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🎶

a day 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

December 28, 2020 at 12:53am

#1000921

It's In The Cards

"30-Day Blogging Challenge ON HIATUS" [13+]:
28. Card Playing Day!
Do you like playing cards? Which card games do you prefer?

Long ago, I used to host a Poker Night at my place. Unlike many such events, this wasn't a boys' club, nor was it merely an excuse for my group of friends to get together. I mean, that's what we were doing, of course, but poker was the purpose of the evening; it was a casual, low-stakes game, but we were still serious about it.

Nor was it a tense Hold 'Em tournament, but good old-fashioned dealer's choice: draw or stud or one of the many variations thereof.

Things drifted apart after a while, so this stopped even before the pandemic made hosting poker games a difficult proposition, but I do miss the games.

I've played a few of the purpose-built card games, you know, like Cards Against Humanity and others that use different decks, and those can be fun too.

When I go out gambling, though, my game of choice is blackjack, not poker. Or, at least, it was when I could still go out gambling.

I realize this isn't a particularly clever or funny entry; I'm just not in the mood to do clever and/or funny right now. But I feel like I should do more than a plain prompt answer, so I thought I'd point out some things about a standard deck of cards.

There are more ways to arrange a deck of cards than there are atoms on Earth

Consider how many card games must have taken place across the world since the beginning of humankind.

That's a stretch. Playing cards aren't much more than 1000 years old

, and the evolution to the modern, standard, French style deck of 52 cards arranged in four suits with aces, royalty, etc. took maybe 5-600 years, so we're not talking about "the beginning of humankind" but roughly "from about the time of Shakespeare."

It seems unbelievable, but there are somewhere in the range of 8x10⁶⁷ ways to sort a deck of cards. That’s an 8 followed by 67 zeros.

What they are describing there is the number of possible permutations of a pack of 52 cards. The number is written as 52!, which is read "fifty-two factorial," and is the result you get when you multiply all of the integers from 1 through 52: 1x2x3x4x5x...x50x51x52.

It is, as the article notes, a really stupendously huge number. Considering that the universe is, by comparison, only about 4x10¹⁷ seconds old, you can maybe start to see how people can say there has probably never been the same particular combination of shuffled cards in all of the history of cards.

Or, as the article puts it with even less clarity, To put that in perspective, even if someone could rearrange a deck of cards every second of the universe’s total existence, the universe would end before they would get even one billionth of the way to finding a repeat.

It is, however, the nature of humans to only notice when such shuffling produces "interesting" combinations of cards -- such as straights, flushes, or four-of-a-kind. When playing poker, we'll notice those (because they tend to be winning hands) far more often that the just as likely 6-10-3-K-8 of mixed suits, for example.

Cards are a good way to teach statistics and probability - but they're also handy tools for explaining the concept of entropy.

But mostly, I just miss playing poker.

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