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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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').

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 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!

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