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 11 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 13 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.

March 5, 2022 at 12:04am

#1028346

Unthinkable

Well, I did promise "posts of substance;" this came up at random, and it qualifies -- though there's not much for me to say about it.

Sick of choosing between the 'lesser of two evils'? Push for electoral reform

It could be argued that anyone with enough arrogance (however well they might hide it) to run for office is usually going to be evil, even discounting that what one tribe considers good, the other considers evil. Oh, sure, you can find counterexamples if you think for a few minutes, but the exceptions don't change the generality. So choosing between the lesser of two evils is almost always a given. Except, of course, if there's more than two people running, and then it's the least of n evils.

As the common adage goes, politics is about addition, not subtraction.

As I have never heard that (insofar as I can remember), I already know I'm probably over my head in this article.

Political addition is the foundation of a healthy democracy, but unfortunately, our current political system leaves many citizens out of the equation. The structure of elections, especially party primaries, contributes to this disparity.

Won't argue with that.

Exacerbating the situation is the fact that more than 80 percent of congressional districts are completely safe for one party.

That's 'cause they're desigerrymandered to be that way.

Anyway, the article spends a few paragraphs explaining the situation better than I ever could.

But is there a simple way to change the party primary system to foster more citizen engagement and more choice, without favoring one party over another?

I'm guessing they're going to say "yes."

An innovative idea that fits that bill is Final-Five Voting, put forth by the Institute for Political Innovation.

I must be psychic.

Final-Five Voting — this specific combination of top-five primaries and instant runoff-voting general elections — would have a powerful, positive impact on our system. Rather than forcing voters to choose the “lesser of two evils,” we’d infuse healthy competition into politics and help to ensure that candidates are more responsive and accountable to all the voters of the state or district they seek to represent.

In the Before Time, when I still had a gym membership, I'd alleviate the incredible boredom of being on an elliptical machine by watching lectures about shit like science, philosophy, and mathematics.

This is where you're tempted to comment, "But wouldn't that make it even more boring?" Well, no; not for me.

Anyway. I wish I could remember the program I was watching or even the slightest hint of the proof they used, but one lecturer made a convincing argument that there is not only no such thing as the perfect system for choosing leaders, but that there can never be the perfect system for choosing leaders. There will always be a downside, a certain level of unfairness.

But just because it is mathematically impossible to implement perfection -- which is usually the case in everything, not just elections -- doesn't mean we shouldn't strive to make something better.

As they say, "the perfect is the enemy of the good." If you resist change because the new system won't be perfect, there will never be change. Of course, a lot of people want exactly that.

The American system of representative democracy is in dire need of constructive reform that increases participation and avoids tipping the scales toward one party or another.

I'm not sure this particular essay has convinced me that this solution is the best of all proposed electoral reforms, but I hadn't heard this one before, so it was worth it to read. I do believe it would be better than our current system in the US.

Of course, it would dilute the power of the only people in the position to implement such a system, which means it'll never happen. It's akin to Congress voting to take a 50% pay cut, which of course is unthinkable.

But it's good to indulge in fantasy every now and then.

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