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

July 19, 2023 at 9:35am

#1052795

Where We're Going, We Don't Need Rules

For the ultra-rare three-in-a-row Cracked hat trick. I promise I have a variety of sources in the queue, but random numbers, like farts, often create streaks.

5 Places Where the Only Rule Is, There Are No Rules

Well, that's misleading. There are always rules.

A world without these rules is the dream of the angriest teenage punks and Libertarians alike.

Angry teenage punks and Libertarians just don't want to face consequences.

Leaving the well-being of your neighbors and the continued functioning of society up to the natural good in people might sound feasible, if the only people you’ve ever met in your life were two nuns.

Have you met nuns?

5. Slab City

The city has no public oversight, or any of the services that comes with it — running water, electricity or the other niceties that are modern requirements for living.

Sounds like a libertarian paradise, all right.

And yet, if you read the article, sometimes the cops do respond there. Hence, rules, even if unwritten ones.

4. Antarctica

Since there’s no owner or governing body, technically, there are no laws on Antarctica.

Except that there actually are. And that's not even going into the whole "Don't go outside without a parka" thing; that's more of a natural law.

3. The Autobahn

If you’re looking to the Autobahn for pure, unregulated freedom, though, you might be disappointed. In order to keep it from being basically a long stretch of twisted metal, there are plenty of other rules, some of which are policed more closely than elsewhere.

"No speed limit" is hardly the same thing as "no rules."

2. International Waters

Those are how certain pseudo-religious groups with lots of money and wide-eyed recruits get around pesky things like "child labor laws" and "regulations against slavery," but still...

Thanks to the U.N. Convention on the Law of the Sea, or UNCLOS, jurisdiction in international waters is pretty cut-and-dry. First, there is international maritime law that applies regardless, and second, legal jurisdiction is decided by the country the vessel in question is registered in.

1. Your Imagination

The cops can’t arrest you for what you do in your dreams… yet.

But for just $19.99 a month of a low annual payment of $149.99, you can subscribe to DreamVPN. Keep those annoying Dream Police outside your head!

4 Like • Comment • View 2 comments

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