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:

tj-Merry Mischief Maker

about 21 hours 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

a day 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🎁

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

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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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.

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August 15, 2024 at 11:40am

#1075303

If Not for That...

We all know I'm a fan of science. Science is the reason I'm alive right now. Science is the reason you're reading this. But, as this Cracked article points out, we could have been even farther along with scientific developments by now.

5 Dumb Stumbling Blocks That Held Science Back
Apparently, we'd have crystal spires and togas if we only let our scientists drink more.

Oddly, all 5 aren't "religion." Well, sort of.

We can all name one big factor holding science back: People’s petty morality, which interferes with our plan to create an army of cyborgs.

That is, indeed, a downside.

And you know I saved this article especially for this first section:

5. Prohibition

All of which means that some — but not all — counties in the U.S. stopped serving liquor starting in 1920. And when we look at those counties that stopped doing so, we see something surprising: People in those counties filed fewer patents than immediately before this time, which is a drop we don’t see in other counties.

On the flip side, there was a meme circulating a while ago that pointed out that, in the Wikipedia article about Irish inventions, you had a few inventions, then "whiskey," then a 300-year gap before anything else. here's a link

to the meme, and hopefully it works.

The shortfall in patents was somewhere between 8 and 18 percent, compared to right before Prohibition. Evidently, alcohol stimulates all inventors’ inspiration.

Necessity is the mother of invention. Laziness, I say, is the milkman. Booze, then, would be the midwife.

4. One Guy Abstaining from Speed

Other illegal substances similarly fuel the scientific mind. Consider mathematician Paul Erdös.

Well, math isn't science. It's an integral (pun intended) part of it, but it's a different universe. Still, advances in math often lead to advances in science, so I'll allow it.

This guy is so celebrated that mathematicians today measure how cool they are by how many degrees of separation they are from him. If you write a paper with someone who wrote a paper with someone who wrote a paper with Erdös, you have an Erdös number of two and are the envy of your whole town.

You might be familiar with this concept as analogous to the Bacon number, which calculates an actor's degree of separation from Kevin Bacon.

As far as I know, only one person has both an Erdös number and a Bacon number, and that's Natalie Portman.

Erdös worked on mathematics for 19 hours a day. He was able to sustain this pace thanks to a diet of amphetamines.

While most of us can be forgiven for thinking this is a Bad Idea, well... maybe for Erdös, but, it turns out, not for society in general:

Some people think daily drug use is bad for you, so a friend challenged him to give the stuff up for one month. Erdös lasted the full 30 days, and he won money on that bet. And at the end, he told his friend, “You’ve showed me I’m not an addict. But I didn’t get any work done. I’d get up in the morning and stare at a blank piece of paper. I’d have no ideas, just like an ordinary person. You’ve set mathematics back a month.”

3. Excel Running Out

Oddly, this one has nothing to do with morality and everything to do with technology. It's a case of technology holding technology back.

The issue was that an Excel fie (.XLS) can hold a total of 65,536 rows and no more. As they input more and more rows, the doctors were accidentally deleting old ones and leaving the total unchanged.

Or, in this case, the issue might have been fixed with a workaround or, and bear with me here, actually updating your software occasionally.

2. One Man’s Ransomware Plot

Lone Asshole Theory: all it takes is one person to ruin everything for everyone. See also: the Tylenol cyanide scare, or every mass shooting ever.

1. Fear of Corpses

We have one more epidemic to bring up — Paris’ cholera outbreak of 1832. It killed Nicolas Carnot, a physicist whose work had great influence after his death. You know our concept of absolute zero, and of measuring temperatures from that value? That came from Carnot’s work. Or what about this whole idea we have called entropy? That, too, originated from Carnot.

Lots more to his career than that, but okay.

When scientists came up with those ideas based on his writings, they were, of course, interested in checking out everything else he’d written. But they couldn’t. Nearly all of it had been buried with him. People thought that was safest, since his corpse had had cholera, and surely all his papers did as well.

In reality, cholera’s transmitted through water and food but not from person-to-person and certainly not from papers-to-person. There was no need to dispose of his writings.

To be fair, our knowledge of disease transmission was fairly limited in 1832.

What discoveries did we lose thanks to this paranoia? Did Carnot figure out how to convert waste heat into fuel perhaps?

No, and for the reason I know it's "no," well, we can thank Carnot for that.

Or did he create a time machine?

Or, ooh, I know, maybe he created a death ray! One that only affects the British.

Anyway, while science is neutral, it can be used for things we consider both positive and negative. We have nuclear power, but also nuclear missiles. We have the internet, but we also have social media and what's misleadingly labeled artificial intelligence. We have beer, but we also have studies saying drinking is bad for you.

I don't blame the science, though. I blame the people involved.

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