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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Show Recent Comments

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 27, 2022 at 12:02am

#1035740

Enablers

Today's article, sent to me a while back by Turkey DrumStik

, is a long read, and despite the partisan title, it has far more general implications than just one political party's shenanigans. Consequently, any political bile spewed at me for this will be summarily ignored.

The Losing Democrats Who Gobbled Up Money
Amy McGrath and other Senate candidates deceived donors to rake in far more cash than their Republican opponents. They got crushed anyway.

There has, historically anyway, been a correlation between money raised and election success. The problem, as is the case with most issues involving correlation, is that it doesn't imply causation. In other words, were politicians who raised more money successful because they raised more money? Or was it because the tactics needed to raise money also translate to garnering votes? Or, another hypothesis which is mine, are the people who raise more money able to do so because they already have a wide base?

As this article points out, spending more money on your campaign doesn't always affect the only poll that matters.

The long-term danger is that small donors, barraged with overheated email pleas that range in veracity from half-truths to outright bullshit, will eventually catch on.

Nah. To do that, they'd have to read this article, and it's about 20 times longer than the average voter's attention span. It's also about twice as long as mine, so I'm not going to bore you by repeating a lot of quotes here. Just the stuff I find most interesting or amusing.

Josh Nelson, a Democratic digital strategist, is among the operatives, most of them from the Democrats’ progressive wing, trying to get the party to abide by more ethical standards. “I just don’t think you can view people on your list as ATM machines, or like they’re idiots,” he said. “It might work for a while, but it’s not sustainable.”

Protip: Do not take any advice from someone who calls it an Automated Teller Machine machine.

In the long lead-up to elections, there are only two metrics for the public to follow. One is polling. The other is the money. Campaigns boast about their one-day hauls and quarterly totals, and the political press covers it as if it’s a sport. And there is useful information in the fundraising numbers. Who’s giving and how much. But the biggest truth gets buried: Money is the most overrated factor in politics.

Full disclosure here: I have never contributed money or effort to a political campaign. You know those boxes on your tax form where you can donate to the Presidential election fund (for y'all foreigners, that's really a thing, but it doesn't change your tax owed)? I've never checked those. I consider it aiding and abetting a politician in the commission of an election.

Yes, I vote. Yes, I'm pretty one-sided in my voting. I just thought the process sucked even before I read this article.

Because who does that fund-raising really help? Not the politicians (at least if they don't skim off the bottom). It's kind of like hiring someone to do SEO for your business. Once everyone does it, search rankings return to what they would have been without optimization. No, the donations contribute to ad agencies, media outlets, consultants, etc. And certainly it doesn't help the donors, because they'll never get that money back.

On the other hand, I don't have any good ideas for what they call campaign finance reform. So I just stay out of the process as much as possible.

Even in the digital age, local broadcast TV still accounts for the biggest share of campaign advertising, as high as 60 percent.

I guess some people assume that I don't watch broadcast or cable TV because I'm some sort of hipster. Nothing could be further from the truth (okay, maybe when it comes to beer I'm a hipster). It's because of ads. 99% of all ads suck, and if I have some burning desire to watch one of the other 1% because people will not fucking shut up about it, it'll be on YouTube. I despise ads with the fiery passion of ten thousand suns; the only thing I hate worse than ads would be having to pay to watch them, and that's the reason I never had cable.

But of all the drooling, mouth-breathing ads out there, by far the absolute worst are political campaign ads. Granted, because of my aversion, I haven't seen a whole lot of them, but those that I have been somehow subjected to (there are often TVs in bars and at friends' houses), they lie even worse than product commercials and, worse, the ones I've been subjected to would rather tear down the other candidate than prop up the one they're supposed to promote. "Vote for Smith! Because Jones has been accused of eating babies and kicking puppies. Do you want a baby-eating puppy kicker representing YOU?"

When campaigns are so flush, they do not have to spend with much discipline. To give one example, McGrath’s advisers were skeptical about the impact of direct mail—figuring that in lots of households campaign literature went straight into the trash.

And that's another example of being treated like subhuman scum by the people running these ad campaigns. I don't throw campaign literature or church fliers in the trash. Hell no! They go into recycling, in the hopes that they'll be repurposed into something actually useful to society. Like maybe joint rolling papers or beer bottle labels.

Anyway, the article's there if you care. I actually managed to read the whole thing. It didn't send me into paroxysms of outrage, not like the ads themselves do, but it did convince me that Something Needs To Change.

As to what that Something is, well... shrug.

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