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!

about 2 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 4 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.

December 3, 2022 at 12:01am

#1041275

Yeah. Right. Sure.

Call me skeptical.

Air Force claims ‘no intent by the pilots’ to draw sky penis near Russian base in Syria

Sometimes miracles happen.

And sometimes, we make miracles happen.

The leadership for U.S. Air Forces in Europe has determined that a KC-135 aerial tanker did not mean to fly in a phallic flight pattern near a Russian base in Syria recently...

You'll have to visit the link to see the actual recorded flight path. Or you could take my word for it that it does, indeed, create the general shape of a dong, complete with balls.

While these adjustments and movements appear to create a vulgar outline, there was no intent by the pilots or the unit to do so.

Sure. I believe that. I also believe in Santa Claus, the Tooth Fairy, and honest politicians.

As we continue to look into this, USAFE-AFAFRICA, AMC [Air Mobility Command] and the USAF will continue to maintain the highest standards of professionalism and airmanship.

...even if we have to lie through our collective teeth to do it.

Earlier this week, the aircraft took off from Chania International Airport and flew east of Cyprus. FlightRadar24, a flight tracking service, shows that the KC-135’s flight path included an oval and two circles, prompting the Italian newspaper la Repubblica to speculate that the plane had drawn a sky penis in front of Russia’s naval base in Tartus, Syria.

Now, keep in mind that schlong drawings are nothing new in Italy (and elsewhere). People have been scrawling schlubs on walls for as long as there have been walls. Here's one

on the Roman Coliseum. The ancient Romans could only dream of being in flying machines that can trace the outlines of a gigantic knob in the sky.

That said, it's not really that close to Syria, is it? It's way out in the Mediterranean, closer to Cyprus, like it's about to stick itself into Turkey. The article acknowledges this:

But Cyprus is roughly 163 miles from Tartus, and while it’s unclear how close the KC-135 was to the Russian base, the aircraft was never in Syrian airspace, an Air Force official said.

However:

While U.S. officials say that the KC-135’s aircrew did not mean to fly in a pattern that resembled the male anatomy, there have been previous incidents in which American pilots intentionally drew large penises in the sky.

Two Navy aviators were administratively disciplined in 2017 for using the contrails of their aircraft to create a large penis-shaped pattern over Washington state, and two Marine aviators were grounded the following year for their aerial rendering of the male sex organ.

It makes me feel all kinds of secure that our pilots, across military disciplines, have the sense of humor of a middle-schooler. Eventually, the Space Force will draw one over the Moon, you watch.

Moreover, in October 2019 U.S. troops left penis drawings for the Russians to find when they moved out of their base in northeastern Syria.

Okay, that's legitimately hilarious.

For reasons that remain not fully understood, phallus imagery has become an integral part of the American warfighter’s psyche.

"Not fully understood," my fat white ass.

“The core of a military identity is tied into this warrior ethos of being strong and tough,” Ramon Hinojosa, a sociology professor at the University of Central Florida, told Task & Purpose in 2018. “This notion that being tough and strong and having a sexual prowess is just a core part of what we as Americans see masculinities are. That translates across institutions, at least as far as the military is concerned, into strong, tough, heterosexuality. Of course, the depiction of a penis just sort of boils that right down to the essence of the masculine norm.”

You want "strong and tough?" That would be the other sex organ. How "pussy" ever became an insult meaning "soft and weak" is what's "not fully understood." Those things are tough, or so I've heard. (I so badly wanted to phrase that "those things can take a licking," but I would never stoop so low.)

They might be a bit harder to draw in the sky, though, I'll grant that.

3 Like • Comment

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