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.

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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.

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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.

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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.

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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.

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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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