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:

tj-Merry Mischief Maker

about 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

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.

2 Like • Comment • View 3 comments

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