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:

Turkey DrumStik

about 8 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

November 30, 2018 at 12:47am

#946577

November's End

I've been hanging around on the internet for a long, long time.

While I can't say I was an early adopter, I was browsing newsgroups and visiting websites at some point before 1995. I don't recall exactly when. One of the first things I got into was IRC, which had bugger-all to do with the World Wide Web, but it was social media before social media was even a thing.

I chatted with a lot of people there, most of us hiding behind aliases for our own protection. I even met some of them "IRL," as we called it. Some of them are still my friends.

"IRL" was early webspeak for "In Real Life." There was, then, this sense of separation between meatspace and cyberspace, that anything that happened online didn't really count.

I knew better from the beginning.

These people I spoke with were just that - people. No different from the folks I'd meet in my job or activities, except that they were spread out all over the world. While the internet wasn't as ubiquitous then as it is now, it was widespread enough so it wasn't just other people from the US that I'd be talking to.

That, to me, was the draw. I think a lot of people go on the net looking for like-minded individuals - and there's nothing wrong with that; I do it as well - but I was always open to another perspective, possibly from people completely different from me.

Inevitably, I'd find that they weren't so different, after all. At least, we could find some common ground. Age, gender, race, nationality, sexual identity, religion, favorite sports team, whatever - all that becomes secondary to who the person is.

Now, it's often different. People identify themselves by how they fit into these categories. That's okay, but it does focus on what separates us rather than on what unites us.

It seems to me that everyone wants to be unique in some way. That's easy enough if you've only got a few other people around. When I was in high school, I was one of about 300 in my graduating class, 1200 total in the school. I went to university and I was but one out of 20,000. After that, when competing for jobs, I was one of 40,000.

On the internet, I'm one of 7 billion. And so are you.

Excelling in a milieu like that is tantamount to winning a lottery. It's not just a matter of not being a big fish in a small pond; it's more like being a piece of plankton floating in the Pacific.

So there's this tension - a sense of competitiveness. A pull between wanting to be unique and wanting to fit in. This woman has 20,000 Tweeter followers. This man's YouTube channel has 50,000 subscribers. What can the rest of us do except follow along?

I figured out early on that there's nothing that I can do that someone else hasn't already done, and better than I could possibly do it. Nothing I can say that hasn't already been said, in a more elegant or satisfying manner. Nothing I can invent that someone else hasn't already patented and started making money on. No jokes that haven't already been told to death. And nothing I can write that would ever stand out amidst the teeming billions of writers.

The best I can hope for is to connect with someone, occasionally. And I'm okay with that. So thanks for reading, and - specifically addressing this to others in the November blog challenge - thanks for writing. I'm supposed to cite one new fact I've learned about a fellow blogger this month. Incidentally, just to see if you're actually reading this, anyone who comments on this entry before the end of today based on website time will get a season ticket chosen at random. Just throwing that in here as a thanks for reading this far. So anyway, one new fact about a fellow blogger?

That's tough, because, as with the early days I mentioned above, I've learned a lot of things about a lot of people. I don't think I really knew most of you other bloggers before this started; of course, I don't think I really know you now, apart from what you've chosen to show us in your writing. If I mention one author, or a few, I feel like I'm shafting the others. But mentioning everybody would be an even more daunting task. Just know that I've appreciated reading all of the different entries and seeing everyone's perspective on things.

So I don't even want to tag anyone here; therefore, I'm not going the {user:name} route. If I have to call out just one person, I'll make it pwheeler. She and I are obviously very different (well, except for the cat thing), and yet I've learned a few things from her entries, notably about food. And food, as you know by now, is one of my favorite things, as long as it isn't too healthy.

Just another example of finding commonality, I suppose. We could do with more of that. Arguments and debates have their place, of course, but in the end, aren't we all just people trying to make our way through the crowd? Don't answer that - someone's bound to say "no, some of us are lizard aliens from Antares 7" or something.

That someone is usually me, so I'm nipping that right in the bud.

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