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 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 9, 2022 at 12:02am

#1028604

Intolerable

Today, I'm going to talk about politics.

Reducing extreme polarization is key to stabilizing democracy

By "talk about politics," I mean that I'm actually talking about talking about politics -- I'm not here to start arguments or promote my own point of view (though I can't promise I'll be entirely impartial).

Political polarization, or the “violence of faction” James Madison warned of in Federalist 10, is as great a threat to democracy today as it was in 1787, dividing voters and their representatives into diametrically opposed camps that are unwilling to compromise or yield power to their opponents.

At some point, it becomes tribal. I've seen people treat politics like they do football, as if it's a spectator sport.

The January 6th insurrection on Capitol Hill over one year ago, and the United States’ ongoing struggle to respond effectively to threats such as the COVID-19 pandemic and climate change, warn us of the dangers posed by unchecked polarization.

As writers know better than anyone, words have power. Though I agree with the sentiment here, I feel obligated to, in fairness, point out that calling the events of 1/6/21 an "insurrection" already gives away at least some of the authors' bias. The other side is trying to frame it as a "protest." One could also call it a "riot," or a "coup attempt," or "trespassing." Also, at this point, even acknowledging that the pandemic is a "threat" (or, in some cases, even exists at all), and asserting that climate change is a) real and b) a problem (let alone whether it's anthropogenic) also results in pushback from the other side.

Given that level of extreme polarization, where some people ignore the existence of certain facts, what hope do we really have?

Well, I have the hope that the article will shed some light on that.

Our contribution, recently published in the Proceedings of the National Academy of Sciences, identifies three key mechanisms for avoiding polarization: (1) increasing tolerance, the range of opinions that individuals find attractive; (2) limiting the radicalizing influence of repulsive extremists; and (3) incentivizing non-extremist policies that align with individuals’ self-interests.

So they're stating their findings up front. Good. The rest is commentary, but important commentary.

We also find that external shocks—such as wars, economic crises, or climate disasters—can consolidate individuals’ opinions into less polarized distributions, but only if they are very strong or if they occur before patterns of extreme polarization have set in.

I should note that this article was published before the Russian invasion of Ukraine, which seems to have promoted at least a little bit of bipartisan cooperation here.

Low tolerance among actors causes flurries of repulsive interactions and the rapid onset of extreme polarization, while moderate to high tolerance does the opposite. How, then, do we foster greater tolerance to dissimilar and opposing points of view?

I still don't see how this can happen the way things are now. And some things simply should not be tolerated -- for example, if group A has the stated objective of killing or enslaving group B, I can't see there being any middle ground.

We need look no further than Twitter to see how exposure to a wide range of ideologies does not necessarily increase tolerance; on the contrary, social media interactions often reinforce deep divisions.

This is one reason I avoid Twatter.

As the saying goes, “the enemy of my enemy is my friend.”

Yeah, that worked out well when the US and USSR were both opposing Nazi Germany in WWII.

Is polarization mitigated—even temporarily—in the face of external shocks such as war, climate disasters, or economic crises? According to our model, it depends.

Of course it does.

Our research highlights the importance of fostering tolerance beyond just exposing people to new ideas. Interventions that focus on empathy and connection, like storytelling, may succeed in bridging ideological gaps where others have caused repulsive backlash.

Naturally, I don't have anything against storytelling per se. But, again, these days, if you tell a true story about someone's experience, the other side is just as likely to mock it and call it fake news as they are to simply dismiss it. As an example, many, many stories have been told about people who refused to get vaccinated against the 'rona, then got sick and wished they'd gotten the vaccine, then died. This doesn't move some people, whose lived experience is people who got sick and then (to some degree) recovered. As further examples, consider all the people who sincerely believe that a certain school shooting was faked with crisis actors, or who screech about how if someone on their side did something heinous, they must have been acting on behalf of the other tribe to make them look bad.

In other words, I think storytelling only works when it aligns with peoples' preconceived ideas. But I'm not a researcher; these authors are.

And then there's the thing I've mentioned in here before: that a person's lived experience is relatively useless for making informed decisions. Those rely on statistical analysis, not things like "Well, my uncle wasn't wearing his seat belt and survived the crash." But people in general don't understand statistics, or they think someone's manipulating the numbers to lie to them (to be fair, this has happened), and they don't understand the concept of risk. "You can get in a car accident even if you haven't been drinking!" Yes, this is true, but that doesn't change the fact that drinking increases the chance of an accident.

With the 2022 midterm elections on the horizon, it is critical that candidates and the media resist the poison of extremism rather than attempting to gain from it.

Yeah... that's not going to happen. Extremism is interesting. It gets views. Whether you agree with the extremists' point of view or not, you're watching the train wreck. News outlets cater to that, on all sides.

So, really, I'd suggest reading the whole article; there's a lot more here that I didn't cover. To sum up my thoughts on it, though, I'd have to say that it doesn't matter how much science you do when a) there's a group of people who reject the very idea of science and b) no matter how much you tell people what to do to change, they won't, because they simply won't be told what to do by anyone not of their tribe.

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