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:

Write_Mikey_Write!

about 11 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

about 13 hours 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.

March 10, 2022 at 12:02am

#1028667

Respair!

Been a while since I made words about words.

From respair to cacklefart – the joy of reclaiming long-lost positive words
We have been bombarded with negativity recently; but the English language is a treasure trove of joyous vocabulary

While I want negative words as well, these will do for now.

“Dwell on the beauty of life. Watch the stars, and see yourself running with them”: words of positivity from the Roman emperor Marcus Aurelius.

Easy to be positive when you're an emperor, I suppose. As long as you're not dodging assassins.

In recent times I’ve made it a mission to highlight a category of English that linguists fondly call “orphaned negatives”. These are the words that inexplicably lost their mojo at some point in the past, becoming a sorry crew of adjectives that includes unkempt, unruly, disgruntled, unwieldy and inept. Yet previous generations had the potential to be kempt, ruly, wieldy, ept and – most recently thanks to PG Wodehouse – gruntled. Some were even full of ruth (compassion), feck (initiative) and gorm (due care and attention).

Well, that explains a few things from previous entries.

It’s not just these negatives that have been lost. The German schadenfreude – pleasure in the misfortunes of others – is now all too familiar. But how many of us know its near opposite in English, “confelicity”, which is joy in another person’s happiness?

Not me. I'll probably forget that word as soon as I'm done with this entry.

But if we are pessimists at heart, it has never stopped us having a laugh.

I mean, sure, that's pretty much my entire brand. That and science. And drinking.

Some of our happiest words are such because they make us smile, and often involve a bit of fun at our own expense. Who can resist such nicknames as “cacklefarts” for eggs, or “bags of mystery” for sausages (because you never quite know what’s in them)? Even the prudish Victorians knew hankies as “snottingers”, and umbrellas as “bumbershoots”.

The Victorians weren't nearly as prudish

as they're made out to be.

Also, I'd never heard of "cacklefart," which is, obviously, a fine word. I've been known to call eggs "cackleberries." And milk becomes "moo juice." I get my groceries delivered now, because I'm lazy (been doing it since before anyone had heard of COVID), and before that I made grocery lists on a note app on my phone. But before that , I'd use the traditional ink on paper. One time, I'd written a grocery list that included cackleberries, moo juice, chick tits (what else are you going to call chicken breasts?), and a few other twisted inventions of my own. By the time I got my groceries to the car, though, I realized that I'd dropped my grocery list somewhere.

I can only imagine the look of shock and/or horror on the face of whoever found that thing and read it. And then I laugh. See? I can experience joy.

The fact that "cackleberries" is a longer word than "eggs" and thus takes more time to write out is immaterial.

It’s true too that other languages dish out positivity a little better than us. They have a splash of what the Italians call sprezzatura, a careless, thrown together nonchalance or indifference to life’s curveballs. As for joy, it would be hard to beat gigil from the Philippines. In a single word it conveys the “irresistible desire to squeeze something cute”.

That concept just gave me diabetes.

But one English word surely stands above all others from the corners of the dictionary. I mention it all the time, because I’m determined to bring it back. Or bring it anywhere in fact, for it never really enjoyed more than a day in the sun. “Respair” has just one record next to it in the Oxford English Dictionary, from 1525, but its definition is sublime. Respair is fresh hope; a recovery from despair.

...nah.

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