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 6 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 8 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.

May 19, 2021 at 12:13am

#1010404

Adventure

You know what else it's wonderful to get back to doing?

I'm sure you know what I'm about to say. Yep, you've got me completely figured out. But I'm going to say it anyway.

Brewery visits.

Yesterday a friend and I visited two local breweries that, somehow, neither of us had ever set foot in before, and they were around in the Before Time so I have no excuse.

As is my usual procedure in such cases, I got a tasting flight of a variety of different beers. And with one exception, they were all really damn excellent beers (the one exception was just a really excellent beer).

Now, I don't want to give anyone the impression that I like all beer. I do not. For example, and I'm sure I mentioned this in here before but what the hell, I was presented with a lime-salt lager a few weeks ago that was utterly foul. But all that does is make me appreciate the good stuff that much more.

Still, not only did I like the beer, but now I'm going to have to go traveling again just so I can find more new-to-me breweries to try. I'm out of local options. Though that doesn't preclude me going back to those.

Edit:

PROMPT May 19th

Write about the most historic world events of your lifetime. How are they (or will they be) remembered? What will the history books focus on?

I know I've written about it before, but during my lifetime, the one historic even that stands out as the "most" is the first moon landing in 1969.

Certainly there have been other important events, and will be in the future, but in my view, the landing of humans and their technology on a separate world for the first time is not only the most significant thing we've ever accomplished, but possibly the most significant thing we can ever accomplish. Oh, sure, maybe we'll plant feet on Mars or soar through interstellar space, but there can only have been one "first time."

Of course, the way history books are trending, they won't see it that way. Possibly they already don't. No, instead of acknowledging this amazing feat of science, technology, teamwork, courage, curiosity, and sheer human determination, they'll probably spin it as:

- It wasted resources that could have better been used for other pursuits (it didn't).
- The effort wasn't inclusive/diverse enough (fair, but irrelevant).
- It was a vast international dick-measuring competition (also fair but irrelevant, as the competition produced results).

Worst of all, though, are the ignorant cockwads who deny that it ever happened at all, that it was a huge hoax. I'm not even going to bother to try to go into the reasons why they're wrong, and I fully admit that at this point some of them are just trolling for attention, same as many flat-Earthers. But I will mention why it's utterly maddening: because to deny that we managed to put a human on the moon not only denies the significance and importance of the accomplishment, but also denies human achievement itself. Denying it is also tantamount to saying: "we are human and thus incapable of great things," so it also denies something basic about our nature and what we are capable of.

Well, I've banged on enough.

I will say, though, that a close second to the moon landing in terms of historical events of great significance was the day in 1978 when Jimmy Carter signed legislation permitting home brewing, which eventually led to a proliferation of commercial craft breweries.

But I fully admit that this one may just be my personal opinion.

4 Like • Comment • View 5 comments

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