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Sunday, Dec. 22, 2024

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Complex Numbers

by Waltz Invictus

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Annette

about 19 hours ago
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

a day ago
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

a day ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

a day ago
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

🌖 HuntersMoon

Dec 19, 2024 at 8:53am
In response to "Five Seconds"

2 Like

Aiva Raine

Dec 17, 2024 at 7:39pm
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

Dec 17, 2024 at 6:14pm
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

Dec 17, 2024 at 9:43am
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 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.

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