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 8 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 10 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.

January 26, 2024 at 10:22am

#1062993

Today is Satireday

Humor is humor... until someone takes it seriously. Crack reporting from Cracked today:

5 Concepts Everyone Takes Seriously But Were Proposed As Jokes

Setting aside my quibble with the "everyone" in the headline...

The world could have been saved a lot of trouble if, centuries earlier, people decided to end every sarcastic suggestion by writing out /s.

Someone should retroactively print that at the end of every copy of A Modest Proposal.

Also, why hasn't my idea of making Comic Sans the Official Sarcasm Font caught on yet?

Sometimes, you see, people aren’t being serious.

Sometimes I am. Sometimes I'm not. Sometimes, both at the same time (see #3 below).

5. Meritocracy

This is one of the few portmanteaux (that I didn't come up with myself) that I can abide. Partly this is because the word is older than I am, and thus, by definition, a real word. And partly because both parts of it come from the same language roots (Latin).

When you toss out that word “meritocracy,” however, you may want to keep in mind where it came from. It entered our lexicon thanks to a 1958 book called The Rise of the Meritocracy. Author Michael Dunlop was a sociologist, but the book was no straightforward sociological text. You might instead call it dystopian sci-fi, or an alt-history novel. It takes place in the year 2034 and looks back at two centuries of increasingly fictional history leading to an imminent revolution.

In summnary, it was meant as a warning, not a reflection of reality or a roadmap for the future. I haven't read the book, but it wouldn't be the first time, or the last, that a book set in the future, intended as a warning, became something people started striving for. I mean, look at half of SF.

4. Daylight Saving Time

This article was from last October, when DST was still active most places. It won't resume here until March 10. So, right now, it's either a bit early or a bit late to start with the endless complaints about time changes, but I work with what the random number gods give me.

Some people, however, credit the invention to Benjamin Franklin, who first proposed the idea all the way back in 1784.

Franklin, writing as he did before Comic Sans was a thing, often proposed things satirically that ended up being taken seriously.

Franklin wrote to The Journal of Paris about how a noise recently awoke him at six in the morning. This was an unusual experience for him, because he normally awoke at noon, having arrived home drunk at around four. Shockingly, the sun was rising when he woke this time, illuminating his room and eliminating all need for candles. When he shared this discovery with several scientist friends, none believed him because they, too, always woke at noon. Perhaps, suggested one, he was confused because he’d mistakenly opened his window and let the darkness out.

See? This, from the man who's also credited with the "early to bed, early to rise" proverb? Guy was an epic troll. You know how you sometimes get questions like "which historical figure would you most like to meet?" in, like, job interviews or dating apps? My answer would be Ben.

3. Schrödinger’s Cat

If this hadn't been on the list, I'd have mentioned it anyway.

Here in the quipping biz, we have a few stock references we drop into commentary to convince readers we’re smart. Some are literary (“The Cask of Amontillado,” The Picture of Dorian Gray), while some are scientific, the chief among these being Schrödinger's cat.

I've known people who thought Erwin had used an actual cat. I'm not judging them, because many a cat has died/not died in the name of scientific advancement, but maybe the rest of us need to be more clear that it was a thought experiment

Of course, a cat cannot actually be both dead and alive simultaneously, but that’s the point of the thought experiment. Schrödinger proposed this to be ridiculous.

While I'm not a quantum mechanic myself, I like to watch them work on my quantum cars. One interpretation of quantum physics is that the state of any particle is unknown until it's observed. Philosophically, this has never sat right with me, because it seems to give primacy back to human consciousness alone, but like I said, not an expert. Problem is, the cat would know it's alive (presumably if it's dead, it wouldn't be in a position, super or otherwise, to know anything).

2. Hiter’s Nobel Peace Prize

The Nobel Peace Prize is a sham, according to cynics who’ve never done anything.

The Nobel Peace Prize is a sham.

To summarize this section, he never actually got the prize, and he was nominated as a joke. Kind of like those epic comedians who write in Mickey Mouse on the voting ballot.

1. The Trolley Problem

There's too much here to pick at any one thing, so you might want to read it at the link, which also contains other links to supporting documentation. Or you might not. Or you could be in both states at the same time.

Suffice it to say that, proposed satirically or not, the thought experiment, with variations, can be useful. Like inspiring this short story I wrote in 2021: "The Trolley Problem"

[18+]

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