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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Write_Mikey_Write!

about 6 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 8 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.

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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').

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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.

August 2, 2020 at 12:26am

#989699

Eruv L'olam

Over at 30DBC, they're still in Antarctica.

Go where your cruise ship can’t — hop aboard a small, sturdy inflatable boat and buzz between the icebergs and around the mountains...

...Another cool and unexpected aspect of this research center is the Vernadsky Station Lounge, one of the southernmost bars in the world. Try the vodka, which has been distilled on site.

But no, I'mma stay where it's warm.

https://www.mentalfloss.com/article/91594/theres-wire-above-manhattan-youve-prob...

There's a Wire Above Manhattan That You've Probably Never Noticed

It's hard to imagine that anything literally hanging from utility poles across Manhattan could be considered "hidden," but throughout the borough, about 18 miles of translucent wire stretches around the skyline, and most people have likely never noticed. It's called an eruv (plural eruvin), and its existence is thanks to the Jewish Sabbath.

Well, NOW people will notice.

On the Sabbath, which is viewed as a day of rest, observant Jewish people aren't allowed to carry anything—books, groceries, even children—in public places (doing so is considered "work"). The eruv encircles much of Manhattan, acting as a symbolic boundary that turns the very public streets of the city into a private space, much like one's own home. This allows people to freely communicate and socialize on the Sabbath—and carry whatever they please—without having to worry about breaking Jewish law.

You know what *actual* work is? Actual work is having to memorize the Tanach, Talmud and Midrash, and be expected to know every detail of not only religious law, but every possible interpretation of religious law. But no, my people don't see it that way; apparently studying these texts is one of the few things you *can* do on Shabbat.

Much of the interpretation of what is and is not acceptable between sunset Friday and sunset Saturday is the result of later texts trying to make sense of commandments in the Scriptures.

But I think they're missing something important, here, and I'm going to change the world with this entry.

New York City isn't the only metropolis in the U.S. with an eruv. They can also be seen (or not seen) in St. Louis, Atlanta, Baltimore, Chicago, Dallas, and numerous other cities across the country.

So, just to be clear here:

There are things that can be done on Shabbat and things that cannot be done, according to law and tradition.

There are things that can be done on Shabbat in the home that cannot be done outside the home.

One can extend the definition of "home" to one's neighborhood, as long as there is an eruv to serve as a boundary.

I think that's cheating, but I'm not a rabbi, so whatever.

With me so far? I might still be drunk from earlier, so let me know if I'm not being clear. It helps to read the linked article and maybe click on the video there.

So, here's my world-changing, earth-shattering proposition, which no one will listen to because I'm not actually a Talmudic scholar.

Take an eruv. That is, imagine that you're sitting in, I dunno, Central Park, and you're surrounded by an eruv.

Now. Imagine that eruv expanding. The area of "home" gets bigger and bigger. It grows to encompass New Jersey, Long Island, Connecticut. But don't stop there. Bigger. Even bigger than that. It stretches until it covers all of North America, South America, the Atlantic. But keep going. Bigger. Pretty soon it's the size of the circumference of the Earth.

But don't stop there, either. Now it starts getting smaller as it moves around to the other side of the planet. You're still on the inside, sitting there in Central Park, and the "outside" of the circle is smaller than the "inside."

You see where I'm going with this, right? Eventually, you've got an eruv in the middle of the Indian Ocean, and everything in the world except for maybe 10 square meters (just an arbitrary number) of ocean is inside the eruv, and then Jews all over the world, unless they are in that particular patch of ocean, can do anything they can normally do at home on Shabbat.

Of course, there's no reason to keep that eruv in the Indian Ocean. It would be tough to erect one underwater. Pick a spot, any spot. Say... I dunno... go to Antarctica (*shudder*). Put up ten poles (ten is an important number in Jewish lore). String an eruv around the poles. Then simply declare that the bulk of the globe of the Earth is inside, while everything else (a few square meters of desert) is outside.

There might be a few penguins who can't carry groceries on Shabbat, but honestly, penguins aren't known for carrying groceries around anyway.

There you go. And no, it's not cheating, any more than the eruv itself is cheating. Okay, so I'm not a rabbi, but I know a loophole when I see one, and that's a loophole. It's a simple matter of spherical geometry: any circle on the surface of a sphere has an "inside" and an "outside" only by convention and declaration. Take the equator, for example: Is the northern hemisphere "inside" or "outside" of the circle of the equator?

And if you think what I just said is a stretch (which, topologically speaking, it totally is), then you've obviously never read the Talmud.

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