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 2 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 4 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.

August 13, 2021 at 12:01am

#1015575

More Cracked Pots

Time for another Cracked link.

6 Stunningly Stupid Moral Panics People Raged Over

I've lived through a bunch of moral panics. I participate in them whenever possible -- D&D for example, which, I don't know, maybe they had a point there because D&D was my gateway to learning about other cultures' gods, but I was never part of the dominant religion here anyway so who knows?

Combine a society where information is disseminated by word of mouth, add in gargantuan generation gaps, account for inevitable fear of the unknown, and you've got a potent formula for pearl-clutching ...

It's worse than word of mouth now, of course. Back when people were freaking out over D&D, there wasn't an internet, and they had to spread their bullshit by literal word of mouth. Now? It's like there's a new panic every day.

5 Communities Banned Jazz for Causing Mental Breakdowns and Spontaneous Miscarriages

Here is where I point out that the headline said "6" but the traditional Cracked countdown starts at 5. The real moral panic should be over innumeracy.

A century before jazz was dying a slow natural death, the '20s and '30s were a crazy time to be a jazz snob. Yet, something was closing up jazz joints. Was some veiled fear of racial integration shutting them down? Well, yeah ... probably. But, in addition to the racial aspect, towns viewed the musical form as an existential threat to their actual health, as if the vibes of people dancing and having fun caused psychic pollution.

I'm not a fan of jazz, but that has nothing to do with racism or believing it poses an existential threat to anything except traditional tonal scales.

Also this paragraph reminds me of a joke: Why are Baptist kids forbidden from having sex? Because it might lead to dancing.

5 Planking, the Effigy for All of the World's Angst

Oh, look, two 5s. I guess there was a tie. That explains the numeric discrepancy, anyway.

Remember planking, that dumb trend where teens laid rigid in stupid places? That one that we absolutely did not participate in. (Any social media evidence to the contrary is clearly Photoshop.)

Yes, it was dumb, but when people complained about it, we got even worse trends like the Tide Pod challenge (beyond stupid, venturing into "fatal") and the ice bucket challenge. Maybe we need to promote the Vaccine Challenge.

4 Steel Drums were Invented In Reaction to a Scheme to Annihilate Pagan Dancing

Associated today with cruises and relaxing by the beach with one too many Pina Coladas as your skin peels off listening to street performers, the steel drum has kind of a depressing history that's ignored.

Is it bad that I think steel drums are kinda cool?

3 Guitar Distortion Was Banned as an Imaginary Catalyst of Gang Violence

Wray's reverb, and suggestive title "Rumble," slang for a fight, apparently convinced those in charge of the radio airwaves across the US that the song would spark an orgy of violence.

To be fair, a decade later the US government got us embroiled into a proxy war in Vietnam, which could well be described as "an orgy of violence." Coincidence? ... yeah, probably.

2 China Banned Puns as a National Security Threat

TOO FAR.

Life in China got even more oppressive in 2014, with the government wiping puns from public display. Wordplay in media like TV and advertisements, as per the government order, is verboten. Their logic? "Uphold[ing] the Chinese spirit" and curbing "linguistic chaos." Punbelievable.

That was So Low of the Han.

1 A Stoner's Prank Caused a National Health Scare in 1970

Amid a time of social panic concerning heroin in 1969 ("Not nice!"), reporters across America issued a stern warning about a brand-new substitute for opiates. The culprit was found in your own house, soon to replace smack. According to a "federal drug expert" named Ernest A. Carabillo Jr., the illicit recipes found in underground cookbooks were children's "culinary escape from reality." Several deaths were reported. (Cue dramatic music.)

When I was a kid, some of the rumors floating around were: You can get high from smoking dried, ground banana peel; a certain popular brand of bubble gum contained spider eggs; and McDonald's hamburgers were made from worms.

I didn't believe any of these things -- I suspect people started them just to see who bites, much like trolls do today on the internet -- but hell, worms would probably be healthier than the actual McDonald's hamburgers. Delicious, tasty, perfectly cooked hamburgers...

Chattanoogan kids found a new use for their lunch. Instead of satiating their hunger, the rumor was that they spent their lunch break shooting up Kool-Aid, peanut butter, and mayonnaise. Mayonnaise to cut the pure Afghani Jif spread, obviously. Why did any sane adult ever believe this? Well, to a generation with no deep familiarity with drugs or counter-culture habits, it sounded legit. Hippies looked weird and did weird stuff with needles; kids were stupid. Sure, whatever, that's all they needed to go on.

Not to sound all racist and shit, but that nonsense is whiter than Mike Pence.

And if you walk away from this article chuckling to yourself about how stupid people were in the past about believing half-baked rumors that were floating around, I want you to remember that a large group of people right now have the sincerely held belief that COVID vaccines alter your body's DNA.

I kind of wish that were true, if it could target the part of our genome that makes us believe bullshit. But, alas, this sort of thing will last as long as humanity.

You know, at least another six months. Probably.

5 Like • Comment • View 5 comments

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