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 "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.

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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This is a private area. Access has been restricted and you may not view it under its current settings. If you feel you should have access, please contact the area's owner and request access.

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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

December 4, 2024 at 8:41am

#1080806

Brainy Day Blues

Getting hit in the head is rarely a good thing, but on the occasions when it is, Cracked has us covered.

5 Unexpected Twists After Accidents Scrambled People’s Brains
The bad news is a huge hospital bill. The good news is you have superpowers now

And hey, if you're in a civilized country, you might even get to skip the "huge hospital bill" part.

If you get hit on the head, you might die...

Occasionally, however, your brain may change in a way no one could predict.

Yeah, I wouldn't recommend getting hit on the head, bitten by a radioactive spider, or falling into a vat of industrial chemicals as reliable means of obtaining superpowers.

5. Turning Into a Math Artist

History doesn’t record exactly what karaoke songs were performed in Tacoma on September 13, 2002. They must have been pretty bad because two men went up to Jason Padgett outside one karaoke bar and kicked his head in.

The only song I knew in 2002 that would inspire that level of rage was by Celine Dion. Anyway, he got a concussion, but then...

Padgett was diagnosed with a variant of synesthesia, where rather than perceiving colors or sounds when confronted with unrelated sensory input, he sees math.

While some people might consider this ability a curse, I call it a superpower.

4. Your Mental Diseases Cured

He told his mother he’d rather die than go on, and she replied (according to George’s account), “If your life is so wretched, just go and shoot yourself.” So, he did.

Mom of the Century award, right there.

As for whether anyone should consider trying something similar as a form of self-medication, doctors said, “No, of course not. What are you, nuts?”

I just wonder if this was the inspiration for the end of Fight Club.

3. Gourmand Syndrome

Doctors associate it with a specific type of damage to the brain’s right hemisphere. It’s not an eating disorder. Those who have it do not overeat (or undereat). They just become very interested in high-quality food.

Hey, I wonder if that's what happened to me, only with beer.

2. Becoming a Chinese Caricature

One of the sillier possible effects of a coma is known as foreign accent syndrome.

Silly, maybe. Not really a superpower.

1. Absolutely Nothing

A Frenchman came to the hospital in 2007 with a seemingly insignificant complaint: His legs felt weak. Doctors gave him some scans and discovered something slightly more serious: He appeared to be missing almost his entire brain.

We have an epidemic of that over here.

As for how he was able to engage in a profession despite lacking a brain, well, it turned out that he worked for the government...

Saw that coming.

Anyway, again, I don't recommend slamming your head into a wall to see if it gives you superpowers. Unless you sing Celine Dion at karaoke.

5 Like • Comment • View 2 comments

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