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 21 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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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

August 21, 2024 at 10:32am

#1075545

Les Lois Bizarres

I figured it'd be important to read this Cracked article, because I'd like to avoid running afoul of the gendarmerie when I go there.

5 Normal Family Activities That Are Illegal in France
Want to take a paternity test? Nope. Straight to jail

France’s official motto is “Liberté, égalité, fraternité.” They’re a little iffy on the liberté part of that, at least by the standards of places who really care about freedom.

Um, have you seen the US?

For example, France bans public school students from wearing religious symbols.

From what I've heard, there were good reasons for the ban.

In the U.S., it would be unthinkable to have any national law on what schoolkids can wear.

Um, not the same thing. France is comparable in size to one of our larger states. Smaller than Texas but bigger than California. Texas and California have state guidelines for schools. That particular ban isn't in place there, but we have other quirks.

But several strange French laws concern families and kids...

I'm absolutely certain that other countries find some of our laws just as strange.

5. Paternity Tests

In France, a court may order a paternity test as part of a legal proceeding, but short of that, they forbid citizens from testing paternity.

Not my problem and, again, we have weirder laws.

4. Naming Your Kid What You Like

In France, courts can rule that parents named their child wrong and order the name changed. In 2015, for example, a court learned that parents were naming a child “Nutella,” and the court renamed the girl “Ella,” to spare her from derision.

Yeah, no, I'm gonna side with France, here.

3. Changing Your Surname

In France, however, there is no simple process for changing your last name. You can petition a court to change your name, if your motive matches one of the few acceptable ones, but you can’t just fill a form and change your name because you feel like it.

This just in: different countries have different laws and customs. Who'd have thought?

2. Disinheriting Your Children

In most places, if you die without a will, your money goes to your next of kin, but you can instead write a will leaving your money wherever you like. Not in France, where you must leave money to your children, no exceptions.

Eh, I don't see this as much different from estate taxes. Either way, there's a law concerning the disposition of a portion of your estate; the only difference is who that has to go to.

But I can see this becoming especially enraging for some men if they spent the kids' entire lives not knowing if they were genetically related, as per #5 above.

1. Swimming Trunks

You may have heard tales of French nude beaches, where every part of your body can go proudly on display. From these, you might think France is a lot more free than other countries when it comes to what you can wear when swimming.

It's not just France. I think the US is one of the outliers here, also known as "the strange one."

When you go to a French pool, you are not allowed to wear swimming trunks. You instead must wear swimming briefs, or what we’d call a Speedo, giving everyone a proper look at the contours of your bulge.

I started adding "Speedos" to the shopping list for my upcoming trip, then crossed it out and wrote NO SWIMMING. You're welcome.

On the other hand, if you want to show off much more of yourself and wear a thong, nope, pools ban that as well.

In summary, lots of rules that make sense in one culture seem strange to others. Part of the reason to travel is to experience these differences. But it's a good thing to have fair warning beforehand, so you don't, say, bring chewing gum to Singapore and end up getting caned.

3 Like • Comment • View 1 comment

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