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:

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.

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

1 Like

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

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

August 7, 2023 at 9:11am

#1053767

Liquid Ick

Another link from Cracked, this time about drinking.

4 Gross Historical Drinks That Show How Good Modern Drinkers Have It

Except that "modern drinkers" quaff kombucha, which is objectively disgusting. Also, veggie smoothies.

Nowadays, even the humblest watering holes are going to have a wealth of beverages available for those looking to tip back a glass of their preferred shape and size.

And yet, it's almost never enough variety for me.

Here are four ancient drinks that show how good modern drinkers have it.

To be sure, the original beer doesn't seem very appetizing to me. And you had to drink it through a straw, because Sumer, for all of its innovations, hadn't figured out filtering.

4. Airag/Kumis

If you were part of the Mongol army and were seeking a pick-me-up after a long day of pillaging, the drink of choice would have been airag, now known by its Turkish name, kumis, which, given its appearance, doesn’t seem like much of an improvement. Now, for such a nomadic people, it wasn’t as if they were able to tote wheat fields or barrels of grapes along, so they had to make do with what they had. And what they had, in great numbers, were horses. Horses that could, to their delight and our disgust, be milked.

That might disgust some people, but really, is it that different from cow's milk? Also, sheep's milk is still a thing. They're all herbivores. No, the origin isn't the disgusting part. The disgusting part is that, since almost any sugar can be fermented into alcohol, and milk contains a kind of sugar, well...

The fermentation wasn’t purely for producing alcohol content, but because straight mare’s milk is apparently nature’s Ex-Lax, and fermenting it kept a little bit more of your insides, well, inside. Making the horse milk a little boozy was just a happy side effect, and the concoction wasn’t only tolerated, but celebrated as one of the sources of the Mongols’ strength.

For some of us, any mammal milk is basically nature's Ex-Lax.

3. Kykeon

Kykeon, a drink that pops up throughout Greek mythology and history, did indeed have the rep of fucking you up, but not by benefit of its ABV. It was instead supposed to have hallucinogenic properties.

One thing that most hallucinogens seem to have in common, not that I have any experience with them, is that they can cause great gastric distress. That keeps me away from them more than the whole "illegal" thing does.

Well, the O.G. recipe isn’t exactly mega appealing, being made from water (okay), barley (sure), herbs (fair enough) and ground goat cheese (hmm).

Goat cheese? Oh no, that's disgusting! Wait, no it's not. Goats are another mammal it's okay to milk (see above).

I'm going to provisionally put this one in the "not that objectively gross" category.

2 Posca

Posca was the fuel of choice for the Roman troops — at least, the ones poor enough that they couldn’t afford to drink anything better.

If they were rich, they wouldn't have been troops.

The sales pitch was that it was great for revitalizing tired troops and fighting infection, which is what you might have to tell someone to get this shit down their gullet. Posca was made from red wine vinegar, water, herbs, spices and honey.

Okay, not something I'd drink, but all of those things are stuff most of us have swallowed.

1. Three Penis Wine

Yeah... no.

Let me disappoint you immediately by clarifying that this is not a euphemism. This is, fairly straightforwardly, wine infused with the tangy taste and power of three different animal penises.

Maybe that's why they invented the word "cocktail."

Then there's that drink with the human toe in it. No, I'm not going to look it up to link it. You can Google same as I can.

Not to fact-check a sitcom’s fake infomercial, but their precise penis combo is a little off, as true three penis wine contains seal, deer and dog penises.

As this is supposedly a magical potion for improving one's sexual prowess, and potions are based on sympathetic magic, you'd think they'd have taken a cue from the Mongols and used horse dong.

I guess maybe that was a steppe too far.

4 Like • Comment • View 4 comments

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