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Sunday, Nov. 24, 2024

10:44am EST

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Complex Numbers

by Robert Waltz

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:

tj-turkey-jobble-jobble-hard-J

about 16 hours ago
In response to "Pursuit"

My grandmother used to say, "Pursuing happiness is like a dog chasing it's own tail."

3 Like

🌕 HuntersMoon

about 22 hours ago
In response to "Pursuit"

Happiness is reading your ramblings daily. True happiness is posting a snarky response.

3 Like

Robert Waltz

Nov 20, 2024 at 2:54pm
In response to "Powers of Three"

🌕 HuntersMoon

- that's pretty much what quantum computing is, from what I've read.

2 Like

Brandiwyn🎶

Nov 20, 2024 at 2:32pm
In response to "Losing Weight Is Easy"

Wordsmitty ✍️

1 Like

🌕 HuntersMoon

Nov 20, 2024 at 1:20pm
In response to "Powers of Three"

With binary, it's either on or off. Wouldn't ternary introduce Schodenger's Cat? On, off, or both?

2 Like

Turkey DrumStik

Nov 19, 2024 at 7:43pm
In response to "Paramount Purpose"

I suspect old web filter attempts

would have combusted when confronted with this article.

1 Like

tj-turkey-jobble-jobble-hard-J

Nov 19, 2024 at 4:04pm
In response to "Losing Weight Is Easy"

Some of them break out in hives if you ask them to add 10% to something.

Yesterday I purchased an item that cost $6.42 after tax was affixed by the register. I handed the cashier a ten dollar bill and watched as she placed it in the cash register, which like most now days, showed her how much change to return. She had to try three times to count out the proper amount, and as I was leaving commented to the bagger, "Why do people have to pay with cash, don't they know how hard it makes my job?"

I didn't say anything, but was trying desperately not to break out in hysterical laughter.

1 Like

🌕 HuntersMoon

Nov 19, 2024 at 9:28am
In response to "Paramount Purpose"

At your age, Bob, take heart in the fact that it can be restored...

2 Like

Wordsmitty ✍️

Nov 18, 2024 at 12:34pm
In response to "Losing Weight Is Easy"

This brings up the old question of whether your weight changes when you inhale a helium balloon to make your voice higher.

Maybe this has to do with the difference of a mug of beer vs a tankard of ale.

The universe has so many weighty/massive questions.

3 Like

Brandiwyn🎶

Nov 18, 2024 at 11:27am
In response to "That's the Spirit"

Maybe you’d go with “Thunder” because that one’s so terrible that only divine intervention can explain its success. Wow. Missed opportunity, here. How could you let this go without a shoutout to mighty Thor, son of the Allfather, wielder of Mjolnir, He with the Long Forehead? (Said in the voice of the Soup Nazi): "No Valhalla for YOU!"

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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