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 8 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 10 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').

1 Like

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.

June 11, 2022 at 12:02am

#1033707

Death Do Us Part

I pick these semi-randomly, and today I get to be controversial. I don't do that often, so fair warning. Won't hurt me if you skip this one.

On This Day
11 June 2001
Execution of a Terrorist

Let's leave aside for a moment how we probably ought not to be memorializing those who commit heinous crimes by plastering their name all over the place and thus giving them oxygen. Everyone knows this guy's name, though I'm not going to repeat it here (you can go to that link; it's from Wikipedia).

How many of his victims' names do you know? Or their surviving family members or friends?

Me? None. Not a single one.

What struck me when reading that page was how every damn thing there presaged a lot of the bullshit we have going on right now in the US. Not inflation, an impossible housing market, or higher gas prices; but the conflicts about legitimate government, guns, mass killings, the media's role in all this, etc. Nothing ever got resolved, even though another terrorist attack just 3 months later (to the day) put most of us on the same page for about 90 milliseconds before everything devolved once again into civil war.

We'll always have disagreements. We're not supposed to resolve them by shooting, blowing shit up, destroying skyscrapers, or attempting to overthrow a legitimately elected government.

But whatever. No, the truly controversial thing this led me to really think about was the death penalty.

I haven't discussed that in here recently. Long ago, once, and my opinions tend to shift. Yeah, I'm inconsistent and sometimes I contradict myself. "I am large; I contain multitudes." But I did find this bit from a very long time ago:

5. Do you believe in the death penalty?
In theory? Yes. There are some people we Just Don't Need. But that assumes a perfect world wherein we knew beyond the shadow of a doubt (not just "beyond a reasonable doubt") that person A committed a capital crime. But - here's the rub - in a perfect world, no one would commit murder. So in practice? No, I don't think the death penalty is a good idea. Every person's death diminishes the rest of us, whether someone believes it's deserved or not.

I still kinda believe that. But since then, I guess my view has evolved. Not that it matters worth a damn, just like all the rest of my opinions. Considering the number of people executed by some government or other for crimes it turns out they didn't commit, or even jailed for a long time while innocent, I still don't generally like it in practice. Hell, on second thought, don't consider the numbers. I'm not even going to bother looking up statistics, because even one is too many. And capital punishment is pretty damn final.

And yet, in the case above, of which today is the anniversary, I can't say I disapprove. We know the guy wasn't innocent. We know, insofar as we can really "know" anything, that he killed a bunch of people. Better if he'd been stopped first, but then we wouldn't have entire Wiki pages devoted to a non-attack, would we?

Problem is, we're pretty sure that the death penalty doesn't serve as a deterrent. Murderers gonna murder. Some of them do it in a bid to take as many people as possible with them when they suicide. So what purpose does capital punishment serve? Don't say justice; that's some Hammurabi shit right there. Vengeance is more like it. And keeping them off the street so they can't do more murder, but that can be arranged with a life sentence too.

There's also the argument that a state-sponsored execution costs society more than a life prison sentence, but that argument always struck me as incomplete. We could choose to make it cost less if we really wanted, and if, as I argue, such a sentence should be reserved for cases where we know beyond any doubt, there's no reason for endless appeals. Arguing on the basis of the cost is a little bit like arguing that you shouldn't smoke weed because it has bad effects. What are those bad effects? Potential loss of civil rights. Make it legal nationally and that argument goes right out the window. Not a great analogy; I know. But the point is, don't conflate physical laws with government laws.

By far the worst consequence, though, of this penchant we have for railroading certain innocent people into confessing to murder or whatever is not only that an innocent person gets locked up and/or executed, but the fucknut who actually did it is still out there wandering around.

But like I said, that's not the case here.

What we do have, though, is a martyr for his cause. Something for other racist bastards to rally around. Like I said, a lot of the bullshit he spouted is still being spouted to this day, and will continue to be spouted. Would this be the case if he'd received a life sentence instead? Well, that's above my pay grade. Maybe. I don't know. We'll probably never know for sure.

But that's why I'm avoiding the name. No more oxygen to these assholes.

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