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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Brandiwyn🎶

a day ago
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').

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 18, 2021 at 8:21am

#1017641

Irregardless

Ready for a dictionary debate? Arguments about words are literally the worst!

Is ‘irregardless’ a real word? We asked our journalists as battle rages on
Merriam-Webster insists irregardless is a word – but not everyone agrees with the dictionary’s take

Disclaimers: Guardian link from last year (but relevant irregardless).

Merriam-Webster has weighed in on the debate over the word “irregardless”, confirming that it is a proper, dictionary-verified word.

Because this is the Guardian, I'm going to go ahead and assume that this sentence uses the British definition of "proper" rather than the American definition. The difference is subtle, and to provide a proper explanation I'd have to go to a dictionary... but the article questions the veracity of dictionaries, so that would be ironic.

The debate over the word is age-old (the word appeared in print as early as 1795) but continues to upset some people – teachers in particular. Evidencing the controversy over the word, Merriam-Webster’s own dictionary definition for irregardless includes a frequently asked questions section, for which the first question is: “Is irregardless a word?”

A dictionary, in English at least, is descriptive rather than prescriptive. That is, there is no Pope of English, no Diction Dictator, no Académie Anglaise. Even the phrase "The Queen's English" is misleading: the British monarch may be extremely influential, but even she (or he; man, it's going to be weird to talk about a king again) doesn't get to decide what's English and what isn't (especially here in the US; we fought a war over that exact thing 250 years ago, and won). No, English is democratic -- except it's not about majority usage, but some lower threshold.

Personally, I think the greatest addition to the language in recent times is the word "yeet." While there exist many synonyms for throw -- hurl, toss, fling, pitch, etc. -- none of them are as intrinsically hilarious as "yeet."

The past tense, by the way, is "yote." As in "That guy just yote a snowball at my car!" Sure, you can say "yeeted," but it just doesn't have the same intrinsic hilariousness.

But many neologisms are abominations that need to be stamped out. I'd provide examples, but that would be the polar opposite of stamping them out.

Anyway, my actual point here is that the only requirement for a word to make it into the dictionary is if "enough" people use it, as the article notes. And, like it or not (I don't), people use "irregardless." It's also, as further noted in the article, far from being a neologism.

I feel more strongly: you can’t just add “ir-” to a word and decree it a new word. “Irreally” is not a word. “Irractually” is not a word. I’m sure there are words that Guardian copy editors don’t know – we’re all always learning, aren’t we? – but “irregardless” is not one of them. Because it’s not a real word.

I have a somewhat vivid memory of learning that "flammable" and "inflammable" are synonyms. I was quite young, but I do remember being corrected on that subject (I mean, it's kind of important to know if something is flammable or not when you have a proclivity for playing with matches). Lots of people have come to this realization; I think The Simpsons even used it in an episode. See, most words, when they take the "in-" prefix, become their opposite, or at least something different. Compatible, incompatible, that sort of thing. Easy enough; kids pick up on that shit right away. So it's only natural to think that "flammable" and "inflammable" are antonyms. But then you realize that "inflammable" comes from the verb "inflame" and it starts to make sense.

Something similar happens with the prefix "ir-," but it's not quite as common. "respective" and "irrespective" have different usages.(amusingly, one synonym for "irrespective" is "regardless," adding to the confusion here).

Someone in the article makes a tangential point to this: "It feels like an accidental word, created by someone stuck between irrespective and regardless." But that's later.

Clearly, if enough people use a word – including irregardless – it is a word. But clearly anyone who uses the word irregardless is an idiot.

Irregardless of nationality or social status.

Apparently an entire country – the United States – is happy to use the phrase “I could care less” instead of the logical, coherent and correct “I couldn’t care less”. So, as the linguists say, all you need to validate a word is a community of speakers. It’s just that they’d all be wrong.

Bite my Yank arse. I never say "I could care less" unless I mean that I could, in fact, care less (that never happens).

If you want to really piss off a language nerd, say "I could care fewer."

And it’s all well having this debate behind a screen, but do those who think irregardless is an abomination think they’d come out looking good or smart if they corrected someone’s use of it in person?

This is true of most internet arguments. Still, if there were a way, in spoken English, to differentiate between "its" and "it's," and someone I was talking to used the wrong one of those, it would be almost impossible for me to not correct them. Unless I feel like they could take me in a fist fight.

Having said all of that, I just can’t abide the word irregardless, I’m sorry. It is a mouthful, there is a word that exists that already does the job better, and it is double negative (the prefix “ir-” and the suffix “-less” are both negatives, so the word undoes itself). It creates more confusion than it does clarity – so really, what is the point in its existence?

There are plenty of words that do the same job; what's the point of any of those redundant ones? I don't like it either, so I don't use it (except when I'm ragging on it), but I also don't use "decimate" to mean anything other than "remove one tenth of" (the word you're actually looking for there is "devastate").

Regardless (see what I did there), arguments about words are almost always fascinating because of what they reveal about the arguers.

Literally.

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