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:

Writer_Mike

24 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 17 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

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.

4 Like • Comment • View 3 comments

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