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:

tj-Merry Mischief Maker

about 21 hours ago
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🎁

a day ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

December 11, 2024 at 12:20pm

#1081035

On the Label

A Slate article from half a year ago takes on an issue I've been wondering about for a long time.

On Both Sides of My Brain
For years now, I’ve been puzzled—and annoyed—by the way people seem to insist on labeling what type of person one can be. I’ve finally solved my problem.

Ah, I recognize that personality type! The author must be a non-labeler!

June 25, 2024 5:40 AM

Yeah, I don't usually copy timestamps in here, but this one gave me pause. Is the writer an extreme night owl, or an extreme early bird? (Or was the article's publication time scripted? Different time zone? Who knows?)

Recently, after I did a silent retreat, I was trapped on a five-hour car journey (long story) with someone who was obsessed with labeling everything. People have “math brains” or “creative brains,” there are “boy chores” and “girl chores,” and in any relationship you will have “the person who reads the map” and “the one who is social.”

Well, there's your problem: you did a silent retreat, and then got stuck as a captive audience while someone spewed out all the words they couldn't during the retreat.

This labeling tic is all over the internet too; indeed, much of the content I see online seems premised upon the idea that everything can be better understood if we simply group it as a type.

Yes, maybe we should call that kind of person a Tag Hag.

The relief in the comments is palpable: Oh, I’m that label! Everything makes sense now.

That's kind of what's been bugging me about labeling, to be serious for a moment. I am what I am (to quote either God or Popeye), so how does putting a label on it help? Like, we all know I'm into science, pedantry, gaming, and science fiction; how does it help me or anyone else if I get put in the "nerd" box?

Usually when I find myself dealing with a “labeler” in real life, it’s because this idea of there being two types of brains has come up.

There are two kinds of people in the world: Those who think there are two types of brains, and those of us who know that's been thoroughly and completely debunked.

It’s not just attachment styles. All over those platforms, you see vlogs and infographics declaring that people can be understood best as bundles of fixed, unchanging symptoms, related to corresponding bundles of trauma, grouped neatly under buzzy labels.

This is, of course nonsense. People can be best understood by their astrological natal chart.

Yes, I'm back to making jokes. But speaking of astrology:

Then there is the enormous popularity of astrology meme accounts. I find it hard to take exception with this iteration of labeling, though, because my star sign is Aquarius, so the @costarastrology account (with its 2 million followers) always presents me with flattering personality reads that position me as a cool, aloof, intellectual sort.

That's not your star sign. It's your sun sign. Your personality is also influenced by what sign the moon was in at your birth, and which one was intersected by the eastern horizon (which is what I said above). As I'm an Aquarius sun and moon (rising sign unknown), I know that astrology is complete horseshit (but sometimes fun horseshit).

On that five-hour car journey with the labeler, though, I could not simply go outside. They were always rushing to finish my sentences too, with an ending they expected might fit with the kind of thing I had been saying. There was a manic, frantic energy to every exchange. As if something terrible might happen if I were permitted to finish a sentence by myself.

Note to self: if I ever do a silent retreat (which I won't), arrange my own damn transportation. Alone.

And speaking about it, I should admit, to my psychoanalyst a few days later helped me clarify my thoughts further. (That’s right, my psychoanalyst. This essay was not eccentric and unhinged enough already.)

Right, because everyone who sees a shrink is eccentric and unhinged? Come on, lady, if you're going to rage against labels, at least stop enhancing the stigma surrounding mental health issues.

But in the wake of the silent retreat, everything seemed bathed in a rosy glow of calmness and goodwill. My thoughts were infused with peace and love and so forth. So, after my frustration had exhausted itself (and, mind you, that did still take a while), I had a sort of epiphany. After all, wasn’t there some of the labeler in me? Even by calling this person a labeler, I was assigning them a type.

I can almost forgive the dig on psychotherapy after seeing this level of self-awareness. Almost. Not quite.

There is, of course, more to the article, including another epiphany about some people needing to maintain control over social situations, or something. I don't know; I'm not sure if that revelation makes things better or worse. Just like with labels. So, I leave the article, my own curiosity unresolved, more confused than ever.

Maybe I should see a shrink.

4 Like • Comment • View 2 comments

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