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:

Turkey DrumStik

about 3 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

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

Nov 14, 2024 at 12:10pm
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

January 11, 2019 at 12:45am

#949385

In Praise of the Vulture

Tell us the story of your name. How and why was it chosen? You can choose to talk about your real name, your WDC username, or even your handle if you like!

I don't know if I've ever explained my username.

As I mentioned a couple entries ago, I'm not going to divulge my legal name here. Suffice it to say that my parents bestowed it upon me.

Nor am I ready to reveal how I chose my pen name.

But I will talk about my login.

It's really quite simple: I admire turkey vultures. You could say they're my spirit animal.

Oh, sure, they're ugly as fuck, and no one wants to be around them. Right there would be enough for me to identify with them. But there's more.

Vultures serve an important purpose in the ecosystem. They remove that which would otherwise contaminate the land - the organic detritus of what was once living. By doing so, they actually clean up the place. Hence their species name - cathartes aura - or "golden purifier." Few appreciate what they do, or even understand how essential it is, but they do it anyway. Yes, they devour the dead - as do we all.

More, they seem to fly just for the hell of it. Because they can. They're awkward getting started, but once they're aloft, they appear to fly just for the pleasure of flying. In doing so, though, they expend the least possible amount of energy for the greatest reward. There's a common misconception, perpetuated by decades of shoddy screenwriting, that vultures circle over that which is about to die. Well, if they're gyrating over a soon-to-be-corpse, it's by coincidence - what they're doing is taking advantage of updrafts, the rising of heated air. They'll seek out these updrafts and use them to gain altitude so they don't have to flap their wide wings. In fact, they try not to flap at all, preferring to glide.

Their propensity to expend minimal effort for maximum gain is another reason I identify with them. Don't flap your wings; find an updraft. Leave the whole "chasing prey" thing to more ambitious carnivores; be satisfied with what's left over. Likewise, why take on the risk of injury, chasing prey that might have defenses or competing with others who have sharp claws, teeth, or beaks, when your whole evolution has enabled you to avoid those hazards?

That which is ugly can, sometimes, become beautiful when you look at it the right way.

Every once in a while, I feel like I should change my username. I've never liked having the "02" tacked on; that was an artifact of having trouble creating an account here, back in the noughties. But you might say I'm too lazy to bother. I'm fine just gliding along, for now.

7 Like • Comment • View 8 comments

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