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:

Elycia Lee ☮

about 9 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 22 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

January 25, 2020 at 12:03am

#974136

Twist In My Sobriety

PROMPT January 25

CREATION SATURDAY! Put on your creative thinking caps

You're headed down to Imagination Station to pick up your imaginary friend.

Tell us about the friend - is it human? Humanoid? Animal? Talking banana? Three-headed monster who's afraid of heating blankets? What's their story? Likes/Dislikes? What name do they answer to? Why are they in your life?

Don't forget to tell us how your friend ended up at the station in the first place!

Obviously, Imagination Station is a bar, and my imaginary friend is a beer.

Not just any beer, of course; that would be silly, because then it wouldn't be imaginary. No, this one is magical -- well, more magical than usual, if that's even possible -- so it never runs out, or away.

Beer has been around since the dawn of civilization. Arguments have been made, compelling ones, that there was a causal link there, that beer is what kick-started civilization. There were probably other factors, notably cats, but the manufacturing of beer required a level of cooperation that only cities could provide. That's what civilization is, you know: cooperation and specialization.

So Beer didn't "end up" at Imagination Station. Imagination Station was built around Beer.

Incidentally, as similar as the names are, there doesn't seem to be an etymological connection between "bar" and "beer." The former is named for the counter where drinks are poured, which in turn was named for the metal rod or bar it traditionally sported, in an example of nested synechdoche (Nested Synechdoche would be an awesome name for a band). The latter is derived from the Latin verb for "drink." "Imbibe" is another derivative thereof.

As old as Beer is -- and it is old, by human standards -- it spent much of its history ignorant of its own provenance. It wasn't until the microscope came into being that we discovered what made Beer beer instead of malted barley soup: yeast. Until then, the process of fermentation was, at best, obscured; it might as well have been magic.

Here I was setting out to craft a story and I got bogged down in history. It happens. It's all fascinating to me, though: both the history and the science, of course, but also the creative spark that I'm just not feeling tonight. Maybe someday we'll invent a tool to examine creativity and find its source, the way the microscope enabled us to discover yeast. That wouldn't end creativity, I think, just as discovering that alcohol is essentially yeast piss didn't end Beer.

For now, though, I'll just raise a glass to science, which may not always make everything better, but certainly increases our understanding -- if not our creativity. And to Beer, which made it all possible.

All God's children need traveling shoes
Drive your problems from here
All good people read good books
Now your conscience is clear
I hear you talk girl
Now your conscience is clear

In the morning I wipe my brow
Wipe the miles away
I like to think I can be so willed
And never do what you say
I'll never hear you
And never do what you say

Look my eyes are just holograms
Look your love has drawn red from my hands
From my hands you know you'll never be
More than twist in my sobriety
More than twist in my sobriety
More than twist in my sobriety

6 Like • Comment • View 7 comments

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