just having an audio transmission, you also have a video screen. Some enterprising individuals even figured out how to make the things work. But people freaked out. What if someone calls when I'm getting out of the shower? What if my hair is a mess? What if my house in the background is a mess? So, even though we had the technology, in spite of science fiction "predictions," the vidphone never caught on.

At least, not until phones were basically replaced by computers, smartphones and other internet - connected devices, at which point people started Skyping and videoconferencing and whatnot. It was the vidphone, manifest at last - but in a way that I don't think anybody foresaw.

It was a failure of imagination all around. The general public couldn't conceive of a way to make a phone call without the traditional "run to the phone to answer it when it rings" routine. Would-be vidphone makers couldn't come up with a way to change that routine, or otherwise alleviate the public's distrust of the concept. It was only when a lot of other factors came together, both technological and social, that we could have the effective equivalent of the videophone.

I see similar arguments playing out around the idea of self-driving cars, right now. Lots of people are distrustful of them, in the same way lots of people are scared shitless of flying because of the possibility of a plane crash, but think little of speeding to the airport if they're a little late. The chance of death or serious injury is many orders of magnitude higher in the latter case, but people focus on the former possibility.

Right now, in the US, about 40,000 people die each year in auto accidents. That number's fuzzy, but let's use it for arguments' sake. You can also find statistics by miles driven if you look for them; I can't be arsed. Logically, then, if we can switch to self-driving cars and get that number down by at least one or two standard deviations - say to around 32,000 - then autonomous vehicles could be shown to be inherently safer than human drivers, and in a perfect world, we'd just switch over.

But an autonomous vehicle killed one pedestrian, and the consensus became "we can't have these things; they're too dangerous." We're apparently perfectly okay with killing 40,000 of ourselves every year rather than reducing that number by changing the way things work.

People, in short, are fundamentally illogical. And that is why you shouldn't trust anyone who says "this is the way things will be in 2030." Not even me. There's no telling how the public will react to a change, either from without or within.

But I will make one prediction that you should definitely take seriously: People are not going to fundamentally change.

And that's why we continue to need science fiction: to prepare us non-changing meatsacks for the things that do change, be it technology, scientific discoveries, unexpected breakthroughs, alien invasion, the zombie apocalypse, 40 foot sea level rise, nuclear annihilation, or the inevitable takeover of all forms of entertainment and government by Disney.

Don't worry, darlin'
No baby, don't you fret
We're livin' in the future
And none of this has happened yet

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