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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Writer_Mike

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

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day 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

a day 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

Sep 21, 2024 at 7:34am
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

Sep 21, 2024 at 6:46am
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

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

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

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
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.

February 4, 2023 at 12:01am

#1044216

Turning Over a New Leaf

Here's a confession:

I'm not entirely confident when it comes to picture prompts.

But I do feel like it's important to try new things, to, as the kids say, step outside of my comfort zone and do something I'm not sure of, like this month's round of "Journalistic Intentions"

[18+].

One thing I continue to do: pick prompts at random. Today, we have this lovely photo of a leaf under a blanket of water: Orange Submersion

I've seen reports that at least one person is having trouble with xlinks, so if that hyperlink doesn't work for you, I can provide the raw URL on request.

Ever wonder why water is (mostly) clear? I have. I mean, apart from when it's murky from suspended particles, or sometimes when mixed with delicious booze (which is itself, in its pure form, clear). And it's clearly (pun intended) not the same as air; you can almost always tell where one ends and the other begins. The surface of pure water is easy to identify, but, for me at least, damn near impossible to render in a drawing. But then, I've never been very good at drawing... but I digress.

The best answer to why water is clear that I've been able to come up with is that this is not the right question to ask.

Life crawled up onto land "only" about half a billion years ago. In comparison to the four billion or so years since life began on Earth, that's a significant fraction of time, but it means that life was changing and evolving for 7/8 of its history underwater. And some of that life, at least the animal portion, found a competitive edge in being able to directly sense prey or predators: in short, vision is a very useful sense to possess.

It's my understanding that eyes evolved several different times. That is, there's not one proto-organism that gradually turned light-sensitive cells (which many organisms have, not just animals) into an eyeball, which then split off into different species. No, the proto-organism might have been mostly blind, and some of its descendants developed vision in different ways: compound eyes like ours, or the simple eyes of arthropods and such, or whatever.

But most of those eyes originally evolved underwater. Thus, they developed in such a way that vision would be an evolutionary advantage, which means being sensitive to a range of the electromagnetic spectrum to which water is mostly transparent.

Water isn't clear because of some innate property of it; it's clear, to us, because our distant ancestors evolved a sense that allowed them to see in it. Water blocks some other wavelengths.

But the other thing it does is change the direction of light at its boundary. Stick your arm into a fish tank, and it'll appear to bend. At some angles, the light doesn't escape the water at all, but reflects off of it like a perfect mirror (this is a function of index of refraction, and it's also how fiber optic cables transmit data). The surface is also partially reflective when viewed from above, which is how you get the artistic dappled effect from ripples, like in today's picture prompt.

Leaf (pun absolutely intended) it to me to get all sciencey about a pretty picture. But I'm of the considered opinion that the knowledge gleaned from science can improve an aesthetic experience. I know the math and physics behind rainbows (or, well, I used to, and can easily find the information again), but that doesn't decrease one's visual impact.

And if today's discussion hurt your brain, just be glad I didn't go into the biology of what makes leaves turn color and fall off as winter approaches.

Another time, perhaps.

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