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 22 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 24 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 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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