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 8 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 10 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.

February 26, 2022 at 12:02am

#1027449

Degeneration

Flight has been delayed until today, so I'm doing one more midnight entry...

Don’t Be So Quick to Stereotype Generations
A new book shows how wrong-headed our assumptions are about different generations and promotes understanding and connection.

Yes, it's a book promotion. As always, that's not going to stop me from posting it here, on a site for writers.

We’ve all heard the stereotypes before. The Greatest Generation is “responsible and hard-working”; Baby Boomers are “selfish”; Gen Xers are “cynical and disaffected”; Millennials are “entitled and lazy”; Gen Zers are “civic-minded.”

To be fair, I'm cynical and disaffected. Hell, whenever some article comes up about the war between Millennials and Boomers, I'm just glad Gen-X is overlooked. As has always been the case.

But, while characterizing generations is a common practice, it’s often counterproductive, says Bobby Duffy, director of the Policy Institute at King’s College London and author of a new book, The Generation Myth: Why When You’re Born Matters Less Than You Think. Duffy argues that assigning cohorts of people particular traits misses the importance of outside factors affecting their attitudes and actions.

Yes, important things like your sun and moon signs.

Plus, it takes us down a fruitless path of pitting one generation against another, creating division.

This is the part I have the biggest problem with. Don't we have enough artificial separators? Political leanings, sports team support, nationality, race?

“Although it is possible to learn something invaluable about ourselves by studying generational dynamics, we will not learn these lessons from a mixture of manufactured conflicts and tiresome clichés.”

I missed "invaluable" the other day when I was talking about words that seem to mean the opposite of what they do, but don't, the other day. "Invaluable" is like "inflammable," even though the "in-" has a different meaning for each: it's not a negation of the rest of the word. No, "invaluable" is basically a particularly intensive form of "valuable." A synonym is "indispensable," which is indeed an antonym of "dispensable."

Another synonym is "priceless" which is the surprising opposite of "worthless."

English is weird.

But I digress.

As a course corrective, Duffy provides longitudinal data on a multitude of issues—from obesity to views on pre-marital sex to car ownership and much more—showing how generations respond to different social, health, and economic trends.

I can accept that there are differences between older and younger people. Always have been, probably will be until we wipe ourselves off the planet. The problem, as I see it, is taking those differences and assuming they apply to everyone.

Let me provide a related example.

Genetically, as I've noted before, I'm as Euro-American as one can be. Pale skin, blue eyes, light straight hair, etc. There's a common stereotype that white people, especially those born in the American midwest as I was, simply cannot abide any spice stronger than mayonnaise. This has led to me visiting, for example, a Thai restaurant, specifically ordering the highest spice level possible, and receiving the culinary equivalent of a few flakes of black pepper.

As far as racist stereotypes go, I realize that's a tame one, but like I said, it's just an example, and one that hopefully doesn't start yet another war. Point is, by applying the "white people can't handle hot spice" stereotype to me, it misses the simple fact that I love me some really hot peppers. My favorite hot sauce is a habanero thing from Belize, and I've also been known to add ghost pepper to dishes.

As an aside, the spice we know as "pepper" that's a staple on restaurant tables across the country (at least before people started worrying about fomites, which, as it turns out, isn't as big a deal as was first thought), isn't closely related to the vegetable fruit known as "pepper." They're not even from the same freaking continent.

English is weird.

Part of what drives generational stereotyping is uncertainty of the future and worry that our children will not do as well as we did in life, says Duffy.

It is clear that, as a group, they will not. Which is one reason I've been so adamant against having kids myself: I saw the writing on the wall as early as 1980.

In other words, our generational stereotypes of selfish Boomers and caring Gen Zers can be misleading. Plus, they can cause people to put too much faith in younger generations, thinking they will solve climate change rather than all of us stepping up to do something.

You want to know how to solve climate change? Well, I've been saying for years that the obvious solution to global warming is nuclear winter. But seriously, though, climate change is human-caused, and making more humans is only going to make things worse.

Another persistent myth—that Gen Xers and Millennials are lazier, more materialistic, and less willing to act responsibly than other generations—obscures more important changes that are happening in society. When you look through the data, it becomes clear these stereotypes are ignoring long-term trends in rising wealth inequality, income stagnation, the need for more (and more expensive) education to compete in today’s economy, and devastating market crashes.

While it's true that I'm lazy and materialistic, that's one data point, and nothing can be learned from that concerning anyone except me. I did want to say one thing about "market crashes" though: my parents both lived through the Great Depression, and every market crash since then, including recent ones, have been peanuts compared to that. Also, at some point, the US stock market became disconnected from the general economy. I can't pinpoint exactly when that happened; I'm sure if you ask six economists, you'll get ten answers.

To solve the problem, Duffy argues, we need to get away from blaming the victims and prioritize affordable housing and rent control for vulnerable young people.

The housing thing is complex and those solutions are probably too simple, but they'd be a start. Of course, they're not going to happen.

Another thing to consider as a partial solution is to stop stigmatizing living with others, as roommates or in an extended family situation. The latter has been pretty normal for most of human history, until at some point they decided that virtue meant going off on your own at 18 or so. My inner cynic (you know, about 95% of me) believes that this was pushed on us in order to boost the economy by selling more houses and building more apartment buildings. It was very effective; people still mock 40-year-olds who live with their parents, even if said adults are taking care of their elderly relatives.

In any event, while it's a good thing to have and stick to a budget, no one today is going to be able to buy a house solely by foregoing their $5 Starsucks lattes in favor of making coffee at home.

Take the current media frenzy over social media’s impacts on Gen Z. This follows a well-worn pattern: Each successive generation has found some kind of new media or technology to blame for the woes of youth, including books, radio, comics, TV, and now social media.

I keep seeing bullshit about people stuck to their phones. When I was in high school, the moral panic was about Walkmans (Walkmen?) -- for my younger readers, it's the thing from Guardians of the Galaxy that Star-Lord always used (this, incidentally, was the most unbelievable part of that movie; a Walkman lasted, at best, two years, and a cassette tape had a lifespan of maybe ten). Older people had the same kind of complaints: "Kids these days and their headphones, tuning out the world around them." Now that the Walkman generation has become older, they're saying "Kids these days and their smartphones, tuning out the world around them."

Incidentally, the "-phones" part of "headphones" and "smartphones" has a different meaning in each.

English is weird.

Anyway, I guess I'm more aligned with the younger people in that regard; there's no difference, to me, between talking to someone online and talking to someone in person -- though it is rude to do the former while also doing the latter. And being an introvert, I tend to limit both.

Though stereotyping is wrong, Duffy does find actual generational differences in attitudes and behavior that might be instructive. For example, older generations attend religious services more regularly than younger generations, with each generation attending less often than the previous one.

I consider this a good thing.

With every successive generation, drinking alcohol has decreased, too—one of the most consistent cohort effects discussed in the book.

I consider this a bad thing.

All in all, if we want to make the world a better place and see thriving future generations, we need to get away from stereotyping and stop pitting generations against each other, which serves no one. Instead, we must find more ways to be together and connect, sharing the necessary work of making the world a better place for current and future generations.

And one way to do that is to promote more connection on the internet -- preferably outside the boundaries of the cesspool that is social media.

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