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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Annette in France

about 11 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 15 hours ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

about 15 hours ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

about 20 hours ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

about 21 hours ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

about 21 hours ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

a day ago
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

2 Like

Cubboo!

a day ago
In response to "Get Baked"

Yes! I discovered this a few years back (slitting the potato about 1/4 - 1/2" lengthwise, then one or two slits across the longer slit.) It came out wonderful! It seems like this was called an English jacket potato, or something like that.

2 Like

Annette in France

a day ago
In response to "Get Baked"

I'll try your oven baked potato in winter when it's sensible to run the oven for 90 minutes.

2 Like

May 14, 2024 at 9:58am

#1071051

Not One Thing

While we spent yesterday discussing the origin of Pop-Tarts, today's article, from Scientific American, touches on something almost as profound.

Is There a Thing, or a Relationship between Things, at the Bottom of Things?
Quantum mechanics inspires us to speculate that interactions between entities, not entities in themselves, are fundamental to reality

Let me start with this: I was, at first, a bit thrown off by the headline. Maybe it's just me, being used to seeing headlines phrased as yes/no questions, which I thought this was at first. Then I realized that they're asking which is more fundamental: things, or relations? Just in case some reader experiences the same confusion, I thought I'd attempt to interpret.

Which makes it still a binary question, but I'll run with it for now.

What’s at the bottom of things?

It's turtles all the way down.

If we keep asking “Why?” where do we end up?

If my childhood was representative, the answer to that is "In our rooms, alone."

The monotheistic faiths assert that our questions must culminate in God, a solitary, supernatural creator.

No, I'm not going to get into that argument today.

Dissatisfied with that hypothesis, physicists postulate that everything stems from a single primordial force or particle, perhaps a supersymmetric string, from which flow the myriad forces and particles of our fallen world.

One could still, presumably, reconcile those two worldviews by assuming, for example, that God created the "primordial force or particle."

I don't buy that either, by the way. But like I said, I'm not here to argue about it.

(From what I understand, supersymmetric string theory is all but ruled out, though I still maintain that a perfectly logical "string theory" is: "The Universe is a big ball of string, and God is a cat.")

Notice that, for all their differences, religion and physics share the ultrareductionist conviction that reality comes down to one thing.

Yeah, okay, but as one of those "things" is supposedly infinitely complex, and the other is infinitesimally simple, that's a pretty damn big difference.

Call this the oneness doctrine.

This may be confusing, too, as, at least to me, "oneness" conjures up images of hippies sitting around going, "Whoa, it's, like, all ONE, man."

So, I’m intrigued by the conjecture that at the heart of reality there are at least two things doing something to each other.

This may also be a bias due to our species having evolved via two things doing something to each other.

In other words, there is an interaction, a relationship. Call this the relationship doctrine.

Another point of clarification: These days, you see the word "relationship," and the implication is sexual and/or romantic. But the word "relationship" is much broader than that, meant to describe how any thing relates to any other thing. Like how the Earth and the Moon relate via gravity.

This word definition creep is the same sort of thing that leads us to make jokes about names like the town of Intercourse. When the town, or village, or whatever, was first named, intercourse described interaction between people. Later, people started talking about "sexual intercourse," and because all we do is think about sex, when this was later shortened to "intercourse," the original meaning was all but forgotten, except among pedants like me.

Point being, erase that connotation of "relationship" from your mind when you think about how the author uses the word. Yes, despite my joke above about "two things doing something to each other."

Anyway, the article cites a bunch of thinkers who saw the "relationship" bit as being fundamental. Then:

Part of me finds the relationship doctrine, and especially Gefter’s you-centered metaphysics, beautiful and consoling, a welcome alternative to mindless materialism. The relationship doctrine also seems intuitively sensible. Just as words must be defined by other words, so we humans are defined, and to a certain extent brought into existence, by other human beings.

If there's only one thing to know about science, it's this: it doesn't care whether you find it beautiful and consoling or not. No matter how appealing an idea is, it needs to be falsifiable, and it needs to be tested.

Moreover, as I mentioned above, I have a long-standing aversion to the oneness doctrine. This antipathy dates back to a drug trip in 1981, during which I felt myself becoming a solitary consciousness, the only one in the universe.

I knew that was coming. I bet you saw it coming a light-year away, too.

I thought: What is the difference between one thing and nothing? One thing only exists in relation to something else.

I mean, I don't disagree with the philosophy, but my main takeaway here is, "Wow, someone actually remembered details of their acid trip."

And yet I have doubts about the relationship doctrine, as I do about all metaphysical systems that privilege mind, consciousness, observation, information. They smack of narcissism, anthropomorphism and wishful thinking.

This is good. Having doubts and expressing them is what saves this article.

In conclusion, for me anyway, the headline question is less meaningful than it seems. Leaving aside the imprecision of the word "thing" (which I've harped on in a recent entry), it's meaningless to consider objects without some sort of relationship between them; and you can't speak of the relationships between objects without acknowledging the objects. Also, as the drug-tripping author notes, if there's only one object, it's not much of a universe at all.

So my philosophy is "both." We have to consider both things, and the interactions between things, to get anywhere.

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