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 2 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 4 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.

August 29, 2021 at 12:02am

#1016296

The Next Generation

Here's something I've been ranting about for a long time, but this time with arguments from... well, someone other than me. In this case, an actual sociology professor; that is, someone who actually studies this crap.

Comment: Bye, boomers; and millennials and Gen X and Z
Naming generations works against understanding societal similarities and differences. Stop it.

I used to wonder what they'd call the generation after Z, which is, of course, the final letter of the alphabet. Then I thought "what does it matter? We're not going to last that long." Boringly, the people whose job it seems to be to slap labels on everything decided to call them Alpha or some shit like that, which will definitely end well, considering what little conversation I've been privy to from the manosphere.

Consider these facts: The tennis champion Williams sisters are a generation apart, according to the Pew Research Center. Venus, born 1980, is part of “Gen X”; Serena, born 1981, is a “millennial.” Meanwhile, Donald Trump and Michelle Obama are both in the same generation. The former was born in 1946 while the latter was born in 1964, making them both “baby boomers.”

I've been saying shit like this for a while now, without those particular examples. Consider, for example, a hypothetical pair of twins born 10 minutes apart -- one at 11:55 pm on December 31, 1964, and the other being born at 12:05 am on January 1, 1965. The former is officially a Baby Boomer. The latter is officially Gen-X. Whatever. Meh.

I'm an early Gen-Xer by any definition in use. The whole "generations" astrology thing proposes that I have more in common with someone born in -- depending on who you ask -- 1984 (my first year of college) than with someone born in 1964 (with whom I could have gone to middle school). This is grade-A cow manure.

People are, for better or worse, born every year. It's a continuum, not a series of quantum jumps. Well, okay, but only very tiny quantum jumps, definitely much finer-grained than this "generations" crap.

Anyway, back to the not-me argument.

Generation labels, although widely adopted by the public, have no basis in social reality. In fact, in one of Pew’s own surveys, most people did not identify the correct generation for themselves; even when they were shown a list of options.

I called it "astrology" above for a reason.

Instead of asking people which group they feel an affinity for and why, purveyors of social “generations” just declare the categories and start making pronouncements about them. That’s not how social identity works.

Quelle surprise.

The practice of naming “generations” based on birth year goes back at least to the supposed “lost-generation” of the late 19th century. But as the tradition devolved into a never-ending competition to be the first to propose the next name that sticks, it has produced steadily diminishing returns to social science and the public understanding.

It's still a thing that exists, so it's helpful to know its origins.

There is no research identifying the appropriate boundaries between generations, and there is no empirical basis for imposing the sweeping character traits that are believed to define them. Generation descriptors are either embarrassing stereotypes or caricatures with astrology-level vagueness. In one article you might read that millennials are “liberal lions,” “downwardly mobile,” “upbeat,” “pre-Copernican,” “unaffiliated, anti-hierarchical, [and] distrustful”; even though they also “get along well with their parents, respect their elders and work well with colleagues.”

They can go sod right off with that "pre-Copernican" bullshit. I don't know exactly what they mean by that, but to me it signifies an abandonment of scientific principles and the belief that the Earth is the physical as well as metaphorical center of the Universe.

But what’s the harm? Aren’t these tags just a bit of fun for writers? A convenient hook for readers and a way of communicating generational change, which no one would deny is a real phenomenon? We in academic social science study and teach social change, but we don’t study and teach these categories because they simply aren’t real. And in social science, reality still matters.

I could dispute that last claim, but I'm wallowing in confirmation bias here and don't feel like it. Besides, I'm cynical and disaffected. I know this because I'm Gen-X.

Worse than irrelevant, such baseless categories drive people toward stereotyping and rash character judgment.

So we're trying to stop judging people by racial stereotypes and now we're judging them by some purely arbitrary chronology.

There are lots of good alternatives to today’s generations. We can simply describe people by the decade they were born. We can define cohorts specifically related to a particular issue; such as 2020 school kids. With the arrival of “Generation Z,” which Pew announced with fanfare, there has never been a better time to get off this train.

If only. No, this crap will continue because people love shortcuts.

When I was a kid, younger folks were blaming older folks for all of society's ills. Now I'm older, and younger folks are blaming older folks for all of society's ills. Specifically, the same older folks who once set out to change the world, and either failed miserably, or succeeded miserably.

An argument could be made by any younger cohort that everything's the older cohort's fault, because, after all, the older cohort has had the opportunity to do stuff or not, while the younger cohort has, for the most part, not. Meanwhile, the older cohort can always blame the younger cohort for not having the life experience necessary to understand why the older cohort chose the things they did.

As far as I can tell, this has been going on since at least the time of Socrates: “The children now love luxury; they have bad manners, contempt for authority; they show disrespect for elders and love chatter in place of exercise. Children are now tyrants, not the servants of their households. They no longer rise when elders enter the room. They contradict their parents, chatter before company, gobble up dainties at the table, cross their legs, and tyrannize their teachers.”

But more likely, since way before then.

I can acknowledge that there might be some value in identifying and naming generational cohorts. But the way it's done seems to me to be counterproductive to anything except finding new ways to divide us at a time when we need to be united.

As for me, I've always had trouble identifying with a group. And when they try to shove me in one, I rebel. Again, it's like with astrology: Us Aquarians don't believe in that stuff.

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