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 6 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 8 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.

July 21, 2022 at 12:06am

#1035507

Punctuated

Today, June 21, is the anniversary of the greatest achievement of humankind (apart from beer): the first dudes on the moon. Civilization has mostly gone downhill ever since, though the internet sure is nice.

The date is sometimes given as July 20. That's because a) the landing was a few hours before the moonwalk, and b) the timestamp for that first boot on the moon was 0256 UTC, which is London time, but as the rocket was launched from Florida, sometimes it's recorded in US Eastern Time, four or five hours earlier (I forget if DST was in effect in 1969).

Today's article has nothing to do with that, but I couldn't let it pass without mention. I wouldn't expect any of my readers here to make such rookie mistakes, but it never hurts to have a refresher. Plus, I have jokes and commentary.

Don't Make These 11 Punctuation Mistakes That ‘Irritate People the Most,' Say Grammar Experts

Want to irritate a young person? Text in complete sentences and put a period at the end. That drives them absolutely up the wall, and it cracks me up.

The article's slant has to do with job hunting, but the rules are universal. Yes, even in texting. As an aside, the reason I mentioned the young person thing is my friend's kid, now all grown up and about to go to college, texted me out of the blue the other day. I was pleased to note that his texts were properly punctuated. I can only assume the college was impressed by his punctuation skills. He even mostly eschewed the use of emojis, but that "mostly" is forgivable; even I use them sometimes. But I get the impression he's an outlier in the texting world.

Good punctuation can go a long way when it comes to job hunting and being taken seriously at work. The average recruiter spends about 7.4 seconds scanning a resume.

Oh, it's worse than that. When I was running a business, I'd share the most egregiously ungrammatical resumes with my business partner, and we'd have a good laugh before hiring someone else. "But Waltz, since when do engineers have to know how to write?" Since forever, because written reports are a thing, and I didn't want to always be the one writing those boring bastards.

Based on comments and responses from our readers and listeners, here are 11 common punctuation mistakes that irritate people the most:

We get these articles from time to time, but bad punctuation lives on somehow. As far as I'm concerned, if you can't get these basic things right, either you don't care, or you're deliberately trolling; either way, I'll ignore you.

I'm not going to comment on every one of these. Check the article if you want to see others.

2. Misusing "it's"

We hear managers rant about this mistake so often that we decided to talk about it separately: Apostrophes show possession, but there's a huge exception: Its and it's.

Really, this applies to any pronoun. My other annoyance is when people mix up there, they're, and their. Or there's and theirs. Or your's. Worse is when Southerners type "ya'll." Y'all oughta know better. And what's up with people typing "loose" when they clearly mean "lose?"

3. Incorrect quotation marks

About a month ago, on a rare venture into the not-so-great outdoors (actually downtown, but close enough), I saw a chalkboard sign that read something like:

"TUTU"
Turkish Street
"FOOD"

I changed the name to protect me from lawyers. Now, I'm sure if they had food, it would be delicious. But they don't. They have "food." Though their happy hour beer prices (also posted on the sign) almost tempted me into the place anyway, before I started to wonder why a Turkish place is selling beer.

6. Using semicolons like commas

I probably overuse the semicolon; I'm aware of this stylistic flaw. But I think I use them properly—most of the time. One of the hardest things about learning French, to me, isn't the words or the grammar; it's that they use commas where we would use semicolons, and omit commas where we would have them. It makes sense within the context of the language, but as the rules are different, it took me a while to figure them out. Also, what's up with the space before the ? or ! ?

7. Putting two spaces after a period

Back in the old days of typewriters (and early computers), there was a hard and fast rule: Two spaces after a period.

You know, I'm kind of a stickler for grammar, but that sort of thing doesn't bug me in the slightest. It's a visual thing, and it's the way I learned it, as I learned how to type before personal computers came out. I did unlearn that, but if someone else hasn't? Whatever. I think the reasoning is that the two-spaces thing was helpful to distinguish the end of a sentence from, say, "Mr. Magoo." And it became obsolete with self-kerning fonts, or whatever you call them. Fonts that don't have the same space for every letter. Like the one you're reading now as opposed to this one.

10. Capitalizing too many words

Sometimes people capitalize a word to make it stand out. But not every word can or should be capitalized.

There's a time and place to do that, mostly for comedic effect—which is not something you want on a resume unless you're applying for a court jester position. Another oddity of French as compared to English: they don't capitalize a lot of words that we do. For instance, we type Monday; they type lundi. We say July, they say juillet. This is in contrast to German, which as far as I understand it, capitalizes every Noun.

11. Using too many exclamation points

Hell, using any bangs in a resume, unless you worked for a company whose listed name is something like Yahoo!, would be a hard pass for me.

I'm going to add one:

Using emojis, or their archaic text equivalents. Again, there's a time and place for that, and it's not on a work document. I once got a design plan comment letter from a municipality which was just riddled with smiley faces and (even more) frowny faces. Stop that. You're a professional. I'm a professional. We can communicate without you trying to ease the sting of your criticism by making funny faces with colons, dashes and parentheses.

Like everyone, I make mistakes now and then. Sometimes I'll catch them on rereading; sometimes I won't. So I don't sweat the occasional mistake of others, because I generally don't hold other people to higher standards than I do myself. And like most grammar rules, there are occasions to ignore the rules.

A resume isn't one of those.

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