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!

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 23, 2019 at 12:17am

#968290

The Right Word

After yesterday's foray into illusionary metaphysics, I think I'll return to the joys of language.

http://mentalfloss.com/article/50698/38-wonderful-foreign-words-we-could-use-eng...

38 Wonderful Words With No English Equivalent

And the header image is a stock photo of bacon. Can't go wrong with that!

Sometimes we must turn to other languages to find le mot juste. Here are a whole bunch of foreign words with no direct English equivalent.

English, as I think I've mentioned before, is an immensely powerful and nuanced language. But it's far from exhaustive when it comes to describing some things.

1. Kummerspeck (German)
Excess weight gained from emotional overeating. Literally, grief bacon.

Hence the reason for the header image, I suppose. Though I think keto followers would disagree with any link between bacon and excess weight. On the other side, vegans' heads have probably exploded by now.

2. Shemomedjamo (Georgian)
You know when you’re really full, but your meal is just so delicious, you can’t stop eating it? The Georgians feel your pain. This word means, “I accidentally ate the whole thing."

This, however, can definitely be linked to excess weight.

5. Backpfeifengesicht (German)
A face badly in need of a fist.

Lots of these around these days. Some of them were inspired by certain Germans of the last century, so... karma?

7. Pelinti (Buli, Ghana)
Your friend bites into a piece of piping hot pizza, then opens his mouth and sort of tilts his head around while making an “aaaarrrahh” noise. The Ghanaians have a word for that.

"Friend." Right.

19. Bakku-shan (Japanese)
Or there's this Japanese slang term, which describes the experience of seeing a woman who appears pretty from behind but not from the front.

I'm the furthest thing from an expert at Japanese, but this sounds like one of those words that they stole from English and made their own, like "waifu." Sometimes, things come full circle.

20. Seigneur-terraces (French)
Coffee shop dwellers who sit at tables a long time but spend little money.

Funny, haven't covered that in Duolingo yet.

29. L’esprit de l’escalier (French)
Literally, stairwell wit—a too-late retort thought of only after departure.

Nor this - but because I am who I am, I've known this one for some time. Why do you think I'm only amusing in writing?

34. Bilita Mpash (Bantu)
An amazing dream. Not just a "good" dream; the opposite of a nightmare.

Had one of these yesterday. Ever have a dream that was so good you were depressed that it ended when you woke up? Come to think of it, I haven't been plagued by sleep paralysis lately. Maybe it's the daily exercise, or maybe it was waiting for me to realize this before it came back.

36. Luftmensch (Yiddish)
There are several Yiddish words to describe social misfits. This one is for an impractical dreamer with no business sense.

37 & 38. Schlemiel and schlimazel (Yiddish)
Someone prone to bad luck. Yiddish distinguishes between the schlemiel and schlimazel, whose fates would probably be grouped under those of the klutz in other languages. The schlemiel is the traditional maladroit, who spills his coffee; the schlimazel is the one on whom it's spilled.

These are far from the only great expressions in Yiddish. "Luftmensch," by the way, can, if I remember correctly, be loosely translated as "airhead," though I think its literal translation is "air person," which makes little sense in English. Some other great Yiddish words have already been stolen by English, such as "chutzpah," "schnoz," and "glitch." In fact, if you want to expand your vocabulary and sound like a yenta, here you go: https://en.wikipedia.org/wiki/List_of_English_words_of_Yiddish_origin

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