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 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 13 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
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

Dec 24, 2024 at 5:16pm
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

April 16, 2024 at 10:19am

#1068893

What's the Difference Between a Duck?

Not every question has an answer, as illustrated by this Cracked piece about riddles.

Five Fictional Riddles That Can’t Be Solved

By "fictional," they mean "appearing in works of fiction." Lots of riddles are fictional in that someone made them up.

The literary, cinematic and funny-paper canon is full of riddles because they’re as fun for the audience as they are dire for whichever hero must solve them to obtain the One Ring, whatever the hell a sorcerer’s stone is, etc.

The Sorcerer's Stone was meant to be the Philosopher's Stone, and it is called that in the book's original language. Of all the ways they dumbed down that series for an American audience, that was probably the worst.

The Philosopher's Stone was, historically, the hypothesized material that alchemists thought they could use to turn lead into gold, or confer eternal youth, or had whatever transformative properties were faddish back before science. Kind of the Unobtainium of its time, and its own unsolvable riddle. The more philosophical alchemists treated it not as an actual material, but a transformative idea for achieving enlightenment or whatever.

5. Alice’s Adventures in Wonderland

Just a few pages after the Mad Hatter asks Alice, “Why is a raven like a writing desk?” in the rant against modern math most beloved by hippies, the answer becomes clear: He doesn’t know.

Likely my first encounter with the subject at hand. Pretty sure it annoyed me back then, but it taught me an important life lesson: Not every question has an answer.

As the article notes, people have stumbled around trying to find an answer, but, like the search for a material Philosopher's Stone, answering it misses the entire point.

4. Life Is Beautiful

Toward the end of the movie, Lessing begs Guido for help with a riddle, translated to English, “Fat, fat, ugly, ugly / All yellow / If you ask me where I am, I say ‘quack, quack, quack’ / Walking along, I say ‘poo poo.’” Sound like nonsense? It is.

In contrast to the Mad Hatter thing, I've never encountered this, which apparently comes from an Oscar-winning movie. But even had I seen the movie, I think I might have accepted that for the near-Brechtian absurdity that it obviously is.

3. Twelfth Night

This takes some explanation; fortunately, the article does just that. But you'll have to see it there, because I can't do it any justice with small excerpts.

Our theory is that Shakespeare wrote fart jokes and never intended anyone to think this hard.

On that point, I have to agree with Cracked.

2. The Hitchhiker’s Guide to the Galaxy

Every nerd’s favorite number is 42, supposedly “the answer to the ultimate question of life, the universe, and everything” calculated by the supercomputer Deep Thought in The Hitchhiker’s Guide to the Galaxy.

It is now absolutely impossible to have a philosophical discussion about the meaning of life without someone shouting that number out.

In fairness, that someone is often me.

1. Monty Python and the Holy Grail

Ah, yes, that other work of art that nerds can't help but quote. You know both this and #2 were created in the 1970s, right? Holy Grail in particular came out in 1974. For the math-challenged, that's 50 years ago.

Wait... Shakespeare's and Carroll's works are much older than that. Some of us nerds like to quote them, too.

This has given comedy nerds a handy call and response by which to identify each other but also an actual riddle: What is the airspeed velocity of an unladen swallow?

It's true. I have been known to select people to hang out with based on whether they can do Holy Grail references. I knew one person who could quote the entire movie, verbatim.

That's dedication.

The real joke: It’s a trick. There’s no such thing as “airspeed velocity.” Sure, you could calculate the airspeed or the velocity of a swallow of any nationality.

"You're using coconuts."

Perhaps comedy is the actual Philosopher's Stone...

5 Like • Comment • View 1 comment

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