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.

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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

July 14, 2023 at 7:02am

#1052589

Can I Have Two Words?

I've said stuff before about etymology. This is more stuff.

Linguistic Siblings: 9 Pairs of Words With Surprisingly Shared Etymologies

Every linguist is familiar with the feeling of delight mixed with vexation when they notice a linguistic connection that had been right under their nose—like that abysmal is the adjective form of abyss.

Does someone not know that? I suppose not anymore, now that you've read this or the article. Wait'll you find out that "terrific" originally meant "causing terror" and comes from exactly the same place as "terrible."

Because somehow, that's not in today's list.

1. Disaster and Asteroid

Doesn't take much knowledge of Latin to know where "aster" came from: "astra"

I think "asteroid" would be a better word for "hemorrhoid."

These star-crossed words remind us of the enduring human fascination with the heavens and our attempts to comprehend their influence.

It would indeed be a disaster if a large enough asteroid crashed into Earth.

2. Galaxy and Lactose

Yeah, this is a fun one.

As you may have guessed, galaxy comes to us from a Latin word for “the Milky Way.”... French chemist Jean Baptiste André Dumas proposed that name for the natural sugar in milk, using the Latin lac for “milk” plus -ose in analogy to another sugar, glucose.

Now I want a Milky Way bar.

3. Company and Pantry

The Latin root panis, “bread,” links the words company and pantry.

I'll just pause here for a moment while you do your Beavis and Butt-Head impression over "panis."

Ready?

Okay. Far as I can tell, neither of those words is related to panty.

4. Sarcasm and Sarcophagus

I have to admit, I never saw the connection here, unlike with previous pairs.

Meanwhile, a sarcophagus is a “flesh-eater,” so named because the limestone used for these coffins was believed to quickly decompose the deceased’s flesh.

Perhaps ironically, sarcophagi (look, it's a legitimate plural) are most closely associated with Egypt, which, as I understand things anyway, used them to preserve bodies.

Skipping a few. Not because they're not interesting, but because I don't have anything to add.

7. Muscle and Mollusk

You might think the shared m and l link these two words, but it’s actually the diminutive -scus suffix connecting them.

And here we have an example of a connection I might never have made on my own.

And before you ask, yes, as far as I can tell, mussels are so named because of their muscle, and they are mollusks.

BONUS: Silence and Silhouette

This, they included as an example of words that aren't linguistically connected. Apparently, per the article, a silhouette is so named because the French dude it was named after was named Étienne de Silhouette.

It's debatable whether Monsieur de Silhouette actually did silhouettes. He apparently had a reputation for being cheap, and silhouettes are cheaper than portraits. This reminds me of the origin of "guillotine," where Monsieur Guillotine didn't actually invent the device.

But. Names have meanings, too. I can't find many references to the origin of the name "de Silhouette," but that form usually makes reference to a place name. An English equivalent would be, like, "Geralt of Rivia." (Yes, I've been watching The Witcher.)

In this case, however, the only reference I can find (a literal footnote on a Wikipedia page,

so it's not necessarily canon, indicates that the name appears to have a Basque origin, and as Basque is, weirdly, unrelated to other Indo-European languages. Or any other languages, for that matter. So yeah, those words are unrelated. But according to this footnote, the place origin of Silhouette is Zuloeta, which apparently translates to "an abundance of holes."

If true, that's something of a coincidence, as well.

But in case you were wondering, yes, English has at least one word of Basque origin (even if through French), which is fascinating enough by itself.

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