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:

Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 8 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

July 22, 2021 at 12:02am

#1014104

Dread

Just a few more prompts from "JAFBG"

[XGC]. This is one of them.

What upcoming event are you dreading and why?

Anyone who's been following along here knows my answer to this one.

When I was somewhere in my 20s, I worked for a home construction and addition business. Safety procedures were lax at best, but I did have safety goggles of a sort, if not a hard hat. One day, I was working on some rich asshole's garage when my glasses fell right down a stack of cinderblock cells, too far down to reach. So I kept going without them, because it turns out that in order to collect a paycheck, one has to work.

Naturally, it was shortly thereafter, as I was attempting to hammer a nail through a nice springy sheet of plyboard, the nail flew up and popped my eyeball.

Surgery to repair torn eyes has supposedly come a long way since then, but at the time, I'm told they put nine stitches in my cornea. This happened under general anesthesia, so all I remember was passing out and then waking up with a patch over my left eye. That was bad enough, but in order to keep the stuff inside my eyeball from leaking to the outside of my eyeball, they sealed the suture holes with cyanoacrylate.

Yes, that's what's commonly known as superglue. I expect it probably had a somewhat different formulation, but it was still runny enough that it leaked onto the sclera, the part of the eye that's usually white but turns red whenever someone drinks or gets cyanoacrylate on it.

I've been in pain several times in my life. Appendicitis, recovery from the appendectomy, bee stings, a goddamned heart attack, toothaches, paper cuts, whatever. Nothing, and I mean nothing, was as unbearable as the pain of the doctor getting CA on my sclera. That shit hurt like someone was rubbing sandpaper on my eyeball. The initial injury didn't hurt that bad. The surgery itself didn't hurt at all (because like I said, I was completely conked out for it). But I remember sitting up in the recovery bed screaming like I was finally ready to tell the enemy all of our troop numbers and deployment.

Eventually, they took the patch off and, in fact, the vision in that eye improved. The ophthalmologist wrote me up in a paper somewhere and I'm told it led to advances in LASIK and other corneal adjustment procedures.

Not worth the pain, though.

Some time after that, I heard a real-life horror story of someone undergoing eye surgery, and the anesthetic didn't work as advertised. The patient was awake for the whole procedure, able to see and hear the surgeons and nurses doing their thing and talking about it -- but unable to move a single muscle, including the ones controlling speech. Can't move, can't breathe (that's handled by a machine), can't do anything but lie there and be tortured. As someone who occasionally enjoys (that's a joke) sleep paralysis, this freaked me out more than clowns, spiders, needles, heights, or tall clown spider needles. As I recall, that patient spent a long time afterwards, perhaps their entire life, with something like PTSD, never right in the head afterwards.

Whether their vision got fixed or not, I don't know.

Perhaps because of this, the one phobia I have that I'm aware of is of something touching my eyeball (though my eyelids get a pass). I never could wear contact lenses because of it, and whenever I need even something as innocuous as eye drops, I have to trick myself into applying them.

So perhaps you can understand why I'm dreading cataract surgery -- an outpatient procedure that's over in a few minutes and has a really good track record of safety and efficacy, but involves only local anesthetic. You're awake for it, with your eyes wide open -- probably via clamps like Alex at the end of A Clockwork Orange

-- and you get to watch as they break up your eyes' lenses with ultrasound, suck out the pieces through slits in the corneas, and replace them with shiny new artificial lenses.

None of which would bother me all that much if it weren't for the "being awake and aware" part. I guess they give you Valium or some shit so you don't freak the fuck out, but I'm not sure that's going to be enough for me.

So, yeah, fine, I'm dreading it. I'm going to do it anyway because being able to see is important to me (it kind of helps with driving), but that doesn't mean I have to like it.

I've already roped a friend of mine into taking me there, even though the surgery date isn't set yet. Hopefully there will be beer nearby when it's over, because nothing says "pass out completely" like alcohol and Valium.

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