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').

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.

July 20, 2021 at 12:02am

#1013969

Opinions Are Like Assholes

A possibly contentious "JAFBG"

[XGC] prompt...

How do you feel about the integrity of the news? Can we believe what we read/see? How much bias is there?

There is always bias in the news.

I mean always. Some things are actually impossible, and one impossible thing is to be without bias. Even if a news source can be trusted to report only the objective facts of something -- keeping opinion completely out of the text -- they demonstrate some bias in the simple act of deciding what to report on.

You hear the complaint all the time. Like, just yesterday, I found out that coal miners in Alabama are on strike,

and it hasn't been reported in the major news networks.

Well, actually, I only have their word that it's not reported on the major networks. I didn't fact-check that. It doesn't matter enough to me, and besides, it's very hard to prove a negative. But let's assume they're correct at that source, for now (at this point someone else may have picked up the story; again, I don't know). It demonstrates the point: One source finds this to be significant enough to report on, and another doesn't. Why? Who knows? Maybe the large corporations running the major networks (both left-leaning and right-leaning) don't want to give attention to workers' rights. Maybe there's too much else going on to squeeze it in. Maybe they don't want to take the chance of losing major advertisers.

In any case, "this is something They don't want you to know" is itself a biased (and clickbaity) tack to take when reporting on an event.

So, yes, there is always bias in the news. That's not the problem, not if you're aware of it and can ask your own questions.

The problem is that sometimes the news tries too hard to be unbiased.

I know I've harped on this before. For a long time, and it might still be going on, a news story about some new finding in, say, the field of climatology would be reported on like "Here is what scientists say is happening," followed, in some stupid attempt at "fairness," something like, "And now here's Joe-Bob insisting that the drought in the West is perfectly normal."

Those two things are not equivalents. It is as if you report on someone circumnavigating the Earth, and then switch to a flat-earther proclaiming that it couldn't have happened because the Earth isn't round. It would be like if you're doing a report on NASA sending another mission to the moon, and in the interest of presenting "both sides," you cut to someone who claims that no one got there in the first place.

We should not be giving equal time -- or any time, really -- to people devoted to denying the facts.

And that's just facts. Opinions don't have equal weight, either.

Like... take my second-favorite subject, beer. I like several different styles of beer, notably stouts and Belgians; I generally dislike West Coast IPAs and I utterly despise mass-produced watered-down piss such as Coors Light and Buttwiper. That is an opinion, and it is mine. Other people really enjoy West Coast IPAs and dislike Belgians. Some people even like Coors Light, or they wouldn't keep selling that crap. Yes, I have strong opinions about it. But my opinion is no more "true" than yours or his or hers or theirs, because what beer someone likes (or even if they enjoy the delicious amber nectar) is a matter of taste. And susceptibility to marketing strategies, but still, people like what they like.

In matters of taste, your opinion is right for you and mine is right for me. We might influence each other, but neither of us is objectively right.

But -- and I'm going to borrow from a comic I once saw -- say you're a passenger on an airplane. You may have an opinion of how the plane should be flown. But you're not a trained pilot. The actual pilot also has an opinion on how the plane should be flown. She or he, in that case, has a better opinion than you do.

They could still be wrong -- sometimes planes crash due to operator error -- but they have a much, much lower chance of being wrong than you do.

Opinions are like assholes: Everyone has one, and most of them stink.

Since a lot of journalists seem to have forgotten this hierarchy of opinion, giving apparently equal weight to evidence-based opinions and ignorant opinions, it's up to us to discern who we should believe when they make a statement about, for example, public health.

And obviously, as a group, we're completely incapable of doing that.

Which sucks. In my opinion.

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