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 11 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 13 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.

June 11, 2020 at 12:04am

#985430

I'm Not Saying It's Aliens

Science is cool.

Science reporting needs some work.

https://www.cracked.com/article_27801_sorry-but-discovery-parallel-universe-bs-t...

Sorry, But 'WE DISCOVERED A PARALLEL UNIVERSE' Is A BS Tabloid Headline

This crap was making the rounds before the riots, and then, understandably, got buried. Though, I suppose, it could be argued that we all woke up on January 1 in a parallel universe, where nothing makes sense anymore. But I still have my goatee, and everyone else doesn't, so there's that argument against it.

Breaking: Scientists have 100% proven that parallel universes exist and that there's one out there where time runs backward, whatever that means. So instead of COVID, they've got a Benjamin Button pandemic ... Except scientists didn't actually prove that at all.

Yeah, that's the story that was circulating, or close enough.

The study revolved around the movement of high-energy particles called neutrinos. They pass through solid objects all the time and even pass through us trillions of times a second (At least buy us dinner first.)

Neutrinos are weird, and they're interesting enough without having to make stuff up about them. For instance, in addition to not interacting very much with the matter we're made of, they're shapeshifters, able to change mass as they travel. I, too, change mass as I travel, but not as quickly, and unlike me, neutrinos don't pick up the extra weight from McDonald's but (probably) from the Higgs Field.

There's this part of a lot of scientific studies where the researchers propose explanations for the unexplainable phenomenon they've observed. Other researchers around the world understand that it's just spit-balling to get everyone's minds churning with possibilities.

Yeah, this is where we get the disconnect between reality and speculation. It's like when you see weird lights in the sky and say "maybe it's aliens" and you tell your friends and they scoff at you for believing in aliens. I didn't say it was aliens; I said that's a possibility, the same way the poop in your yard is maybe from unicorns but it's way more likely it's from the neighbor's dog. Does it look like a rainbow and smell like roses? No? Okay, discard the unicorn hypothesis and put up a fence already.

Tabloid newspapers don't understand that, so they publish the purely theoretical musings as established fact.

It's not just tabloids. Even some respectable publications like Cracked make this mistake sometimes.

Don't get me wrong; parallel universes are not outside the realm of possibility, if you pay attention to certain interpretations of quantum mechanics, or start wondering why it is we're here and not somewhere else or nonexistent. Hell, it might even explain why neutrinos shift they way they do (but then again it might not).

I was reminded of this recently when, plastered all over some site that Google keeps sending me to because it thinks I'm interested in science for some odd reason, was another "Astronomers Discover An Earth-Like Exoplanet!!!" headline.

No. No, they did not. First of all, to an astronomer, Venus is Earth-like. But reporting on it like that makes the reader think they've found Gallifrey or Risa or some shit, and we finally have a goal for when this planet becomes unbearable to live on, sometime later this month. But a little digging told me that, while the planet in question does indeed seem to orbit within its star's habitable zone and might be relatively Earth- or Venus-sized, it might not exist at all, and there's nothing in there about showing any signs of actually harboring life. Right now it's just a blip in the data, and that's all.

I've little doubt that, somewhere in the galaxy, there's at least one other planet with conditions we'd think of as Earth-like. But no, we still haven't found one.

This sort of thing only increases the general public's distrust of science, and that shit needs to stop. I get that you want readers so that you'll get ad revenue, but in the long run it's going to bite you in the ass.

But hey, then you can always blame aliens.

4 Like • Comment • View 2 comments

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