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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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').

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

June 12, 2024 at 11:01am

#1072557

Phrase Shift

I tend to be skeptical about the sort of thing featured in this article from Mental Floss, because there's a lot of fauxtymology out there, and I've called out a bunch of it in here before ("tip," for example, which most definitely did not start out as an acronym for "to insure promptness" but that crap keeps getting repeated as truth). So I'll just note, by way of disclaimer, that I can't guarantee that all of these are true.

The Racist Origins of 7 Common Phrases
From ‘cakewalk’ to ‘no can do,’ the origins of these common idioms and sayings are surprisingly dark.

Even the most nonsensical idioms in the English language originated somewhere.

I could get pedantic about it and note that the true "origin" stretches all the way back to whatever proto-language our distant ancestors used to hunt antelopes or whatever, but I think I can stick to the intent of the article, here.

While these common phrases are rarely used in their original contexts today, knowing their racist origins casts them in a different light.

Let's also point out that meanings and connotations change over time, like how the swastika was once a symbol of peace and harmony and now... it's not. It can go the other way, too, from bad to innocuous or good. If someone calls you "nice," that doesn't mean they think you're foolish. Necessarily.

1. Tipping Point

No, this isn't directly related to the gratuity kind of tipping.

In the 1950s, as white people abandoned urban areas for the suburbs in huge numbers, journalists began using the phrase tipping point in relation to the percentage of non-white neighbors it took to trigger this reaction in white city residents.

I'm still not sure if that's racist in itself, or just calling out racism. Given that it was the 1950s, it could go either way.

Tipping point wasn’t coined in the 1950s (it first appeared in print in the 19th century), but it did enter everyday speech during the decade thanks to this topic.

So it's not really an origin. And if you use "tipping point" in relation to, say, climate change, that doesn't make you racist, either.

It's a fairly common and descriptive term in other areas of science, too. Like, maybe, the stress at which steel stops acting elastically and starts to deform permanently.

2. Long Time, No See

In contrast, this one is clearly and deliberately racist.

According to the Oxford English Dictionary, this type of isolating construction would have been unusual for the indigenous languages of North America. Rather, it originated as a way for white writers to mock Native American speech, and that of non-native English speakers from other places like China. By the 1920s, it had become an ordinary part of the American vernacular.

Other sources emphasize the Chinese immigrant stereotype over the Native American one.

3. Mumbo Jumbo

In the Mandinka language, the word Maamajomboo described a masked dancer who participated in ceremonies.

Hardly the first, or last, word or phrase mangled from another language, though other sources peg it as "uncertain origin." While it does reflect a disdain for African language on the part of colonials, if you want to get technical about it like I did way up there, all our words have African origins.

4. Sold Down the River

Before the phrase sold down the river meant betrayal, it originated as a literal slave-trading practice.

So, yeah, not great, and there's more about it here.

A roughly equivalent expression of betrayal is "thrown under a bus," which, given the implied violence, I'm not sure is much better.

5. No Can Do

Similar to long time, no see, no can do originated as a jab at non-native English speakers.

This one was more explicitly mocking Chinese-Americans.

6. Indian Giver

Merriam-Webster defines an Indian giver as “a person who gives something to another and then takes it back.”

I think even Kid Me recognized that this phrase was culturally inappropriate. It's pretty damn obvious from the "Indian" part of the name, which didn't refer to Asian Indians.

7. Cakewalk

In the antebellum South, some enslaved Black Americans spent Sundays dressing up and performing dances in the spirit of mocking the white upper classes. The enslavers didn’t know they were the butt of the joke, and even encouraged these performances and rewarded the best dancers with cake, hence the name.

As this one was about getting one over on the oppressors, I'm not sure it's such a bad thing.

Anyway, I've tried to avoid using the obvious ones. As writers, I think it's important to know the tools (words and phrases) we're working with, and make more deliberate decisions about them. Hence my sharing of this list. Besides, a lot of them are cliché anyway.

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