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:

Wordsmitty ✍️

about an hour ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 4 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day 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

Dec 23, 2024 at 4:55pm
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

June 10, 2024 at 9:56am

#1072422

Betelgeuse 2

The person (P.Z. Myers) who writes the blog I'm linking today is a scientist—but he's a biologist, not an astronomer. So his take on astronomy probably isn't any more reliable than mine. This doesn't stop either of us from writing about it.

Betelgeuse, Betelgeuse, Betelgeuse

I'm sharing this because 1) I would love to see a supernova before I kick it (though hopefully not one that's too close) and 2) I'll take any opportunity to call out misinformation.

Something funny is going on 650 light years away…or should I use the past tense? Something funny was going on 650 years ago.

Yes, we know light has a speed, and we don't have good grammatical tenses for it. The way I look at it is: what matters is when we see the event; the distance (or time) when it happened is immaterial.

The star Betelgeuse is/was acting up, dimming and then brightening (well, it’s always been flickering a bit, but this was a greater reduction in brightness than usual.)

This is cool by itself, because it contributes to astronomers' understanding of how stars work.

And now some people are saying it’s about to go supernova!

Well, it is. For cosmological definitions of "about to."

There is a real-time deathwatch on YouTube. “LIVE Betelgeuse Supernova Explosion Is Finally HAPPENING NOW!” it says.

And that's where the misinformation, or at least misunderstanding, comes in.

I'll let PZ explain there at the link, or at least quote actual star-gazer type scientists.

Awww, but it sounds like it will be spectacular when we do get the Giant Space Kablooiee, and not spectacularly dangerous, the best kind of spectacular there is.

The Giant Space Kablooiee is not to be confused with the Horrendous Space Kablooie,

which was, to the best of our knowledge, what kick-started this enormous thing we call the Universe. It's not even in the same ballpark. To quote Jules from Pulp Fiction, it's not even the same sport.

But yeah, the science-talkers who should know stuff about these things don't think it'll do much, if any, harm to Earth when it happens. But "spectacular" is an appropriate adjective.

I do wonder if that guy running the live video feed is prepared to keep it going for 10,000 years.

No, but he's probably prepared to collect ad revenue from YouTube, taking advantage of the misinformed. Which is why I'm not going to click on that video.

Perhaps I am more cynical than PZ.

To be clear: we don't know enough about stellar lifecycles to predict just when Betelgeuse, or any other star of appropriate mass and age, will become a supernova. When it happens, next year or 10 millennia from now or somewhere in between, if there are still scientists around at that point, we'll learn more.

Hell, even the "650 light years" thing might not be correct. For whatever reason (it's discussed at the link I'm about to provide, though with an overwhelming amount of math), it's incredibly hard to estimate that star's distance with high certainty. Wikipedia

puts it at 400-600 ly, which, even on the low end, is still supposed to be comfortably far for a supernova (though close enough to look awesome).

Now, I've written about the probably-not-a-supernova-anytime-soon bit before, notably here: "Stardust"

. Today's update is more about the "live feed" nonsense. And maybe a bit of a distance correction.

If you want a live feed that's actually interesting, I usually remember to check this one

out occasionally around late spring/early summer. Bears!

5 Like • Comment

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