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!

a day 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.

October 4, 2019 at 12:43am

#967224

Watch your Language

While we're stealing from The Guardian...

https://www.theguardian.com/news/2018/jul/27/english-language-global-dominance

Behemoth, bully, thief: how the English language is taking over the planet

No language in history has dominated the world quite like English does today. Is there any point in resisting?

Now, you're probably already aware of this, but The Guardian is a British source, so there may be some bias here.

On 16 May, a lawyer named Aaron Schlossberg was in a New York cafe when he heard several members of staff speaking Spanish. He reacted with immediate fury, threatening to call US Immigration and Customs Enforcement and telling one employee: “Your staff is speaking Spanish to customers when they should be speaking English … This is America.” A video of the incident quickly went viral, drawing widespread scorn. The Yelp page for his law firm was flooded with one-star reviews, and Schlossberg was soon confronted with a “fiesta” protest in front of his Manhattan apartment building, which included a crowd-funded taco truck and mariachi band to serenade him on the way to work.

Is that what I have to do to get actual tacos and a personal mariachi band? I... I don't think I could be that big a hypocrite. This is America, dammit; speak Navajo. Come to think of it, I wouldn't mind a lifetime supply of Navajo frybread. That stuff's delicious.

Elevating English while denigrating all other languages has been a pillar of English and American nationalism for well over a hundred years.

Yeah, look who's bloody talking. Okay, so the author is American. This is still a British rag.

Behemoth, bully, loudmouth, thief: English is everywhere, and everywhere, English dominates. From inauspicious beginnings on the edge of a minor European archipelago...

And all the British readers just spat out their tea.

One straightforward way to trace the growing influence of English is in the way its vocabulary has infiltrated so many other languages. For a millennium or more, English was a great importer of words, absorbing vocabulary from Latin, Greek, French, Hindi, Nahuatl and many others.

I saw someone comment once: "English isn't a language. It's three languages in a trenchcoat pretending to be one." But the best description was from some other anonymous internet commenter: "I always thought of English as the bastard child of an orgy of languages ending with a huge bukakke 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."

...the Dictionary of European Anglicisms, which gathers together English terms found in 16 European languages. A few of the most prevalent include “last-minute”, “fitness”, “group sex”, and a number of terms related to seagoing and train travel.

And I'm starting to understand why the rest of the world thinks Americans are weird.

After all, what a work is English, how copious in its vocabulary, how noble in expression, how sinuous in its constructions, and yet how plain in its basic principles. A language, in short, with a word for almost everything, capable of an infinite gradation of meanings, equally suited to describing the essential rights of mankind as to ornamenting a packet of crisps, whose only defect, as far as I know, is that it makes everyone who speaks it sound like a duck.

Quack.

It’s not that English is bad. It’s fine! A perfectly nice language, capable of expressing a great many things – and with scores of fascinating regional variants, from Scots to Singapore English.

Okay, other sources have called Scots a separate language that happens to have a lot in common with English. I'm not sure where the distinction between that and "regional variant" is drawn, but I've heard it said that Scots is probably close to what English would look like if it hadn't absorbed entire dictionary sections from Romance languages. Singlish, on the other hand, I know little about apart from its existence.

Several Japanese speakers say that it’s easier to express anger in English, especially by swearing.

In my first abortive attempt to learn Japanese, when I was in high school, I studied at the feet of my sensei, a smoking hot Japanese exchange student. Naturally, all I really wanted to know was how to curse in Japanese. I did learn some kanji and some dirty words, but I've forgotten all of that and I never did ask her to go out with me.

In some ways, the worst threat may come not from the global onrush of modernity, but from an idea: that a single language should suit every purpose, and that being monolingual is therefore somehow “normal”. This is something that’s often assumed reflexively by those of us who live most of our lives in English, but historically speaking, monolingualism is something of an aberration.

I've always known, on an intellectual level, that I was missing out on a lot by being monolingual. But my best effort at learning another language - four years of Latin in high school - didn't make me fluent in that dead-but-zombified language, so I just assumed I would forever be crap at learning languages. Now that I'm actively trying to learn one, I wish I hadn't waited so long.

One interesting attribute of English is that it's pretty damn versatile. Of course, I don't have much to compare it to, not directly, but from everything I've seen, the way we can go through neologisms or words stolen from other languages, discarding the ones that don't really work and incorporating the ones that do, makes English fairly flexible. And there's no one true central authority for English - dictionaries, which are as close as we come, are, as I've noted here before, more descriptive than prescriptive.

It's also, from what I've heard, the only language that needs a thesaurus.

So I don't know if I fully agree with the author, here. Yeah, some languages are dying, but as far as I can tell, that's part of the natural way of things. I'm not convinced that the languages are dying because of English - though some, of course, have gone extinct because of the colonialism of native English speakers. Attempts to make other languages universal - whether existing languages like French, or synthetic ones such as Esperanto - have met with only limited success. If you're going to have a universal language for people to speak in addition to their own tongues, you could do worse than English - even if the structure, spelling and vocabulary are a bitch to learn.

And face it, it's either that or Klingon.

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