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.

January 7, 2022 at 12:03am

#1024291

Bad Advice

And again from "JAFBG"

[XGC]:

What widely-accepted piece of advice is actually total bullshit?

Whoa. Like, where to begin?

I suppose "Don't get that vaccine, just eat horse dewormer if you get sick" would be too obvious. But it fits the prompt.

Perhaps something a little more general, like "Be yourself?"

No, that one isn't total bullshit -- if you're attractive, rich and charming. If you're not, "be attractive, rich and charming" would be better advice. If only there were some way to successfully follow it.

Oh, I know one! "When you get to prison, find the biggest inmate and punch them in the face to establish dominance." Yeah -- that's not great advice unless you want to avoid your long prison sentence by being dead. But it also won't apply to most people.

Let's see... "Put all your cash into cryptocurrency?" Nah, not widely-accepted enough to qualify. But total bullshit. Well, I suppose if you like to gamble. But I like to gamble and I still stay away from that nonsense.

So I'll dig deep, go against the grain, and follow several other clichés to suggest that one widely-accepted piece of advice that is total bullshit is...

Never give up.

A couple of years ago, there was a guy out in the desert who was absolutely, totally convinced that the Earth was flat, and that anyone who said otherwise was part of a vast, overarching, global (snort) conspiracy. What he believed the purpose of the conspiracy was, if anything, I don't really know, but it's irrelevant.

So he built a rocket. I guess because the rockets that other people built and used to orbit the really very obviously round Earth were all part of this grand conspiracy or something.

In case you've forgotten this little incident, here's the report of his final flight

, which failed to prove or disprove the planar nature of the planet, but did effectively provide further evidence to support the theory of gravity.

"Mad" Mike Hughes, 64, crash-landed his steam-powered rocket shortly after take-off near Barstow on Saturday.

1. Appropriate nickname.
2. Steam-powered rocket.
3. I can't emphasize this enough: steam-powered rocket.

A video on social media shows a rocket being fired into the sky before plummeting to the ground nearby.

You know, as mulishly ignorant as this guy is, I don't want you to get the impression I'm amused by or celebrating his death. Nor do I have any interest in viewing a snuff video. But I will admit to making "not exactly a rocket scientist" jokes. At least when Musk played with rockets that demonstrated lithobraking, they weren't crewed.

Hughes was well-known for his belief that the Earth was flat. He hoped to prove his theory by going to space.

4. Again... other people have done this and reported extensively on it. Complete with photographic evidence.
5. Moreover, one does not need to go into space to demonstrate the curved nature of the surface of the planet.
6. Specifically, we've known this since at least 500 BCE, thanks to math. You know, the stuff you need to know to do rocket science.

With the help of his partner Waldo Stakes, Hughes was trying to reach an altitude of 5,000ft (1,525m) while riding his steam-powered rocket, according to Space.com.

7. 5000 feet?
8. I will remind you that he did this in or near Barstow, California.
9. Barstow is approximately 140 miles from the highest mountain peak in the continental US.
10. Said mountain (Mt. Whitney) has a summit approximately 14,500 feet above sea level.
11. Therefore, he could have fucking climbed Mt. Whitney and exceeded his goal altitude by a factor of nearly 3.
12. Or gotten into a jet that would cruise at around 30,000 feet.
13. And that still wouldn't have been enough to convince him.

My point in going through all this is not to make fun of idiots or the stubbornly and willfully ignorant, though I'm certainly not above doing so. No, my point is that if he'd believed and listened to actual science, Hughes might well have given up. And possibly still be alive today.

Except he probably wouldn't be, because I guarantee you he'd also deny the existence or severity of COVID, and was old enough to get hit hard by it.

Had he lived, though, I don't know, maybe Bezos could have given him a ride in that giant phallus of his. I'd have loved to hear what he had to say about it afterward. My guess, though? For people like that, no amount of evidence would be enough, and he'd have found a way to double down on the flat-earth assertion.

The moral of this story is: It's great to have a dream. It's great to pursue it. But at some point, if people keep telling you that it is not possible for a human to breathe water, you can either give up on proving it for yourself by sticking your head in a bucket, or you can die drowning.

Yes, there have been instances in history of people who insisted that they were right, while the established beliefs were wrong. But in nearly every one of those instances, it was a matter of scientists doing science and pushing back against ignorance and superstition. Hughes was doing precisely the opposite of this.

Want to know how to spot a crackpot? See if they say something like "No one believed Galileo either!" If they do, back away slowly.

In other words... give up.

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