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 5 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 7 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 6, 2022 at 12:02am

#1034755

Propaganda: A Socially Correct Goose

Sent to me a while back by Turkey DrumStik

, today's article is not one I have a lot to say about. It's an opinion piece, and I'm not sure how much I agree or disagree with it; it's a point of view to consider.

Why a push to exclude Russian state media would be problematic for free speech and democracy

Admittedly, I haven't been following the Ukraine situation as much recently. But I think there are other issues to consider here.

In the wake of the Russian invasion of Ukraine, Russia has excluded media outlets based in the U.S. and allied countries. Facebook, Twitter, Instagram, Radio Free Europe, Deutsche Welle, BBC, and other news sources are no longer legally available there. A self-imposed digital iron curtain has descended across Europe, enforced in part by the new draconian Russian law criminalizing the distribution of “false information” about the Ukraine conflict.

Here, "false information" translates to "actual facts." Russia is no stranger to suppressing anything their government doesn't want you to know.

Russia doesn't have an equivalent to our First Amendment (or if they do, they roundly ignore it like they ignore most of their post-Soviet Constitution). The difficulty for us (by which I mean not just the US but similar democracies) is that free speech can also lead to the protection of fake news, like what you get on Fox. And sometimes by our own governments.

But the closing of the information space has not been one-sided. The European Union has banned Russian state media in Europe and U.S. companies have made it more difficult to access these media outlets in the U.S., thereby reducing the range of information sources available to the public here in the West.

It's easy to say "they're lying anyway, so screw 'em." But the chilling of any speech runs contrary to our values.

Senator Mark Warner sent a public letter to tech company CEOs urging them “to prevent misuse of their platforms by Russia and Russia-linked entities.”

Yes, he's one of mine. Pretty sure he's overstepping his bounds here.

Part of the stated basis for blocking Russian state media organizations is that they are state-controlled and engage in propaganda to further Russian foreign policy aims. But these organizations have always been state-controlled and have always echoed Russian government propaganda. The new element is the emergency of the Ukraine war, where Russian propaganda could aid its war effort by undermining the domestic will to fight in Europe and Ukraine. This suggests that the legal ban on RT makes sense only as a huge exception to a background policy of openness.

It was another senator, over 100 years ago, who said something like, "Truth is the first casualty of war." This was before the big conflicts of the 20th century, but it seems to have held up. In this case, though, it's lying propaganda they're trying to suppress, not "truth." But how are we to know the difference if we don't check multiple sources and have some idea how what they say matches reality?

Still, isn't it in our best interests to have some external confirmation of the truth or falsehood of what someone's saying? Well, the linked article points out one possible effect of this:

Shutting out state media from foreign adversaries also makes it easier to treat domestic critics of U.S. policy as agents of a hostile power who are amplifying the enemy’s talking points. If it is legitimate to silence state media as agents of foreign propaganda because of what they say, commentators and policymakers are likely to think that it must be legitimate to label anyone who says the same thing as also an agent of a foreign power.

Taken to an extreme, it can be used as a mouthpiece for whatever propaganda our governments want to promote -- the polar opposite of freedom of speech. We saw some of this with the McCarthy (another senator) hearings in the 50s. Well, I didn't; I hadn't been born yet, but that whole episode always struck me as anti-American. We're supposed to consider all viewpoints.

For generations, U.S. communications policy has been premised on the idea, articulated in the Supreme Court’s 1945 Associated Press decision, that “the widest possible dissemination of information from diverse and antagonistic sources is essential to the welfare of the public.”

It's hard sometimes to tell truth from fiction. Or even from opinion (and I say again, the linked article is an opinion piece). But I don't think that suppressing free speech makes that any easier. As soon as you have one source that's effectively a mouthpiece of the government, you immediately run the risk of having that mouthpiece just regurgitate whatever lies they want you to believe. And then the "enemy" wins because we've become just like them.

So anyway, draw your own conclusions if you read the article. Or, like me, don't conclude anything, but at least think about what the author is saying.

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