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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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

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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 18, 2023 at 9:37am

#1052753

Facts... In... SPACE!

The random number gods have blessed us with back-to-back Cracked links.

5 of the Most Useless Facts Scientists Have Deduced About Space

Not really useless, though, are they? Apart from my assertion that there's no such thing as useless knowledge, they at least get the site some clicks.

For me, and most of us, space is like retirement: It seems cool, but there’s little to no chance I’ll ever personally experience it.

Maybe stop buying lattes and you'll be able to afford a trip on the Muskrocket? Or retire. But not both.

Along those lines, here are five of the most useless facts science has provided us with about outer space…

Which might all someday be useful, if we don't blow ourselves up like a Muskrocket first.

5. Parts of It Smell Like Rum

There are way worse things to smell like.

There’s a cloud known as Sagittarius B2 that’s floating out in our own little Milky Way galaxy.

In space, "cloud" is relative. From what I understand (I could always be wrong), it's even less dense than the Martian atmosphere.

Specifically, it’s the chemical that gives rum its distinctive smell, meaning that, somewhere out in our galaxy, there’s a space cloud that, if you smelled it, would make you retch remembering your 21st birthday party.

No, for that, it would have to smell like tequila.

There is also an alcohol cloud

in space; unfortunately, it's mostly methyl (the kind that makes you go blind if you drink it). But space is big, and in my headcanon, somewhere out there is a cloud of Everclear.

SF idea: a spaceship dives into one of these clouds to replenish its fuel supply.

4. A Year on Venus Is Shorter Than A Day on Venus

A year on Venus is shorter than a day, even though that is not what we have agreed those words mean like 99 percent of the time. It’s because Venus spins incredibly slowly on its axis, so much so that it completes a full revolution around the Sun before it rotates a full 360 degrees.

This fact may indeed have little use—after all, on the surface of Venus, it doesn't much matter whether you're facing the sun or not; it's still hotter than actual hell. Not to mention corrosive and under more pressure than an intern on a deadline. But for a long time, we didn't know what the rotation rate of the planet was. All we had to go by was cloud top rotation, and that can be different. So the remarkable thing isn't that Venus rotates more slowly than it revolves, but that we know it.

3. There’s A Huge Diamond Out There

That was an entire Doctor Who episode.

The diamond is actually a huge, dead star known as a white dwarf.

It's probably wrong of me to yawn at the idea of a white dwarf. It is, after all, an example of just how scary outer space can be.

Larger stars, like our Sun, which so generously provides us with melanoma and the ability to burn ants with a magnifying glass, end with a supernova, one of the coolest things I hope to never see.

Wrong.

Our sun will also leave behind a white dwarf remnant. Supernovae start at, can't be arsed to look it up, a star much bigger than our sun.

Once the sun starts fusing helium, however, it's likely to expand to Earth's orbit, so the distinction doesn't much matter to us. Also, whatever happens will happen billions of years from now. Probably.

It’s so far away there’s nothing we can possibly do about it, but maybe, someday, we’ll figure out a way to send poor people there to harvest bits of it at great bodily danger to themselves.

Another SF plot that writes itself. Unfortunately, one would have to first overcome the crushing gravity, not to mention the heat.

2. We’re Eventually Going to Crash Into the Andromeda Galaxy

That's certainly not useless knowledge for a writer. It's just that this will happen just about when our sun expands; that is, billions of years from now.

You tell me we’re about to collide with another galaxy, and when I ask, “So Earth and all the planets we know are just going to smack into the Andromeda ones?” I get hit with a “well, not exactly.”

That's because space is largely made up of—you might want to sit down for this revelation—space. We've looked at galaxy collisions; the biggest effect is gravitational fuckery.

1. It’s Infinite

Er... maybe.

Probably not. Likely, it's very, very big. Maybe it's looped in four dimensions, the way the surface of the Earth is looped in three. Very, very big is just as far from infinity as 1 is.

But from a practical standpoint, "very, very big" might as well be infinity, as there will always be things we don't know.

And that's awesome.

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