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:

tj-Merry Mischief Maker

about 9 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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.

2 Like

🐕GeminiGem🎁

a day 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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day 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.

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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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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.

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September 13, 2024 at 11:21am

#1076742

Journey Man

Ran across this road-tripping article from Condé Nast Traveler, and, being no stranger to road trips, I couldn't resist snarking on it.

The Golden Rules of Road Tripping
Veteran road trippers (including a gas station food evangelist) share their savviest tips and tricks.

Maybe I'm just sour because they didn't bother asking me.

Ahhhh, the great American road trip. A chance to put rubber to pavement along our millions of miles of roads and see the nooks and crannies of the country we’d never otherwise see.

Translation: "I wonder what's really in flyover country?"

Yes, this article is US-centric. It's called the great American road trip for a reason.

Along with all that good stuff, road trips have the potential for some big pitfalls, too, and in order to make sure your road trip is more life-changing than life-ruining, there are some basic guidelines first-time and veteran road trippers should follow.

Because you never know when you're going to be driving some empty stretch of country road and total your car into a deer.

Book your hotels ahead of time

There are certainly times you should do this, like if you know you're going to spend a couple of days in, like, Kansas City, and you care about location. Or if you're traveling during peak times to popular destinations like, I don't know, the Grand Canyon. But I've never once found a hotel that didn't have any vacancy. I suppose it helps if you decide to schedule your stops so that you find a hotel right after check-in starts.

It also helps not to be picky. I've stayed in 5-star hotels, and I've spent nights in sketchy roadside motels. Doesn't much matter to me, as long as there aren't bedbugs, which don't really care how fancy the hotel is, either.

Search out your roadside meals

It might be tempting to make a highway exit franchise row your lunch stop. But settling for processed tacos or sub sandwiches can mean missing out on some fantastic regional food.

You know what's good for that? Google Maps. I can read a paper map as well as anyone, and better than most, but GPS is the best thing that ever happened to road trips.

“Some of the best expressions of regional food flavors and unique culinary styles can be found in gas stations,” says Frank Beard, a traveling sales rep who’s become an evangelist for the convenience store industry, and once spent an entire month eating at only gas stations.

So this is the "gas station food evangelist" from the subhead. I have a challenge for him:

Go to Salt Lake City. Head west on U.S. Route 50, through the Utah and Nevada desert, destination: Reno. Once you pass Ely, Nevada, the road becomes signed as "The Loneliest Road in America," which it really isn't (that would be some roads in Alaska, or, if you want to limit your scope to the contiguous states, Route 6 south from Ely to Tonopah), because you can actually find gas stations along it occasionally.

The challenge is: drive this route, stop at one of the aforementioned US50 gas stations, buy one of their corn dogs, and eat it in the car as you drive off.

You will quickly lament the state of gas station food, and the lack of restrooms on US50.

Why, yes, I am speaking from personal experience. Why do you ask?

Make sure your tires are filled and your fluids are topped off

I mean, that's way more basic advice than some of the other stuff. It won't guarantee you won't get stuck somewhere, but it does improve your odds.

Don’t depend entirely on your phone’s maps

While I agree with this (I always carry a road atlas as a backup), there's no need to fear technology, either. GPS can only lead you into a lake if you let it. People act like no one ever got lost or turned around using paper maps, but they absolutely did, and way more often, I'd bet. And like I said above, it's great for finding nearby places of interest, not all of which can afford to advertise on the main roads.

Anyway, lots of the advice here is aimed at people taking road trips in groups. Since it's a documented fact that no one can put up with me for more than two hours at a time, I tend to do road trips alone.

But there's one piece of advice I feel should have been added:

Stay off the interstates.

Look, interstates are great. They're an everyday masterwork of civil engineering. Sure, there are some issues with them, especially in urban areas, but, overall, if your goal is to drive from Point A in the US to Point B elsewhere in the US, you can't beat an interstate for efficiency, speed, and convenience.

But road trips, in my view, shouldn't be about those things. If you have limited time, and you care more about the destination than the journey, I'm not going to knock you for taking the fast route. I've certainly done it, else I couldn't make the comparison. But, with a few exceptions, you can't see the US from interstates.

Taking secondary roads, also known sarcastically as "the scenic route," really gives you a better experience seeing the country. It's a bit harder to find services, but that's what GPS is for. I think of it like playing a video game in hard mode: somewhat trickier and more challenging, but more satisfying, too.

To quote from a Brandon Sanderson novel: "Journey Before Destination."

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