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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Brandiwyn🎶

about 22 hours ago
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.

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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').

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

July 30, 2024 at 10:42am

#1074579

The Habits of Nones

Today's article, from The Conversation, pushes me into the territory of one of the Forbidden Subjects. But ultimately, it's about science, albeit the fuzzy branch known as "social science."

The number of religious ‘nones’ has soared, but not the number of atheists – and as social scientists, we wanted to know why

One problem with polls is that, often, they don't ask the right questions. Or there's implicit or even explicit bias, like how a political party might send out a poll like: "Who do you plan to vote for at the next election? a) Our fine, upstanding candidate; b) The other party's incompetent fool; c) Not sure; d) Not planning to vote.

There, now, I've put politics out there, too. While we're on Forbidden Subjects.

The number of individuals in the United States who do not identify as being part of any religion has grown dramatically in recent years, and “the nones” are now larger than any single religious group.

Well, now, that's going to depend on how one defines "religious group," doesn't it? I don't have the exact numbers, but if you broke it down by "Christian, Muslim, Jewish, Hindu, Buddhist, Pagan, Other, None," I strongly suspect Christians would be in the majority here in the US. But if you further subdivide the categories, say, by separating Christian into "Catholic, Presbyterian, Baptist, LDS, Lutheran, Episcopalian, Methodist, Pentecostal, Mennonite, etc." then okay, I could believe the bit about Nones. Hell, some of those sects don't even consider some of the others to be Christian; it's only my outsider's perspective that would lump them all under one heading.

Not to mention "None" could be extraordinarily confusing in conversation, thanks to homonyms.

According to the General Social Survey, religiously unaffiliated people represented only about 5% of the U.S. population in the 1970s. This percentage began to increase in the 1990s and is around 30% today.

"Religiously unaffiliated" doesn't imply "atheist." People can be religious without identifying with a particular group, or not religious while identifying with a group.

At first glance, some might assume this means nearly 1 in 3 Americans are atheists, but that’s far from true. Indeed, only about 4% of U.S. adults identify as an atheist.

As noted above, how these things are worded makes a big difference. "Do you believe in God?" is, at first glance, a yes/no binary question, but it can get complicated, what with varying concepts of God and all. Also, "I do not believe God exists" is an entirely different assertion than "I believe God does not exist." Add to this the demand for self-labeling, and things get really murky (one reason I called social science "fuzzy" above). Hell, if someone were rude enough to ask me flat-out what my religion was, I don't think I could boil it down to one word. Unless I was in a joking mood, in which case I'd answer, "Beer."

According to the website of American Atheists,

"Atheism is not a disbelief in gods or a denial of gods; it is a lack of belief in gods."

Unfortunately, by strict application of that definition, monotheists would also be covered, as they do not believe in "gods" but in one "God."

So you see where some of this confusion might be coming from?

There is even diversity in how religiously unaffiliated individuals identify themselves. When asked their religion on surveys, unaffiliated responses include “agnostic,” “no religion,” “nothing in particular,” “none” and so on.

Break down the categories of belief/unbelief finely enough, and you get to the point where each individual is their own religious category.

Our recent research examines two questions related to atheism. First, what makes an individual more or less likely to identify as an atheist? Second, what makes someone more or less likely to adopt an atheistic worldview over time?

"Examines" is a good verb there. It would be wrong, I think, to take their findings as the one definitive, true answer, but it's interesting to see how they're looking at it.

Said examination follows, and it's brief, so I'll just skip to near the end:

On the other hand, we find that adults with more income... are more likely to adopt the stance that they do not believe in God...

This could be a function of income providing a buffer against any stigma associated with holding an atheistic worldview...

Some social scientists have suggested that both wealth and faith can provide existential security – the confidence that you are not going to face tragedy at any moment – and therefore a higher income reduces the need to believe in supernatural forces in the first place.

Which I think a lot of people have internalized, hence why religious groups tend to demand money. Keep the people poor and relying on them, not on their own financial independence. Your first clue that you're being fleeced should have been when they called you a "flock."

Agree or not, "In God We Trust" is on our cash. I think many houses of worship should have signs like "In Cash We Trust."

I want to try to be clear about this: I'm not ragging on religion in general with those comments. Only on the ones that are such naked money grabs that strippers are going "Okay, now that's a naked money grab."

In conclusion, I'm not coming down for or against the ideas presented in the article. But my skepticism moved me to comment on it.

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