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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:

Wordsmitty ✍️

May 13, 2024 at 1:18pm
In response to "The Breakfast and the Breakfurious"

And for those who rather not read the Smithsonian article, the History Channel episode that covers Pop Tarts (if you have a cable TV account):

https://www.history.com/shows/the-food-that-built-america/season-4/episode-1

Annette

May 10, 2024 at 12:05pm
In response to "Meditation Clause"

Everytime a stranger asks me to close my eyes, I keep my eyes open to see what they are up to.
Meditation is not for me.

But I think most people would have 45 minutes every day if they took that time away from social media or TV watching.

2 Like

Wordsmitty ✍️

May 9, 2024 at 1:15pm
In response to "Super Stew?"

But . . .

What's the difference between one being in the soup or in a stew? Is one thinner or thicker? How do writers use soup and stew?

1 Like

Elisa the Bunny Stik

May 9, 2024 at 10:11am
In response to "Super Stew?"

"Served in a pot? Maybe in your country."

Honestly, in the US I'd think soup would be served in a pot due to the quantity that ends up being consumed.

1 Like

Lazy Writer est 4/24/2008

May 8, 2024 at 3:02pm
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

Annette

May 8, 2024 at 10:52am
In response to "Be Positive"

Asking doctors for advice has its own pitfalls. Because there is no money in recommending a sensible nutrition pattern, they will say things like: Look it up online.

2 Like

🌕 HuntersMoon

May 5, 2024 at 8:50am
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

May 3, 2024 at 11:58am
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

October 18, 2018 at 12:38am

#943662

Alone

https://www.livescience.com/63843-look-harder-for-aliens.html

I like Star Trek and Doctor Who as much as anybody, and more than most, but the more I learn, the less I think we'll find spacefaring aliens anytime soon.

What it comes down to for me is only one factor in the Drake equation.

If you don't know, the Drake equation isn't anything like a real mathematical equation. It doesn't describe anything in physics or astronomy or any other science. It is merely a starting point for thinking about the possibilities for extraterrestrial life.

Let's take a look at it (from Wikipedia

):

The Drake equation is:

N = R_∗ ⋅ f_p ⋅ n_e ⋅ f_l ⋅ f_i ⋅ f_c ⋅ L

where:

N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);

and

R_∗ = the average rate of star formation in our galaxy
f_p = the fraction of those stars that have planets
n_e = the average number of planets that can potentially support life per star that has planets
f_l = the fraction of planets that could support life that actually develop life at some point
f_i = the fraction of planets with life that actually go on to develop intelligent life (civilizations)
f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space

Before I go any further, I'm going to call "intelligent life" anyone who can build spaceships. Like us. So we can stow the inevitable jokes about not finding intelligent life on Earth for a bit. And we're not talking about dolphins or cats or other superior species; just the ones who build spaceships.

My point of contention is f_i. We have exactly one data point for this: us. And it took 4-5 billion years from Earth's formation for us to appear, and Earth came into being about 10 billion years after the universe began. That's a long damn time, especially compared to the 60 or so years that we've been building spaceships.

And here's the thing that bugs me: some humans like to think of their species as the pinnacle of evolution, but the evidence for that is lacking. It's a matter of perspective, you see - certainly we've been successful from a species survival standpoint, but so have roaches. Intelligence - technological prowess - is obviously not a requirement for species success, and given the existence of nuclear weapons, I'd argue that it's no guarantee of it, either. The ecosystem would do just fine without us (some would argue it would do better, but that's not my point right now).

Nor is there anything that mandates the evolution of technological ability. Life went on long before that particular trait came into being.

So we don't know. It could happen a lot. Maybe it only happens rarely. With only one data point, we have no way of knowing. Perhaps it's a little like winning the lottery - you only have a one in (some astronomical number) chance of winning it, but once you've won it, your chance of having won it is unity. And we're looking at the cosmic lottery from the perspective of having won it.

Therefore, it hardly matters what the other terms are, if the chance of intelligent life developing is only one in 10¹⁰⁰. That's as close to zero as doesn't matter, even in a universe as inconceivably large as ours. And the Drake equation is focused on our galaxy alone. Even if every other fraction in the equation is effectively 1, and even if there's a trillion stars in the galaxy (estimates tend to be lower than this), if that one factor is 1 in 10¹⁰⁰, we could easily justify the number of technological civilizations in our galaxy to be 1.

Do I think there's other "intelligent" life out there, somewhere? Yes. I've said so before. Big universe and all that. But if it's not in our galaxy - nay, even if it is in our galaxy and, say, on the other side of the core - it might as well not exist for all that we'll ever be able to contact it or vice-versa. Absent, of course, revisions to the known laws of physics.

To be clear, even finding a goddamn microbe of non-terrestrial origin would be of earth-shattering importance (hopefully not literally). I think it likely that this will happen, given how short a time it was from the formation of Earth to the first stirrings of life thereon. But it hasn't happened yet, so we are still in the realm of speculation.

I hope I'm wrong. I really do. And I think it's important to keep looking, if only for the other things we can learn from the search. But maybe we also need to work on becoming the superior beings we wish we could believe in.

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