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!

a day 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

a day 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.

September 25, 2019 at 5:29am

#966760

Musing

PROMPT September 25th

Another prompt from bobturn!

What manner of creature is your Muse and how do you feed it? Where does it live and does it ever escape your control?

I don't personify what little creativity I have. Sometimes I'm inspired, sometimes I'm not, but usually once I start writing I can keep writing. There's nothing in my head but me.

I do enjoy drinking, from time to time. It gives me different ideas, different perspectives. Sometimes I can work with these perspectives after I sober up; sometimes it's not worth it.

Last night, whilst drunk, I found out that my favorite bar is closing forever. This plunged me into a deep despair. Nothing matters. It's all for nought in the end. I'm not even going to get a chance to visit it one more time to say goodbye; it's just gone. I'm still working through the classic stages of grief, here - which I don't even remember what they are. Denial, bargaining, anger... whatever.

There's no one I can talk to who can relate to this. It's a supremely personal thing, something that has hammered home to me the knowledge that, no matter who reads my stuff, no matter who I can find to talk to on occasion, in the long run, I am completely alone. As I said, nothing in my head but me. That's liberating and terrifying all at the same time. No gods, no monsters, no muse, no whispering agents, no angels, no devils. And then there's the school of "thought" that insists that there is no Self, that even consciousness is an illusion.

With this loss, I'm inclined to accept that. Nothing is everything. Everything is nothing. A swirling accretion disk around a naked singularity, temporary and eternal.

The place where we humans can live is a tiny sliver of atmosphere, proportionally thinner than an eggshell. Anywhere outside that veil, anywhere at all, and you're dead in three minutes or less - usually less. I've been told by well-meaning spiritual folk that the universe loves us and wants us to live and thrive, but this is demonstrably untrue. Pick a random spot, anywhere in the observable universe. Pick another. And another. Pick hundreds of spots. Millions. Billions. Trillions. Pick a googol of spots: 10 raised to the power of 100. Hell, pick a googolplex number of random spots: that's an enormous number, one with a googol number of zeroes after it.

The chance of any one of those spots wanting to keep you alive is nonexistent.

Hell.

There are other bars, I know.

But this? This hurts. On the one hand, at least I feel an emotion. On the other hand, why does it have to be this emotion?

As always, Leonard can say it better than I ever can.

And I lift my glass to the awful truth
Which you can't reveal to the ears of youth
Except to say it isn't worth a dime

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