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 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day 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

Dec 23, 2024 at 4:55pm
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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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

July 17, 2024 at 11:35am

#1074060

Turtles

I consciously saved this article from Big Think a while back. There were only a couple of things I had saved from that source; it's random coincidence (operating on a limited set of possibilities) that I got both of them to pop up in the same week. Considering my issues with the last one, which I discussed two days ago, it's just as well that I've cleared my queue of these; I'm not sure this one's much better.

Can the known particles and interactions explain consciousness?
At a fundamental level, only a few particles and forces govern all of reality. How do their combinations create human consciousness?

Again, I'm not sure that they're asking the right questions.

How odd is it that these particles and forces fit together so precisely as to enable conscious beings like us to exist? It’s a cutting-edge question, but one that we’re closer than ever to answering definitively.

We're also "closer than ever" to developing usable fusion power. Instead of 20 years away, as we were back in the 1960s, we're now only 19 years away.

Living creatures can be divided into cells; cells themselves are composed of organelles; organelles can be broken down into molecules; molecules are made up of atoms; atoms are comprised of electrons and atomic nuclei; electrons cannot be broken down further, but nuclei themselves consist of quarks and gluons.

As I've noted before, every hard science discipline is, at base, physics.

We should, therefore, be able to take these fundamental constituents of matter — quarks, gluons, and electrons — and assemble them in various ways to explain everything that we encounter in everyday life.

I'm not sure that follows. There are probably things we can never fully explain, but that's a good thing in my book.

The question that inspired that article is quoted within it, and it goes like:

“It has always puzzled me how come the particles & forces in the original quark gluon soup fit precisely to form:

nuclei, and when joined by electrons,
atoms (each with their distinct properties),
countless molecules (each with their distinct properties),
capable of forming life,
which can achieve consciousness,
and ultimately lead to astrophysicists?

This precise “fitting” can hardly be a pure coincidence.”

And that sounds to me like the question of a disingenuous religious person, trying to tie scientists up into a big knot of explanation to try to shore up his (the name is Ottho Heldring, so I'm assuming gender based on that) belief. Maybe I'm wrong about that, but believers have been known to do exactly that, with baseless questions like "if humans evolved from chimps, why are there still chimps?" Which is a question that rests on several false assumptions.

Coincidence or not, if such a history as he briefly described in the question had not taken place, we wouldn't be here to argue about it, now, would we? As I've said before, it doesn't much matter what the odds were after you've won the lottery.

The article itself isn't that long, but it manages to provide a brief history of the physical processes that, ultimately, led to the universe as we knew it, up to and including cats. I won't quote from the meat of it; it's there at the link, and while I have some quibbles, that's going to happen when you take something it requires a PhD to describe (which I definitely don't have) and boil it down to a couple thousand words.

Then, in summary:

But this is not necessarily miraculous. So long as there are a few simple rules and properties to nature:

And a restatement of what I said above:

If the laws of physics were so different that we couldn’t have come into existence, we never would have arisen to find these things out.

Pretty sure both of those are a version of the Anthropic Principle,

which has its own problems because it doesn't really explain anything.

Ultimately, we know that complexity can absolutely arise from simplicity. There's a random element to it, but it's not completely random; it's subject to physical (in the sense of "physics," not "physicality") rules.

And isn't that a more likely hypothesis than complexity arising from even greater complexity? The latter just raises the question of where the "greater complexity" comes from. "It was always there" is deeply unsatisfying, akin to "it's turtles all the way down."

3 Like • Comment • View 2 comments

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