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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🌝 HuntersMoon

Sep 20, 2025 at 12:12am
In response to "Numbers End"

Boy, am I behind thee times. I thought you just put this on hiatus. Of course, in March I wasn't a complete devotee of your brilliance yet.

I'm more than happy with Carrion Luggage, however.

1 Like

Elisa, Snowman Stik

May 1, 2025 at 7:09pm
In response to "Shhhhhhh"

Anytime I hear/read about the Quiet Zone, I am always reminded of this video.

TJ says, "keep on keeping on!"

Mar 12, 2025 at 8:16pm
In response to "Penultimatum"

As my Grandmother used to say, "Everything changes except that which stays the same." I never did understand why she said it, but it sounds good.

1 Like

TJ says, "keep on keeping on!"

Mar 12, 2025 at 8:13pm
In response to "Numbers End"

Looking forward to the new book and hoping you don't encounter a meteor.

1 Like

Lazy Writer est 4/24/2008

Mar 12, 2025 at 4:19pm
In response to "Numbers End"

Sorry this one is closing..I will look for the new one tomorrow! Looking forward to it.

2 Like

Aiva Raine

Mar 12, 2025 at 3:45pm
In response to "Numbers End"

Thanks for a great 3000 entries.

Looking forward to the new blog, whatever it might be.

2 Like

Wordsmitty ✍️

Mar 12, 2025 at 1:36pm
In response to "Numbers End"

OK. What have you done with the real Robert Waltz?

Also, for the scientist engineer in me, have you actually tried to make entry 3001 and been rejected. Until there's proof, it's not real.

Thank you for your indulgence and sharing of wisdom.

R.I.P. "Complex Numbers"

3 Like

Anni Pon

Mar 12, 2025 at 11:20am
In response to "Numbers End"

Sorry that this one is coming to an end, but I'm excited to see what you have in store for the new one!

2 Like

Wordsmitty ✍️

Mar 11, 2025 at 1:42pm
In response to "Penultimatum"

It's been an enlightening run. We wish you well in your next adventure.

It will be interesting to see if after the next entry that WDC actually does stop you from adding another.

We're all waiting on pens and pencils for the unveiling of the new blog and to discover what it will be called.

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BIG BAD WOLF is Merry

Mar 11, 2025 at 10:46am
In response to "Penultimatum"

I'm reminded of this scene from The Princess Bride.

(Vizzini has just cut the rope The Dread Pirate Roberts is climbing up)
Vizzini: HE DIDN'T FALL? INCONCEIVABLE.
Inigo Montoya: You keep using that word. I do not think it means what you think it means.

3 Like

January 30, 2007 at 12:29pm

#484463

The Father of Invention

elusive ennui warm inside

asks: what do you think is the most important invention/discovery of the last 100 years? and why?

As a matter of boundaries, this leaves out fire, roads, the wheel, aqueducts, electric lights, radio, telephone, automobiles, powered flight, x-rays, radio, and AC power - among other things.

That leaves a whole lot of territory - the last 100 years has witnessed an explosion of technology, from the small (microchips) to the huge (Saturn V, the rocket that put a dude on the moon).

As for discoveries, well, there's relativity, quantum mechanics, galaxies, the Big Bang theory of origin, and Bruce Springsteen.

So I got to thinking - what's really important, and to whom?

They say that necessity is the mother of invention. To that, I say that laziness is the milkman. For every invention that resulted from perceived necessity, there's at least one that was designed to help us stop doing crap work: washing machines, cars, water pumps, boat engines, and so on arose not so much out of necessity, but some man or woman going, "I'm tired of doing this shit. I'm going to design a machine that can do it for me." Even the personal computer, which could be argued to be the invention with the most far-reaching effects, falls into this category.

But even the personal computer relies on earlier technologies: microchips, printed circuits, CRTs and so on. Those in turn relied on earlier discoveries and inventions, and on all the way back to fire and the wedge.

No, if I had to choose the most important invention of the last 100 years, it would have to be one that enabled other inventions, provided a foundation on which other discoveries and/or inventions could stand. And ultimately, what drives invention?

Necessity. Laziness. And dreams - both the night kind and the day kind. Without dreams, we are mere automatons, going through our daily routines without thought or hope.

A big feature of the Industrial Revolution, and one that continued long afterward, is the industrial image of people working as cogs in a giant, well-oiled machine. We all have our function, says this myth, and we must perform it for 40 years, then retire and die. We must get up at six, get dressed, go to work, come home, eat dinner, go to sleep, rinse, repeat. Progress depends on you! Don't call in sick! We're depending on you!

This stifles dreams, aborts inventions, resists discoveries, smothers curiosity. Cogs have no free will. The great inventions of the industrial age, paradoxically, destroyed our individuality, caused us to be ruled by the tyranny of time.

But there was subversion and rebellion against this chronocracy. Of course, all open rebellion was quickly squashed, its practitioners relegated to the fringes to live in boxes under bridges. But there was guerilla warfare, gradual erosion of the soul-numbing industriousness.

And what was it that was at the root cause of this insurrection? A rebellion that got to the root of the chronocracy, striking a blow against the oppression of the measured life while at the same time giving us more time to dream, to think, to plan... to invent.

I'm unable to determine the name of the rebel-saint who blew up the Parliament of Industry. Perhaps she or he is destined to go through history unappreciated, unsung, unrecorded. But the invention has forever changed the way we approach work and dreams.

It is.... the snooze button.

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