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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 23 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

a day 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.

December 30, 2019 at 12:53am

#972221

The Peasants' English

I haven't talked about it much recently, but I'm still working on learning French on Duolingo. 123 day streak as of yesterday. I can now ask a French person what color his or her cat is, and maybe find a hotel. Yay?

I'm under no illusion that I could go to Paris tomorrow (well, I could, but that's another story) and carry on a conversation in anything but English. Still, linguistics has long been one of my interests, and I'm kicking myself that it's taken me this long to learn a useful language (Latin doesn't count, and besides, I've forgotten most of it).

So today's link is about English.

English is not normal
No, English isn’t uniquely vibrant or mighty or adaptable. But it really is weirder than pretty much every other language

https://aeon.co/essays/why-is-english-so-weirdly-different-from-other-languages

English speakers know that their language is odd. So do people saddled with learning it non-natively. The oddity that we all perceive most readily is its spelling, which is indeed a nightmare. In countries where English isn’t spoken, there is no such thing as a ‘spelling bee’ competition. For a normal language, spelling at least pretends a basic correspondence to the way people pronounce the words.

Um... have you seen French?

But English is not normal.

No, it's Norman. Get it? Norman invasion of... oh, never mind.

Our language feels ‘normal’ only until you get a sense of what normal really is.

Which generally requires learning other languages.

The closest an Anglophone can get is with the obscure Northern European language called Frisian

Some would say Scots, but others call that a dialect of English. I don't know enough to make a case either way. What I do know is that the Frisians were closely related to the Angles, and it's the Angles who gave their name to English, so, okay. Fun fact: the French word for England is Angleterre. You probably already knew that. I didn't. I do now.

Thinking about it, I'm surprised their name for England isn't much ruder, what with their history and all.

We think it’s a nuisance that so many European languages assign gender to nouns for no reason, with French having female moons and male boats and such.

Well, again, Latin training here - I'm used to words having gender. What I'm not used to is such assignations not making any damn sense. I know that language gender doesn't necessarily correlate with human gender, but still. The word for "cow" is feminine, okay. "Milk" is masculine, what? "Car?" Feminine. And don't get me started on body parts. No, get your minds out of the gutter; Duolingo didn't teach me the word for "breasts" (I figured that one out from *ahem* other sources *cough* long ago). I mean, like, arm is masculine, hand is feminine, leg is feminine, foot is masculine, mouth is feminine, head is feminine... nose is masculine. Quoi le fuck?

As an aside, the French word for "shower" is douche. (Also feminine, while "bath" is masculine.) I mean, I knew that's what the English word "douche" came from, but knowing and having to translate it are two different things. One day, I will stop reflexively snorting like a 12 year old every time that comes up on Duolingo. That day is not today. It's probably not tomorrow, either. Hopefully it will be before I'm in a hotel in Paris and have to call down to the front desk to complain that the shower isn't clean.

La Réception: "Bonjour?"
Moi: "La douche *snicker* n'est pas propre... *snort*."
LR: "Pardon, monsieur?"
Moi: "La douche- heeheeha"
LR: "Est-ce que vous êtes américain?"
Moi: snicker "Oui" snort
LR: *hangs up*

Anyway.

Why is our language so eccentric? Just what is this thing we’re speaking, and what happened to make it this way?

I saved a quote from the dark recesses of the internet once: "I always thought of English as the bastard child of an orgy of languages ending with a huge bukkake leaving German covered in the messy splooge of all the others. German is seeking a paternity test while Latin fled the scene and French is denying everything." No idea who first posted it, but I stole it fair and square.

The first thing that got us from there to here was the fact that, when the Angles, Saxons and Jutes (and also Frisians) brought their language to England, the island was already inhabited by people who spoke very different tongues.

Ah, yes, the old "We're taking over now." "Screw you and your stupid language."

The second thing that happened was that yet more Germanic-speakers came across the sea meaning business.

By "business" he means "war."

Finally, as if all this wasn’t enough, English got hit by a firehose spray of words from yet more languages. After the Norse came the French. The Normans – descended from the same Vikings, as it happens – conquered England, ruled for several centuries and, before long, English had picked up 10,000 new words. Then, starting in the 16th century, educated Anglophones developed a sense of English as a vehicle of sophisticated writing, and so it became fashionable to cherry-pick words from Latin to lend the language a more elevated tone.

"Firehose." Right. You know he wanted to use the Japanese term from the anonymous quote above.

The multiple influxes of foreign vocabulary also partly explain the striking fact that English words can trace to so many different sources – often several within the same sentence.

And that's why I've been interested in linguistics.

Thus the story of English, from when it hit British shores 1,600 years ago to today, is that of a language becoming delightfully odd.

I think actual linguists have a different definition of "delightfully" than you or I do. Or than people who have to learn English as a second (third, fourth, whatever) language do.

The article provides an interesting perspective, I think. Nothing surprising in it, no world-shattering discoveries, but an excellent overview of our sometimes-maddening language.

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