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:

🌖 HuntersMoon

about 23 hours ago
In response to "Paint in the Neck"

Given the current situation, maybe they'll update their logo... by changing the paint color to orange.

1 Like

tj wanderlust-words-in-motion

a day ago
In response to "Revisited: "Bloomington""

I have been to Bloomington a few times myself, the one in Minnesota, not Illinois or Indiana.

2 Like

🌖 HuntersMoon

a day ago
In response to "Revisited: "Bloomington""

You and Jim look just the same as I remember. Hmmmmmm. I was hoping my memory was fading but no.

1 Like

tj wanderlust-words-in-motion

Feb 8, 2025 at 4:58pm
In response to "What's In a Name?"

It seems no matter what, someone will be offended by it. Even if no one is offended, someone will become offended that no one's offended. Maybe we should bring back All In The Family.

2 Like

🌖 HuntersMoon

Feb 8, 2025 at 9:30am
In response to "What's In a Name?"

Shhhh! Let's hope one of the Great Pumpkin's minions doesn't see this. You may get banned on the basis of... promoting transgenderism, exploiting children, or whatever the hell they come up with.

2 Like

tj wanderlust-words-in-motion

Feb 7, 2025 at 4:26pm
In response to "Roll Model"

Most commercial-grade wipe is like trying to clean one's bum with super-thin wax paper! When I see "Mega-Roll" I instantly realize it will not fit into the holder. I don't put a lot into buying butt-wipe, I stick to one product I know will fit the holder, clean the crap, and my fingers won't bust through.

1 Like

tj wanderlust-words-in-motion

Feb 6, 2025 at 3:24pm
In response to "So Much Energy"

I always thought electroweak{i/} was a bad choice of terminology. It sounds like the batteries in you flashlight are almost dead! They should have used a term like "electronuke" which sounds like some super-hero stuff. There's no need to fear, Electronuke is here!"

Also, when crossing Higg's field, beware of Higg's boson (cross between a boar and a bison) or you may get killed; unless of course, Electonuke shows up and saves your ass.

Alright, maybe I should be writing a story instead of leaving a comment...

1 Like

Graham B.

Feb 6, 2025 at 12:04pm
In response to "So Much Energy"

The only energy-carrying particle that concerns me is the Joeson, as in "I need my morning cup of Joeson to get me up." It's responsible for the strong nootropic force (for me, anyway).

3 Like

Lazy Writer est 4/24/2008

Feb 6, 2025 at 11:36am
In response to "So Much Energy"

I need to find some of that energy..because mine is worn out!

2 Like

tj wanderlust-words-in-motion

Feb 5, 2025 at 4:33pm
In response to "That Healthy Glow"

Yes, they do still make black lights, most are LED. We have one above my wife's plants, it helps them survive through the winter with our lack of sunlight. They seem to be more purple than the black lights from days gone by. Interestingly, they make our Australian Shepard glow purple (her black areas) and they make our Husky's blue eyes glow, but not the black on his fur.

1 Like

February 8, 2025 at 9:21am

#1083521

What's In a Name?

This SciAm opinion piece from last summer supports the idea that words (including names) have power and should be chosen carefully.

Astronomers Should Take a Deeper Look at Naming
We should—and must—take careful measure of what we name cosmic objects and the terms we use throughout science

For full disclosure, I'm personally acquainted with the author.

To summarize the opening, the James Webb telescope (you know, the one out in space that's been taking pictures of the farthest reaches of the observable universe) was named after a controversial figure, and who wasn't even a scientist or astronaut.

For these and many other reasons, a large contingent of astronomers, including myself, prefer not to use the telescope’s official name and instead refer to it by its initials, JWST.

For similar reasons, I continue to call the airport across the river from DC "National Airport."

The observatory is the most visible example of this naming issue when it comes to science; the fight over statues of Confederate figures and buildings and roads named after such individuals is probably the most well known among the public.

While this article isn't even a year old, it seems to me that this particular controversy has faded already. We have short attention spans here.

For example, astronomers have a lot of nicknames we use for cosmic objects. That’s not surprising; “the Whirlpool galaxy” is a lot easier to remember than NGC 5194, its more official catalog name.

It's true that I can't remember official catalog names for most astronomical objects. To be fair, there are quite a few astronomical objects.

Not all names are so benign. NGC 2392 is a gaseous nebula, a favorite of amateur astronomers for its brightness and location on the sky, which make it easy to find. For decades, though, it was known as the Eskimo nebula, a term that is considered offensive by many Indigenous people in Canada and Greenland.

Because of this, at some point, they tried to rename it to the Clown Face Nebula, which, for whatever reason, some people still found offensive. To whom? Clowns?

It's also known as the Lion Nebula, but to the best of my knowledge, no African feline monarchs have come forward to claim offense.

Another example is a popular piece of astronomical software used to extract sources from an image. It was given the unfortunate name SExtractor. That should be pronounced “Ess Extractor,” but I don’t think it’s too much a stretch to see where the problem lies.

Okay, that's genuinely funny. And it's not even as perverted as the popular image of a business called Kids Exchange, the problem with which should be obvious when removing the space and capitalizing all the letters.

Unfortunately, this is one of those times when an article has been hanging around in my queue long enough for it to lose relevance, which it did last month. I mean, I agree with Phil here, don't get me wrong. It's just that, officially, we don't do "inclusivity" or "diversity" or "concern about someone taking offense" anymore.

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