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:

Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 8 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

August 29, 2020 at 12:03am

#991792

Rare Earth

Okay, whoever keeps asking the question, "How could this year get any worse?" -- I want you to stop that shit right now. Never ask that again.

I learned of Chadwick Boseman's death a bit over an hour ago, so I'm getting drunk and watching Black Panther. But that doesn't mean I can't do a quick blog entry at my usual time.

Today's ties in with the entry from a couple of days ago, and it's written by a guy I know.

https://www.syfy.com/syfywire/do-aliens-exist-probably-are-they-intelligent-prob...

Do aliens exist? Probably. Are they intelligent? Probably not.

Just to be clear, my own speculation on the subject, as seen in "No, It's Not 36."

, was based on stuff I'd figured out before I saw Plait's article. As you'll see, we are in fairly close agreement on these things.

I've been asked if I think life exists elsewhere, and my answer is always that I do. This is based upon a single fact: Life got a toehold on Earth very rapidly after the planet formed, just a few hundred million years.

Only in cosmology is "a few hundred million years" considered "very rapidly."

Maybe life usually takes billions of years, and we were lucky. We only have our one example of life arising and so it doesn't really tell us anything about how easy it is. All it tells us is that the chances of it happening are not 0.

The chances could be one in billions, trillions, or even more. Or it could be close to 1 in 1. We simply don't know, and we won't until we find other examples of life and study them closely. If it seems unreasonable for the chance to be 1 in a trillion, just remember: your odds of winning the lottery may be 1 in a billion when you buy the ticket, but once you've won, prior odds are irrelevant. Maybe we won the lottery. We'll find out eventually if this year would stop fucking with us.

A new study using a sophisticated form of statistical analysis shows that life is actually quite likely to arise if conditions allow it. Hurray! However, it also shows intelligent life is far less likely. Boo.

Which is fine and all, but any statistical analysis, sophisticated or not, boils down to guesswork in this sort of thing. And yet, I'm inclined to believe it. Perhaps because of confirmation bias. But just because I'm inclined to believe it doesn't make it true.

But they found that when they ran their calculations, changing the prior assumptions strongly influenced their result, giving wildly different answers. That's frustrating.

That also supports my "don't put too much weight on this analysis" suggestion. Phil goes on to explain some of the math involved, but I've already had too much wine to know what to paste and what not to paste here, so you'll have to read the article.

What about intelligence? Assuming life arose quickly, the odds of intelligent life evolving are actually slim. Looking at whether the probability is very close to 0 (meaning extremely rare intelligent life) or 1 (very common), Kipping's work favors the low probability at odds of just 3:2. In other words, it's more likely intelligent life is extremely rare.

Phil uses approximately the same definition of "intelligence" as I do; to wit: the ability to use technology to go into space. Those are the species we'd be most likely to encounter, if they exist. In my own opinion, sure, we could find there's a highly advanced squid-like civilization under the ice of Jupiter's moon Europa, but our own squids are pretty damn clever and they're not building rockets. More, we can't seem to communicate with them.

But if you like to gamble, it implies that the best bet is that life is common, but intelligence is rare.

This study, this article, and my own thoughts on the subject are not new. There's a thing called the Rare Earth Hypothesis, which, in a nutshell, asserts that the particular combination of conditions the Earth has experienced and allowed creatures to evolve who could build rockets is probably rare in the universe.

He hasn't proven anything, because that's not how this kind of probability works. He's just shown that, given these assumptions, life is likely to be common and intelligence rare.

Leaving this here because it's important. As I noted before, I could be wrong and I kinda hope I am. Also, again, it's a really big universe, and lots of things are possible... somewhere.

Anyway, if you have the time and desire, read the article and watch the video embedded in it. There's also apparently a Tweetstorm at the link, but I've arranged my browser to pretend that the blight on the Universe known as Twitter doesn't exist, so I can't see it.

And now, back to getting drunk and watching Black Panther.

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