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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 3 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 5 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.

April 4, 2021 at 12:01am

#1007712

Past Intense

It always bugged me when people would romanticize the past. I mean, okay, technically, there was a period called the "romantic" period, and look, let's face it, "romantic" has WAY too many definitions, mostly incompatible with each other.

What I mean is the people who look at the past all starry-eyed and breathless and go, "I wanna live THEN." And look, I've had friends in the SCA, I read and write fantasy, and I play fantasy role-playing games. Know what all of those things have in common? "Fantasy." I can still take advantage of modern sanitation, electricity, and medical facilities.

Leave it to Cracked to make a numbered list of exactly why the past completely sucked syphilitic ass.

55 Facts That Show The Past Was Absolutely Gross

Aren't you happy to live in the modern, hygienic world that exists today? A world where you can easily wash your hands anytime you like, and must disinfect yourself with sanitizer every time you enter a building, and wear a mask to limit germ transmission, and submit to continual temperature checks in order to walk around publicly ... uh, all right, some of this might feel a tad excessive. But wow, is it infinitely better than the alternative you'd have to deal with if you lived centuries ago.

Okay, so maybe the present isn't exactly what you'd call "perfect."

I won't be copying all 55 of these. Of course. Go check out the link to find examples even more egregious than the ones I'm highlighting here. Especially the first one, which I'm going to skip right over.

2. Cat Piano

Johann Christian Reil arranged cats so hammers hit their tails, and they would cry out. Somehow, this was supposed to jump-start patients back into sanity.

I don't believe in an afterlife. But if I'm wrong, I want it to be run by cats, with this guy hit eternally by hammers. On his "tail."

11. Splinter Butt

Before we settled on toilet paper (or the convenient anal hose), people as recently as the 1930s were wiping themselves with rough stuff that gave them splinters.

About a year ago, this was looking like it was going to make a resurgence.

17. Fire Toilets

Roman toilets would occasionally erupt in open flames.

Huh. I didn't know they had Chipotle back then.

21. Castration Cults

Rome had a cult based on the Greek goddess Cybele, and adherents modeled them after her consort, a eunuch.

Yeah, that's a truckload of nope right there.

30. Gong-Farming

Someone had to clean out medieval cesspits. That job fell to the gong farmer, who would have to jump into the pit to shovel the stuff into a bucket.

And I thought my job, which sometimes involved investigating sanitary sewer pipes, could suck.

33. Werewolf Trials

Along with the very reasonable fear of witches, Europe in the 15th and 16th centuries suspected people might be werewolves. Thousands were executed for this offense. Circumstantial evidence included proximity to tomatoes.

Hey, I was just trying to enjoy my Big Mac in peace.

42. The Big Stink

In 1858, the Thames got clogged with sewage, a heatwave set in, and the smell was so bad, Parliament had to relocate to a rural area upstream.

Odd how one never sees that year on Doctor Who.

Now, I'll admit that there are some things from the past that we might do well do emulate. After re-reading that list, however, I'm hard-pressed to think of a single one. So as much as this past year has been no fun for just about everyone, at least we're not taking shits in exploding Roman latrines and wiping our char-broiled butts with splintery wood.

Yet.

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