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:

Annette in France

about 8 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

about 12 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 12 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 17 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 18 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 18 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

June 14, 2024 at 9:44am

#1072638

And Time Again

There are lots of subjects we don't fully understand. This is a good thing and it makes life interesting.

We Still Don’t Fully Understand Time

Interesting, but also amusing, because here's an article about time from Time. Or is that from Time to time?

In our everyday lives, time is a precious commodity. We can gain or lose it. We can save, spend or waste it. If our crimes are revealed, we risk having to do time.

Notice the phrasing in that last sentence: "If our crimes are revealed..." Not "If you do crime and get caught," no, straight to "You know what you did even if the authorities don't, yet."

To scientists, time is something we can measure. Clocks have, over the centuries, been the high tech artifacts of their era—the water clock, the pendulum clock, Harrison’s chronometer, and so forth up to the incredible precision of atomic clocks—marvels of modern technology, albeit without the evident aesthetic quality of more traditional timepieces.

You know, it occurred to me the other day that the first computers were clocks. When you think about it, they have the ability to compute the current time based on an input value (the time you set it at). I got this insight when I saw an astrolabe—a device that computes the movement of sun, moon, and planets—described as an early computer. But what's a clock besides a simpler astrolabe, one that only focuses on the sun with respect to the earth?

I guess it's a categorization thing, like with hot dogs or Pluto.

The article spends a few paragraphs going into the history of our conception of time, delving into philosophical, religious, and scientific matters, but not telling me anything new.

Then:

The traditional view, even among those who accept Darwinian evolution, is that we humans are necessarily the culmination of the evolutionary tree.

Yeah, that may be the "traditional" view, but there's no evidence for this egocentric idea. Not that there's any compelling evidence against it, besides pedantically wondering what "culmination" means, when we could be wiped out while other species (cockroaches being the most popular example) continue to exist.

But even in the immensely concertinered timescape that modern cosmology reveals, extending billions of years into the future as well as the past, this century is special. It’s the first in the 45 million centuries of Earth’s history when one species, ours, can determine the entire planet’s fate. We’ve entered what’s sometimes termed the ‘anthropocene.’

Actual scientists have denied the idea of the anthropocene,

but the name's stuck in public consciousness now, and good luck digging that tick out.

Perhaps our remote descendents will have a much-enhanced lifespan; they might even become near-immortal.

Or perhaps they'll lose capacity for cognitive function. We're already seeing that happen. Assuming, of course, that the collective "we" will have descendants. That "traditional view" up there assumes that evolution is about gaining functions such as what we call intelligence, but really, it's about genetic survival (that's simplistic, of course, but close enough for my argument). When "intelligence" stops being a survival trait, evolution starts to select against it.

So this is more of a philosophical essay on the state of humanity than anything meaningful about time. That's okay, though. I'd comment on more of it, but it seems I'm out of time.

3 Like • Comment • View 2 comments

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