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 8 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.

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

September 6, 2024 at 1:47am

#1076349

Secretion

"Blog Week Birthday Bastion 2024"

[13+] by WakeUpAndLive

Prompt 6. Sept 6.

Nothing makes us so lonely as our secrets. Paul Tournier
Can you share a secret? Small or big, tell us. No? Tell us why not.

No.

...well, now I have to come up with at least 299 more words.

I'm pretty open in here, I think, though there are some things I edge around in an effort to avoid potential problems. This blog is public, though, so once I share something here, it can no longer be considered a "secret" in a literal sense of the word. So asking me to share a secret in here is the same as asking me to destroy it.

Some things were told to me in confidence by other people, and those aren't mine to disclose. If I'm ever in doubt about whether someone else's words to me are confidential or not, I assume they are. So those aren't getting discussed.

Ever wonder, though, why "secretary" contains the word "secret?" That's not an accident. While the word has largely fallen out of favor in the business world due to sexist connotations, replaced by "office assistant" or some similar term, a secretary is, historically, someone entrusted with secrets. The word survives in government administration, like with the Secretary of State in the US, or the UN's Secretariat. I suppose we didn't want to go with "minister" like in the UK or other countries, despite its inclusion in "administer."

How the word "secretary" got applied to desks with top shelves is still a mystery to me. Perhaps it was because secretaries tended to use such desks for organization, while a boss's desk is traditionally shelf-free, as they had secretaries to do all the filing and such. Or it might be used there in the sense not of "secret," but of "secrete," as in hiding, filing, or secreting stuff away. In yet another example of English's weirdness, "secrete" can also mean to ooze forth, like... well, you know what I'm talking about; unless you're a biologist, it's usually considered disgusting.

But the noun and verb are etymologically related.

I suppose it's no secret that when I can't think of anything else to say about a subject, I retreat into etymology. Most word origins are traced back to Latin or Greek, but even Latin and Greek built on earlier languages. It's just that the further back in time you go, the fewer written records survive, so it's tough to trace everything back to what linguists call Proto-Indo-European, or PIE, the apparent source of most Western languages.

But language probably got invented long before PIE, and it's even tougher to peer back into that ancient time. The past holds many secrets, and we will probably never expose them all.

We get to give out a free Merit Badge every day this week. Want one? Anyone who comments here before 11:30 pm WDC time today could get today's. (I'll need that extra half-hour to pick a winner and send the badge before midnight.)

To clarify:

When I say "comment," I mean comment. Not review. Though reviews are always welcome.

I also mean "here," not on the newsfeed post.

MB recipient will be chosen at random.

Maximum of one MB per commenter for the week.

If I don't get comments, I'll pick a previous commenter, and maybe not at random.

The MB will be the one I commissioned two years ago, "Complexity," which is a publicly available MB.

I appreciate all comments; this is just a little incentive.

1 Like • Comment • View 2 comments

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