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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 6 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

1 Like

🐕GeminiGem🎁

about 10 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

about 12 hours ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 5, 2024 at 10:17am

#1079535

Anomalies

Ah, yes, November 5, and Election Day in the US. The UK will be celebrating an attempted terrorist attack, while over here, we're trying to avoid a terrorist attack.

The Random Number Gods have chosen to bless us with another Cracked link today.

5 Spots on Earth Where the Laws of Physics Fall Apart
There’s a gravity hole in the ocean

Okay, but, no, there are no spots on Earth where the laws of physics actually fall apart. Well, maybe, sometimes, at CERN, but they do it on purpose.

Still, these are rather interesting.

There are some places on the planet where things get weird. For instance, you ever heard of the Bermuda Triangle? Well, it turns out there’s nothing weird about that bit of the ocean at all — it sees a lot of traffic, but vessels that travel there are no more at risk than those anywhere else.

Like Bigfoot, that's not going to stop humans from making shit up about it.

5 Gravity Drops Near Sri Lanka

Once you learn that the force of gravity is slightly variable across the planet, it stands to reason that there are some spots where it's less and others where it's more. Finding out where it's less, though, that's what science does.

Gravity varies from place to place — and in some places, it varies a lot. In the ocean near Sri Lanka, gravity is so much weaker than in the rest of the world that the sea level is more than 300 feet lower than it would otherwise be.

Hey, I just came up with a fix for rising sea levels! Just increase the gravity of the Indian Ocean, and presto!

To know why gravity’s so low there, we’d have to burrow deep into the planet, and possibly cut it in two, which is inadvisable.

Awww.

Now, look, what that article's not telling you is that it's a minuscule effect. The variation from average, above or below, is about 0.5%. We'd never feel it. Sure, it has a profound effect on sea level, but look at what the comparatively really very tiny effect of the Moon's gravity does to that on a daily basis.

4 The Toasty Bit of Norway

In the Norwegian Sea, we have one disturbing bit called the Lofoten Vortex, where the water stores an unusual level of heat.

Lofoten Vortex can be the name of my Bjork cover band.

Yes, yes, I know Bjork is from Iceland, but come on, look at a globe. (A flat map inevitably distorts the distance between Iceland and Norway.)

3 On Top of Paraguay, Magnetism Disappears

I recently saw an argument for why the Earth's magnetic field isn't as important as we thought for maintaining our atmosphere. But this isn't about that.

Without looking at the map, if you had to guess one spot where the magnetic field gets weird, maybe you’d point your finger at one of the poles. But the planet’s rotational axis, which defines where we put the north and south poles, isn’t the same as its magnetic axis (which creates the magnetic field). As a result, we have this belt of radiation around the globe that dips down and comes close to us at this unlikely spot above Paraguay.

So, it doesn't "disappear." It just gets weaker there. How much weaker, I can't be arsed to look up. I can forgive a comedy site for hyperbolic headers, but they made it sound like, I don't know, compasses won't work in Paraguay or something.

2 The Tulsa Center of the Universe

You might not have known that the center of the universe is in Tulsa, but that’s what this spot is named, and Oklahoma wouldn’t lie to us.

Can't be the center of the universe; I don't live in Tulsa.

We’d go investigate ourselves, but that would mean having to spend time in Tulsa.

Good reason.

1 The Cave Where Energy Comes from Rock

I mean, technically, coal is rock. Uranium ore is rock.

But then you have Movile Cave in Romania. The interior is totally cut off from the outside world, and the creatures in there get no energy through photosynthesis, either directly or indirectly. Instead, the producers of this food web are bacteria that get their energy through chemosynthesis.

Here, though, they're talking about an entire cave ecosystem that doesn't rely, at its base, on solar energy input (apart from, you know, it not being frozen solid and all). Which is outside our normal experience, and definitely doesn't break the laws of physics any more than the other examples do, but is really interesting. Because now we know for sure that life can exist without photosynthesis.

Exist, yes, but imagine crawling out of that cave to go vote. Who are the candidates, now?

3 Like • Comment • View 2 comments

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