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:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
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

Dec 24, 2024 at 5:16pm
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

March 15, 2024 at 9:38am

#1066306

Why So Selfies?

Yes, I used to get paid to do photography. No, that doesn't make me an expert. Not being an expert has never stopped me from posting stuff here. This one's from, surprisingly enough, Business Insider.

I take such good travel photos of myself that people swear I have a secret photographer. Here's how I do it.

Like this author, I tend to travel alone and take photographs. There's one important difference, though: never, in the history of the world, has even one photograph been improved by my presence in it, and, more often, it ruins the whole shot. At least once, it literally cracked the camera lens. So, sure, if you just gotta be the focus (pun intended) of every picture, and you're attractive enough to justify it, great. Otherwise, there's absolutely nothing wrong with taking general landscape shots to prove you've been somewhere.

Now, honestly, the article's three-bullet summary should be enough, though even then, the first one is superfluous:

As a solo-travel content creator, I've learned lots of tips for taking great photos while alone.

I always travel with a smartphone tripod and use my smartwatch as a remote shutter.

If I have to ask someone to snap a photo of me, I always take a photo of them first.

So that's all, folks.

...okay, no, I have a few more things to say about the article.

First, the example photos really are good, so she's not just blowing wind, here. The last one, especially, with the cacti? It's the first time I've looked at a nature shot and said, "Wow, this picture really is improved by having a human in it." Mostly because the human is wearing something deep red, which nicely complements and contrasts the other colors in the picture.

Second, she uses her smartphone for the pictures. These days, there is nothing wrong with this. Phone cameras can be remarkably good, now. Fifteen years ago, I might have scoffed at the idea, but not now. There are things you can do with a standalone camera that you just can't with a phone, but they don't involve composition.

On to details:

A smartphone tripod is the No. 1 thing I pack on any trip or hike.

I would think that the phone would be "No. 1," but whatever. Really, that's it. That's the secret. That, and...

I use the Bluetooth connection between my smartphone and smartwatch to create a remote shutter and snap a picture.

No matter the camera, selfies, being by definition shot at arm's length, distort features. Most people aren't going to be that close to you, excepting crowded subways, concert pits, and intimate situations, so you're going to look different. Also, it's rather difficult to get a whole body pic (assuming you have a body worth photographing, and/or an outfit you want to emphasize) that way, even with a selfie stick. So the remote thing is a good idea, I think. I used to use a manual cable. I never could get Bluetooth to ever work reliably on anything, so best of luck with that if you try it.

A few other tips, and then:

There are times when it's not possible to set up a tripod, but I still want to get some photos in a beautiful location.

In those instances, I may ask someone to snap a photo or two of me — and, of course, I return the favor.

One, that's always been kind of a gray area in my knowledge. If someone else uses my phone to take a photo following my basic requested directions, should they get the photo credit? Like I said, I'm not an expert.

And two, I hope you have a backup camera/phone, because not everyone you meet is trustworthy. Most are. But sometimes, you get unlucky.

Now, in a way, this article is an ad, not only for the author's material, but also for the products she mentions. That doesn't mean there isn't something useful in there.

3 Like • Comment

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