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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

August 24, 2023 at 8:39am

#1054518

Cool It

A certain comedy site has trained me to see numbered lists in countdown order, and also to expect comedy from them. This, however, is not Cracked, but Mental Floss. Consequently, I'll have to supply my own comedy.

9 Creative Ways People Kept Cool Before Air Conditioning
People have tried everything from funky fan chairs to enormous ice chambers to survive a sweltering summer day.

Full disclosure: the house where I spent my childhood was located in one of the hottest and most humid parts of the mid-Atlantic (Chesapeake Bay area), and did not have air conditioning; plus, it was located about 100 meters from a massive swamp. This might have "built character," as my dad would have put it, but it did not; it ensured that I would never, ever go without air conditioning, ever again.

Not going to quote all of them here.

1. Fan Chairs

At first I thought of those movies and stage plays where the important person sits in a chair while servants wave giant feather fans. The important person stays cool; the servants get hotter. Basic thermodynamics.

People used their feet to operate the fan that moved above their head, much like someone would power an old sewing machine.

What? No servants? That de-feets the purpose.

2. Sleeping Porches

Also known as "baby disposal chutes." (For the full effect, go to the article; there are pictures.)

Another president interested in keeping cool was William Howard Taft, who had a “sleeping porch” erected on the roof of the White House in 1910.

He must have had servants reinforce the roof, or it would have collapsed. Taft made Chris Christie look like Kate Moss.

4. DogTrot Homes

The cats can just fend for themselves.

On a more serious note, one wonders why this architectural innovation is featured here, but the shotgun house is not. Despite the name, these have nothing to do with firearms, but were designed to funnel any breeze through the home's three rooms. My childhood home, noted above, started out as a shotgun.

Oh, right, because that style is associated with servants' dwellings.

5. Punkahs

These hand-operated ceiling fans have their origins in colonial India. Each year, thousands of poor seasonal workers were contracted, or otherwise compelled, to spend monotonous days pulling a cord that swept a piece of fabric back and forth across a room for the country’s elite.

AHA! I knew we'd get to the servants. "Otherwise compelled," my ass.

6. Drinking Buttermilk

EW! I'll just sit here and sweat, thanks.

The Indian subcontinent gave the world another refreshing idea for keeping cool in the searing heat: drinking buttermilk.

I guess it worked in the long run; they just landed a robot on the moon. We now have two worlds known to be inhabited entirely by robots, with just one known to be inhabited by carbon-based biological life.

Robots, incidentally, are okay to use as servants. For now.

9. Not Stressing About the Heat

How did I handle the oppressive heat and humidity when I was a kid? Well, there was a giant body of water in my front yard, and I learned how to swim.

No servants required.

4 Like • Comment • View 3 comments

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