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.

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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.

August 10, 2020 at 12:03am

#990401

The The

Today's article is about an article.

https://www.bbc.com/culture/article/20200109-is-this-the-most-powerful-word-in-t...

Is this the most powerful word in the English language?

Personally I would argue that "fuck" is the most powerful word in the English language. I mean, consider how people react to it, and its versatility. Here on WDC, it's the dividing line between 13+ and 18+ ratings. Well... along with certain other words, but I've already used up my sexually-derived word quota now. So on with the article.

‘The’. It’s omnipresent; we can’t imagine English without it. But it’s not much to look at. It isn’t descriptive, evocative or inspiring. Technically, it’s meaningless. And yet this bland and innocuous-seeming word could be one of the most potent in the English language.

Long ago, I looked up the definition of the word "the." Since I've been drinking and it was long ago, I just looked it up again. The dictionary entry is entirely too long to quote here, so here it is.

Words are split into two categories: expressions with a semantic meaning and functional words like ‘the’, ‘to’, ‘for’, with a job to do. ‘The’ can function in multiple ways. This is typical, explains Gary Thoms, assistant professor in linguistics at New York University: “a super high-usage word will often develop a real flexibility”, with different subtle uses that make it hard to define.

One thing I've noticed in my studies of French, as well as in my very surface skimming of linguistics in general, is that different languages use articles differently. The link above from BBC goes into this to some extent, but, for example, if you ever want to fake a really bad Russian accent, just eliminate articles entirely. "Is glorious day in America, comrade." I mean, come on. Just try to say that without sounding like a cartoon Russian.

Helping us understand what is being referred to, ‘the’ makes sense of nouns as a subject or an object. So even someone with a rudimentary grasp of English can tell the difference between ‘I ate an apple’ and ‘I ate the apple’.

This author might be surprised at what even native English speakers are entirely ignorant of, at least over here on this side of the pond.

There are many exceptions regarding the use of the definite article, for example in relation to proper nouns. We wouldn’t expect someone to say ‘the Jonathan’ but it’s not incorrect to say ‘you’re not the Jonathan I thought you were’.

I have a vague memory of the first time I saw or heard "the" used before a proper name. I don't really remember what the name was, but the context was something like: "So, you're the James T. Kirk." It sounded weird then, and it sounds almost as weird now.

This simplest of words can be used for dramatic effect. At the start of Hamlet, a guard’s utterance of ‘Long live the King’ is soon followed by the apparition of the ghost: ‘Looks it not like the King?’

Of course my mind immediately went to a Monty Python bit from Holy Grail: "Hey look, it's the King." "How do you know it's the King?" "He hasn't got shit all over him."

‘The’ can even have philosophical implications. The Austrian philosopher Alexius Meinong said a denoting phrase like ‘the round square’ introduced that object; there was now such a thing. According to Meinong, the word itself created non-existent objects, arguing that there are objects that exist and ones that don’t – but they are all created by language. “‘The’ has a kind of magical property in philosophy,” says Barry C Smith, director of the Institute of Philosophy, University of London.

Don't get me started on the philosophy of linguistic paradox and the nature of reality. Oh, hell, I have plenty of blog entries about that, in the past and probably in the future as well. But I'll say this here: "a round square" brings the paradoxical object into being, at least Platonically, just as thoroughly.

The British philosopher Bertrand Russell wrote a paper in 1905 called On Denoting, all about the definite article. Russell put forward a theory of definite descriptions. He thought it intolerable that phrases like ‘the man in the Moon’ were used as though they actually existed.

All due respect to Russell, he of teapot fame, you can also speak of Santa Claus, sans definite article, as if he actually existed. And there is a huge difference between "god" and "the god."

Scandinavian languages such as Danish or Norwegian and some Semitic languages like Hebrew or Arabic use an affix (or a short addition to the end of a word) to determine whether the speaker is referring to a particular object or using a more general term. Latvian or Indonesian deploy a demonstrative – words like ‘this’ and ‘that’ – to do the job of ‘the’. There’s another group of languages that don’t use any of those resources, such as Urdu or Japanese.

My vague, distant memories of learning Latin tell me that Latin didn't have a definite article, either. I could be wrong about that, though. But it does make me wonder why French, a Romance language, uses one (actually three: le, la, les) in ways we'd never dream of in English -- and then omits it in ways we'd never dream of in English.

French is a mess. From an English-speaking perspective, of course. Objectively, it makes so much more sense.

Conversely, Smith describes Russian friends who are so unsure when to use ‘the’ that they sometimes leave a little pause: ‘I went into... bank. I picked up... pen.’

We go to party, drink vodka, yes?

In some ways, this cartoon Russian accent makes Russians sound uneducated, kind of like a Southern accent in the US. Nothing could be further from the truth. They just don't have the dependence on articles that English does.

Even within the language, there are subtle differences in how ‘the’ is used in British and American English, such as when talking about playing a musical instrument. An American might be more likely to say ‘I play guitar’ whereas a British person might opt for ‘I play the guitar’.

One of the things that messed me up in English when I was a kid was that I'd devour British books right alongside US ones, so I was always confused about which side of the pond some word or phrase, or spelling, comes from. Didn't help that some of my favourite bands were British. Dammit, there I go... "favorite."

They invented the language. We perfected it. Still... there are many British expressions that we might do well to adopt.

According to Culpeper, men say ‘the’ significantly more frequently. Deborah Tannen, an American linguist, has a hypothesis that men deal more in report and women more in rapport – this could explain why men use ‘the’ more often.

Oh, shit, here we go. Okay, no, I'mma stop this right here and move on to the next subject.

The letter y in terms like ‘ye olde tea shop’ is from the old rune Thorn, part of a writing system used across northern Europe for centuries. It’s only relatively recently, with the introduction of the Roman alphabet, that ‘th’ has come into being.

I have other vague memories of doing a whole newsletter on thorn (þ) a while back. Fantasy newsletter, maybe? Of course, its use wasn't just in the word "the," but it was, like, the Nordic version of the Greek theta. In Iceland, if I recall correctly, it's actually still a letter in their alphabet, whereas in English it's been completely dropped from the alphabet and its sound replaced by "th."

What intrigues me most, though, about "the" is this: you can pronounce it "thee" or "thuh," depending on what comes next, but the rules for that are not nearly as clear-cut as the ones surrounding the indefinite article (where it's "a" before a consonant and "an" before a vowel).

Never did fully understand that, but then, if you devote too much time to thinking about "the," you might just wrap your head into a pretzel. And that would be a thorny situation, indeed.

Oh, and just because it would be wrong of me to leave this blog entry here without some explanation for the title:

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