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:

Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

1 Like

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').

1 Like

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.

January 28, 2023 at 12:02am

#1043778

In the Cards

Today's article is a few years old, but it's not like the subject matter has an expiration date.

Tarot Mythology: The Surprising Origins of the World's Most Misunderstood Cards

With their centuries-old iconography blending a mix of ancient symbols, religious allegories, and historic events, tarot cards can seem purposefully opaque. To outsiders and skeptics, occult practices like card reading have little relevance in our modern world. But a closer look at these miniature masterpieces reveals that the power of these cards isn’t endowed from some mystical source—it comes from the ability of their small, static images to illuminate our most complex dilemmas and desires.

Symbolism is a powerful thing, and there's nothing supernatural about it. It's not necessary (or desirable, in my opinion) to "believe in" the divinatory aspect of Tarot to appreciate the art that goes into it—just like you don't have to be religious to admire the art in the Sistine Chapel, or the architecture of Angkor Wat.

The article, as with the one a couple of days ago, contains illustrative pictures, which are a pain (and probably a violation of something) to reproduce here. But, as with an old issue of Playboy magazine, it pays to read the article in addition to looking at the pictures.

Even the earliest known tarot decks weren’t designed with mysticism in mind; they were actually meant for playing a game similar to modern-day bridge. Wealthy families in Italy commissioned expensive, artist-made decks known as “carte da trionfi” or “cards of triumph.” These cards were marked with suits of cups, swords, coins, and polo sticks (eventually changed to staves or wands), and courts consisting of a king and two male underlings. Tarot cards later incorporated queens, trumps (the wild cards unique to tarot), and the Fool to this system, for a complete deck that usually totaled 78 cards.

The relationship between Tarot decks and the common French playing cards used for casino games and solitaire is a bit murky, but there are clear parallels: the Fool corresponds to the Joker; there are three court cards instead of Tarot's four; and cups, swords, coins, and sticks have their equivalents in hearts, spades, diamonds, and clubs.

The rest of the article deals with the history of Tarot, both factual and speculative, and it touches somewhat on other decks. Again, the illustrations are what makes this really interesting.

I find randomness appealing in part because it can provide a needed break from one's thinking habits. You randomize a deck of cards by shuffling them; you then draw something that's unexpected, though within the parameters of the deck. It's kind of like the system I use to pick topics here, selecting from a curated list. Being random ensures I don't always pick the easy ones, or stick with a theme for very long. Randomness isn't mysticism, of course; it's just that, sometimes, it can help you jog your mind into a different direction.

We see patterns in the randomness, and perhaps meaning, but the meaning is what we decide it is.

And sometimes it's fun just to look at the art and see all the details.

3 Like • Comment • View 2 comments

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