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 2 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 4 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.

August 24, 2021 at 12:02am

#1016040

A Word, Please

There's a scattering of actual writing-related articles in my queue, and as luck would have it, we get another one today.

25 Words That Don’t Mean What They Used To

As usual, I'm not going to reiterate all 25 of them here, but I have a few highlights to comment on.

When the English clergyman Thomas Fuller used the word unfriended in a letter dating from 1659, we can be pretty sure he wasn’t talking about his Facebook page. Instead, Fuller used the word to mean something like “estranged” or “fallen out,” a straightforward literal meaning that has long since “fallen out” of the language.

Yeah, I don't know, I think Shakespeare was more of a Twitter guy.

How many times have I noted in here that language changes over time? I can't recall. It's been a lot. Some of those changes are happening before our eyes right now, like the word "woke" and the use of "literally" for emphasis (WHICH IS STILL WRONG).

1. ALIENATE

Alienate, like alien, is derived from the Latin word alienus, which was used to describe anything that was unfamiliar, unconnected, or foreign. And when alienate first appeared in English as a legal term in the mid-1400s, it meant to transfer ownership of some property over to someone else, so that it is now “foreign” or “unconnected” to you. It’s from here that the modern meaning of “estrangement” or “distance” eventually developed.

Pretty sure the use of a version of that word in the Declaration of Independence was an example of the former usage.

3. BUNNY

Bunny derives from bun—which was an old English word for a squirrel, not a rabbit.

Fun fact: "bunny" isn't the term for a baby rabbit (at least not anymore), but rather an alternate word for rabbit. But "rabbit" used to mean young examples of the species; adults were called "coneys" (as in Coney Island, which I was very disappointed as a child to find wasn't an island filled with interesting three-dimensional geometric shapes, or even rabbits). Baby bunnies are now called - no shit - kittens.

7. FANTASTIC

The link between fantastic things and absolute fantasy was once much closer than it is today. Fantastic originally meant “existing only in the imagination,” or in other words “unreal” or “based on fantasy.”

You still see the adjective used in the old way occasionally; its meaning can usually be deduced from context. Or be a deliberate pun, as in "Fantastic Four" (the movies featuring that team have been less than fantastic).

17. NAUGHTY

Naughty is etymologically related to nought, and meant “to have nothing” when it first appeared in the language around 600 years ago. Soon afterward, it came to mean “to have no morals,” and, by extension, “wicked,” “depraved,” or “vicious,” before its meaning softened in the late Middle Ages.

Well, I've often suspected that, but never could be arsed to look into it. My word for the first decade of this century - the "noughties" - plays on the similarity of those words.

19. NICE

Nice derives from a Latin word, nescius, meaning “ignorant” or “not knowing”—and that was its original meaning when it was first adopted into English from French around the turn of the 14th century. Over the years that followed, nice was knocked around the language picking up an impressively wide range of meanings along the way—including “wanton,” “ostentatious,” “punctilious,” “prim,” “hard to please,” “cultured,” “cowardly,” “lazy,” “pampered,” “shy,” “insubstantial,” and “dainty”—before it finally settled on its current meaning in the early 1700s.

This one, I basically already knew -- but I am moved to add that "nice" is often used sarcastically to mean "shitty."

21. QUEEN

The word queen apparently started life as a general name for a woman or a wife, before its meaning specialized to “the wife of a king” in the middle of the Old English period. It has remained unchanged ever since.

Another fun fact: "queen" is also the term for a female housecat. Female cat: queen. Female dog: bitch. And they ask me why I'm a cat person.

I would also be remiss if I didn't note that "queen" can also refer to the sovereign of a country, independent of whether she's married or not. And y'all know the other use of the word.

24. THRILL

To thrill originally meant “to piece a hole in something”—your nostrils, etymologically, are your “nose-thrills.”

Now this was new to me, and I had to get off my arse (figuratively; I've literally been sitting on it the whole time) to check. Are "thrill" and "drill" related? Sources point to "no." But there may be an older linguistic connection that the etymology doesn't go into. The words are just too similar; it may be a coincidence, but it's not impossible for a Þ sound to morph into a d sound. Or vice-versa.

Anyway, obviously there are more at the source. It's fun to learn new words, but it's also fun (well, for nerds like me anyway) to learn how some words changed over time.

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