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 20 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

October 26, 2024 at 9:26am

#1078956

Moon Illusion

From SciAm, and an author I hung out with for a week a few years ago:

Why Does the Moon Look Bigger Near the Horizon?
The rising moon looks huge on the horizon, but it’s all in your head

I can smugly assert that this is something I'd already figured out.

I remember watching the full moon rise one early evening a while back.

You'd think astronomers would spend more time watching moonrises.

As it cleared the horizon, the moon looked huge!

Eastern Colorado, where this occurred, is basically a continuation of Kansas: Flat as the proverbial pancake. There's an actual horizon there.

Anyone who is capable of seeing the moon (or the sun) near the horizon has experienced this effect.

I don't have an unobstructed view of the horizon here, but I've spent time on the coasts, so yes.

But it’s not real. Simple measurements of the moon show it’s essentially the same size on the horizon as when it’s overhead. This really is an illusion.

As opposed to time, which really is not (yes, another time article is in the queue).

Attempts to explain it are as old as the illusion itself, and most come up short. Aristotle wrote about it, for example, attributing it to the effects of mist.

As smart as Aristotle was, he got lots of stuff wrong. This was one of those things.

A related idea, still common today, is that Earth’s air acts like a lens, refracting (bending) the light from the moon and magnifying it.

The "simple measurements" noted in an earlier quote refute that bit instantly. Refraction does make it appear redder than usual, though, just like it does with the rising or setting sun.

Another common but mistaken explanation is that when the moon’s on the horizon, you’re subconsciously comparing it with nearby objects such as trees and buildings, making it look bigger. But that can’t be right; the illusion still occurs when the horizon is empty, such as at sea or on the plains.

But, see, coming up with explanations like that, even when they can be disproven... that's part of science.

So what’s the cause? Like so many things in science, there are two effects at play here.

Phil indeed goes on to explain the cause, but no need for me to quote it word-for-word. The first part of the answer is related to this well-known optical illusion.

The second part is devoted to showing that we don't see the sky as a hemisphere, but as a nearly-flat surface. This is the harder part to accept, I think, but next time you're out on the plains or the ocean on a partly cloudy day, note how the bottoms of the clouds make the sky seem flat. This is for the same reason that the Earth itself appears (but only appears, dammit) flat in such locations.

So you've got two (nearly) planes, earth and sky (the temptation exists to make a plains/planes pun, but I'm not in the mood), but only one moon. Still, presumably, you've seen the moon high in the sky quite often, so then when you see it near the horizon, your brain goes back to remembering seeing that.

Moon illusion misconceptions still abound, and like so many myths, they likely won’t go away no matter how much someone like me writes about them.

Yes, this is, indeed, the curse of fact-checkers.

1 Like • Comment • View 1 comment

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