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:

Wordsmitty ✍️

about an hour ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 4 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 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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