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:

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 15, 2024 at 12:00pm

#1073994

Pray Tell

By random coincidence, hot on the heels of yesterday's recapitulation of a philosophical blog entry from the noughties, we have a philosophical question from Big Think:

Everyday Philosophy: Is praying fundamentally egotistical?
God is not a vending machine, but is it wrong to treat him like one?

Based on the subheadline there, the article's focus would seem to be Western Christianity, not one of the many other religions that involve prayer. Being an English language website, though, I'd expect most of their target audience to be familiar with that religion, so, okay, I can ignore the lack of universality in this case.

To answer this week’s question, we look at the work of Sigmund Freud on religion and the theologian Friedrich Heiler on prayer.

And already, I spot red flags. Freud was, famously, the prototypical psychoanalyst; while psychology and religion are necessarily intertwined, pretty much everything Freud actually said was later debunked, so it's worse than citing Wikipedia (which is usually factual but needs to be checked). And while I'd never heard of Heiler, the entire field of theology is suspect because, unlike with psychology (for instance), you can't design experiments to support or falsify the hypotheses.

I'm not saying the question isn't worth examining. Just that I'd probably have looked elsewhere for inspiration.

The whole thing starts with a question purportedly posed by a reader:

Is it egotistical to think your prayers will be answered or that what you wish for will come true?

The question itself is framed in a way that bugs me, but I can't quite put my finger on why. Is the implication that "egotistical" is necessarily bad? Do a lot of people hold the worldview that what they wish for will necessarily come true?

Mostly, I hear about people who claim their prayers were answered in an unexpected or twisted way, meaning they didn't get what they wanted but were okay with that on some level.

It’s 1590, and two armies stand in the fields just outside Ivry in France. Both armies are a rag-tag shamble of children, old men, cripples, and foreign mercenaries. After 30 years of civil war, that’s all that’s left. In this latest, macabre chapter of the French Wars of Religion, priests walk up and down the lines. They tell everyone standing that “God blesses you,” “This is all for God,” and, most importantly of all, “Please God, give us victory this day.” Protestants and Catholics are praying to the same God to give them both victory. God will have to disappoint a lot of people.

I don't know enough about history to know who won that battle, and can't be arsed to look it up. But we all know that religious wars happened, and they happened with alarming frequency, and in each case, both sides thought that the Supreme Being was on their side, and at least one of those sides were apparently wrong.

Still, I might have used a different example. A football game, say. A few months ago, a bunch of 49ers fans fervently prayed to God to make their team win. A bunch of Chiefs fans fervently prayed to God to make their team win. And yet, after the Chiefs won, as far as I've been able to tell, the 49ers fans didn't suddenly become atheists en masse.

Sure, that's lower stakes than an actual fighting war, but it's probably more relatable to an audience. You could also use the example of elections, where presumably people are praying for their candidate to prevail, while only one can win. My point being that it's blindingly obvious that it's impossible for [deity] to answer every prayer. Unless, of course, [deity] is actually the interpretation of quantum physics that splits the outcome of every probabilistic event into different, separate universes, and elsewhere outside of our spacetime, there's a universe where San Francisco won—but that's irrelevant, because that's not visible in our reality.

Not only that, but because it's demonstrably impossible for [deity] to fulfill every desire due to inherent contradictions, anyone who thinks [deity] does isn't egotistical, but delusional. That, or extremely, unbelievably powerful.

I look at it this way:

Does [deity] grant your every wish? Then [deity] is your bitch, which makes you the true ruling force in the universe. Please don't smite me.

Does [deity] grant none of your wishes? Then why bother wishing? Alternatively, you can use that to your advantage by praying for the exact opposite of what you want. Please smite me.

Does [deity] grant some wishes and not others? Then how is that distinguishable from random chance operating within the structure of physical laws?

So I won't quote the rest of the article. It does provide different perspectives, but, as with all philosophy... draw your own conclusions.

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