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 10 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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This is a private area. Access has been restricted and you may not view it under its current settings. If you feel you should have access, please contact the area's owner and request access.

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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 14, 2024 at 12:01pm

#1073950

Revisited: "More On Philosophy"

Today, my random time machine took me all the way back to the beginning of February, 2007, with this entry that turned out to presage many of my later musings: "More On Philosophy"

In case it's not obvious, one thing I clearly remember after all this time about that title (while forgetting most of the contents) was that I deliberately crafted it to be a pun on "moron philosophy," a bit of self-deprecating humor because the entry was about my views on existence.

The entry begins with, and was inspired by, a quote from a comment made by a former site member: "Would you agree that nothing is truly 'discovered', that it is just a question of a new awareness of this or that, which may have existed all along?... we live in a bigger world than what we are 'aware' of, proven by many 'discoveries' starting with the planet not being flat... being newly 'aware' of its real shape does not take away the fact that it always was round."

I should emphasize here that I have exactly no formal training in philosophy. But my philosophy is that someone shouldn't have to in order to draw their own inferences about life, the Universe, and everything, any more than one needs a PhD in art history to decide whether they like a certain painting or not.

The danger, of course, is that you're just restating what Plato or Camus or Nietzsche or Springsteen, or whoever, has already said (or sung).

I like to think that my philosophy is pragmatic and realistic; that is, I don't concern myself much with the question of whether all y'all exist or not. Solipsism is an easy trap to get into, especially on the Internet, but it's also easy to get out of: I'm real, you're real; my computer is real.

I have, of course, joked around about solipsism many times since then.

I'm also not "a butterfly dreaming I'm a man;" that observation strikes me as some straw-grasping from a guy who doesn't want to deal with reality.

I'm not sure I'd put it exactly that way today, but I still don't think it's nearly as profound as some people make it out to be.

I'm willing to accept variant definitions of reality, but they all have to embrace the fact of me sitting here on my lunch break and typing in my blog; the idea that this is all illusion is simply preposterous, not worthy of consideration outside the walls of University philosophy departments and Buddhism.

With the cushion of time between then and now, I can see how this might come across as a rag on Buddhism. I doubt that was my intent. I think it was a reference to the "butterfly dream" above, which was first promulgated in Daoist philosophy, which at the time I confused with Buddhism, because I never did get everything right and never will. Even so, to "get" the reference, one must be at least glancingly aware of Eastern philosophical traditions, which contradicts my assertion above that one shouldn't need to study philosophy in order to do it.

Self-contradiction is a key component of human thought, in my view.

Now, that doesn't mean that we don't all run around with illusions, or that these illusions don't affect our reality. That's the symbolism of all the math discussion in my blog header, if you haven't figured it out by now: the idea that there's a real part and an imaginary part, and that most of what we see isn't purely one or the other but some mixture of both.

I can't be sure if that was my first attempt at explaining the title of this blog, but it might have been.

Finally, I get around to addressing the past member's comment:

So, yes, to me the definition of "discovery" is finding something that existed before you knew about it (as a species or an individual) AND incorporating that into your worldview. Like, for instance, I have a "Discover" card, so named because you have to run around and discover places that actually take it. They took it before I found them; I merely found what was already there, and others already knew about. Once I find them, they're on my mental list of "places that take Discover cards."

Again, I'd probably change the first part of that paragraph now (while keeping the funny bit about the Discover card). I'd emphasize that something had to exist before it could be discovered. Sometimes that existence is "real" as I view it: an exoplanet, maybe, or a fountain in a city I'd never been to before. The exoplanet discovery is universal to the human race, while the fountain discovery is unique to me; the people who live in that city would have already known about it.

In the former sense, Columbus never discovered America, as humans already lived there. In the latter sense, he did, as it was new to him. This is the source of much confusion and righteous indignation, but it turns out to be partly an issue of semantics.

A third sense of discovery is when someone discovers something that doesn't have a physical reality. One example would be the discovery of complex numbers. Another is Einstein's famous equation; E has always equaled mc²; but no one articulated that relationship before Al did.

All of which makes it difficult, sometimes, to separate invention from discovery. When the Wright brothers built and flew that first airplane back in 1903, they brought something new into the world, but was it invented, or did they draw from the Platonic realm of ideals? That's why we have discussions about these things, preferably while on mind-altering substances.

I won't bother with the rest of the blog entry. Suffice it to say that I saw in it the seeds of several later discussions, but, as is proper, my views have changed a bit over time.

2 Like • Comment • View 1 comment

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