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:

Turkey DrumStik

about 7 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

November 10, 2018 at 1:41am

#945262

Essay It Ain't So

https://www.thoughtco.com/write-the-perfect-personal-essay-3858745

The assignment is really quite easy, it's about you after all, so this is your opportunity to shine!

The darkness does not shine. The darkness broods. And maybe smoulders a little bit, but mostly broods.

I always hated writing essays. Probably that's why I went into engineering instead of pursuing a liberal arts degree. No... wait... that was the money.

1. Understand the Composition of an Essay

The five-paragraph essay form was the bane of my high school existence. Well, that and jocks. Well, mostly jocks, but also the five-paragraph essay form. It was especially bad when I had to write one in Latin. Semper ubi sub ubi, et non illegitimati carborundum.

2. Find Inspiration and Ideas

Ever see one of those "how to draw" sketches? Like, "How to draw a puppy."
1. Draw an oval for the body.
2. Draw a smaller oval for the head.
3. Fill in the minor details to make it look like a puppy.

That's what I think of whenever someone urges me to "find inspiration." Like it's that easy. I can't draw worth a damn, so I get stuck on Step 3 and it ends up looking like a Jackson Pollock puppy. I can write well enough, I think, but... "Find inspiration and ideas." Yeah... right. You know what I noticed today? My deck is covered with leaves. I can either let them sit there, or I can rake or blow them off. If I let them sit there, eventually a wind will come by and remove them, or they'll rot away. If I remove them, more will take their place and I'll have to repeat the chore; it's still fall.

That's the extent of my inspiration and ideas for today: justification for being lazy.

(As an aside, it's both heartening and disheartening that this section of the linked article recommends doing something I've been doing for a while, the "core dump" I outlined two entries ago. Heartening because someone else thinks it's a good idea. Disheartening because it was MY IDEA DAMMIT.)

3. Freshen up Your Grammar

My grammars are dead, you MONSTER. Oh, wait, that kind of grammar. Well, that's always been pretty easy for me. That and spelling. This is despite my predilection for incomplete sentences in blog entries, which, in my defense, I know I'm doing; I use them in an attempt to strike a conversational tone.

Several years back, I heard a teacher - an actual, real-life, primary school teacher - say that she doesn't teach spelling or grammar, because "spelling and grammar checkers will take care of any mistakes." No wonder so many people can't tell the difference between "its" and "it's," or "there," "their," and "they're." Fuck off, teacher - spelling and grammar checkers still don't catch those kind of errors, and now they're ubiquitous. The errors, I mean. Not checkers. Damn pronouns and their ambiguous antecedents.

4. Use Your Own Voice and Vocabulary

No, this is a really bad idea. Well, they're right about active vs. passive voice, but to me one of the great things about writing is being able to stretch one's vocabulary, learn new words, and figure out new ways to use old words.

You know what I haven't used for 30 years or more? A thesaurus, that's what. I don't even know any synonyms for thesaurus. Unfortunately, this shows in my writing. My weakness isn't spelling, grammar, or punctuation (though I know those are not perfect), but repeated words in my first drafts. They're hard for me to catch, since I wrote them in the first place. Since I suck at editing, they tend to stay there.

5. Be Specific with Your Descriptions

This... well, this is actually good advice for writing anything.

6. Be Consistent with Your Point of View and Tense

So is this. Goddammit, get back to stupid shit so I can snark on you.

7. Edit, Edit, Edit

Ah, that's better. There is such a thing as overediting, you know. Even if my spell checker doesn't recognize the word "overediting." Anyway, I'm not one to give editing advice. But I know an ad for EssayEdge when I see one, and this is an ad for EssayEdge. Seriously, does everything have to be a freaking commercial?

8. Read

And if you've gotten this far, a) congratulations and b) you already know to do this.

Fortunately, blogging is usually utterly distinct from the personal essay format, at least for me. Here, I'm just trying to make jokes and a point. I'm not trying to sell myself or prove how clever I am (though, to be fair, I am rather clever - at anything except selling myself). If people like it, great; I appreciate that. If not, oh well - I can't please everyone and I quit trying a while back. In any case, I'm fine with improving my skills and learning new ones, but if I'm forced to write in a certain format, it feels like work.

And I'm allergic to work.

Now, to get back to brooding...

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