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:

Write_Mikey_Write!

about 5 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 7 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.

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...

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