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 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.

June 3, 2022 at 12:02am

#1033272

Like a Version

Note: Tomorrow's entry will be delayed. I'm going to a beer festival in DC tomorrow. Assuming no one opens fire into the crowd (this is America, so it's a nonzero chance), I'll return home on Saturday and then blog. Between expecting to be drunk and having a drunk friend to deal with too, I doubt I'll be able to get online at my usual time.

So, like, time for an article about language.

Why do people, like, say, ‘like’ so much?

Link is from the Guardian so the spelling is, like, British.

Why do people have such a problem with “like”? Is it because it simply won’t go away? In 1992, Malcolm Gladwell wrote a robust defence of the word and the way it carries “a rich emotional nuance”, responding to what had already been a decade of criticism. This did nothing to settle the debate.

Some people get annoyed over the stupidest things. This is one of them. Who cares?

You know what other word gets overused and we don't think about it until it's pointed out? "Up."

Scores of recruitment specialists and public-speaking coaches have publicly bemoaned the word’s rise and say those who use it prevent themselves from getting opportunities.

I don't know what a recruitment specialist actually does, but I think public-speaking coaches are the people that try to get us to stop saying "ummm..." too. Which does nothing to actually promote public speaking; sometimes a speaker has to pause for thought without giving the audience an opportunity to think she's stopped speaking.

Sure, if it becomes a vocal tic, perhaps then it's truly annoying, as with any other word or phrase. Until then, to me, using "like" as an intensifier is merely the sign of a casual conversation. Hell, I even use it in my writing in here. Deliberately.

In 2010, Emma Thompson complained to the Radio Times that she “went to give a talk at my old school and the girls were all doing their ‘likes’ and ‘innits?’ which drives me insane… I told them ‘Just don’t do it. Because it makes you sound stupid.’”

Well... "innit" is unique to England, as far as I know, and it really does make you sound like a chav.

There’s certainly an element of sexism here and the detractors of “like” say it makes you sound girlish and stupid, arguing that this is a newish tic said mostly by women and that it’s a meaningless “filler” word that doesn’t add anything to a sentence’s meaning. But they are, in fact, wrong on every count.

As the article eventually points out, the liberal sprinkling of "like" was popularized in the US in the early 80s, the whole Valley Girl thing. But who made men the gatekeepers of language?

The first of these is the quotative “like”: “He cooked a spag bol for me last night, I was like, that’s delicious.”

Maybe this is because I'm American, but "spag bol" would enrage me way more than the use of "like" there.

The other hated “likes” are as a discourse marker, “What did I do last night? Like, had dinner, hung out”; an adverb to mean approximately, “It was super quick to cook, like 30 minutes”, and what’s known as a discourse particle, which goes in the middle of a phrase, rather than at the end of it, “This dinner is like the best I’ve eaten.” But there are more uses than that, for example the Geordie tradition of finishing sentences with a like. “He cooked dinner for me, like,” and increasingly “like” is also used as a noun because of Facebook and Instagram, “I gave it a like.”

Always good to know what part of speech you're using. As for the social media aspect, hate the media, not the noun.

In Shakespeare’s Twelfth Night, written at the start of the 17th century, Valentine says to Cesario, “If the Duke continue these favours towards you, Cesario, you are like to be much advanced.” The linguist Anatoly Liberman says that this version of “like” was being used as a shorthand for likely, and may be the beginnings of our contemporary usage.

Here's where they lose me. I'd need a bit more support for that hypothesis. The dialogue isn't "You're, like, gonna get promoted," but as noted, a shorthand for "likely," perhaps to fit better in iambic pentameter.

But the biggest lie about “like” is that it’s stupid; that it adds nothing to the meaning of a sentence. “People say language is random. But language is almost never random. You can’t just stick that like in anywhere,” says Fought. “So for example, if I say, ‘Oh look at that boy over there. He’s wearing a top hat. And he’s like, 10.’ That makes perfect sense. But if you say ‘How old is your brother’? And I say ‘He’s like, 10’ that’s a little more unusual. Or if I said, ‘My, like, grandma died.’ That’d be a very strange context to hear it. So there’s patterns. There’s ways to do it more grammatically.”

English evolves, but as with any evolution, there are constraints.

So if linguists are largely agreed that “like” is, at least in some contexts, no bad thing, why does society still bristle at it? Katherine D Kinzler, the author of How You Say It, a book about linguistic bias – which she argues is one of the most persistent prejudices in our society – says that taking someone to task for the way they speak is one of the last societally accepted ways to exercise our prejudices.

I can buy that. I'm in the South, but near the cusp of North, and even around here, a lot of people think a Southern accent marks someone as having lower intelligence. This ain't necessarily so. Sure, plenty of Southerners are dumb as dirt. So are plenty of everyone else.

...the best linguistic studies today suggest people who say “like” may actually be more intelligent than those that don’t. One, published in the Journal of Language and Social Psychology, which examined 263 conversational transcripts, found that “conscientious people” and those who are more “thoughtful and aware of themselves and their surroundings” are the most likely to use discourse markers such as “like”.

"May" is the key word there. (And don't get me started on may vs. might

.) One study does not science make. But it's at least a falsification of the premise "all likers are dumb."

Anyway, just another thing I thought I'd, like, share with y'all.

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