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:

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.

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.

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

1 Like

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

October 29, 2020 at 12:02am

#997051

But Mama, That's Where the Fun Is

Science is still surprising us. This is a good thing.

The Sun Is Stranger Than Astrophysicists Imagined
The sun radiates far more high-frequency light than expected, raising questions about unknown features of the sun’s magnetic field and the possibility of even more exotic physics.

Though, actually, it shouldn't be that surprising, since it was only 100 years ago that someone

had the idea that maybe the sun (like other stars) generates its power through the process of nuclear fusion. That was only an idea at the time, but it turned out to be -- probably -- the right one. Point is, 100 years isn't that long in the grand scheme of things, and we should still be figuring stuff out.

A decade’s worth of telescope observations of the sun have revealed a startling mystery: Gamma rays, the highest frequency waves of light, radiate from our nearest star seven times more abundantly than expected.

Just don't tell Marvel Studios, or they'll reboot Fantastic Four again.

The surplus light, the gap in the spectrum, and other surprises about the solar gamma-ray signal potentially point to unknown features of the sun’s magnetic field, or more exotic physics.

Again, this is a good thing. Oh, not for astronauts, obviously; gamma rays suck. My understanding is that our atmosphere and magnetic field generally keep the gamma rays from wreaking havoc here on Earth, but once you get beyond Low Earth Orbit, you don't want them zipping through your body. No, you won't turn into the Hulk; you'll just die horribly. But for scientists, hell, here's some potentially new physics to explore.

“It’s amazing that we were so spectacularly wrong about something we should understand really well: the sun,” said Brian Fields, a particle astrophysicist at the University of Illinois, Urbana-Champaign.

And why, exactly, is it that we "should" understand the sun really well? I mean, that's why we point instruments at the accursed thing. Because we don't understand it as well as we could.

Not only is the gamma-ray signal far stronger than a decades-old theory predicts; it also extends to much higher frequencies than predicted, and it inexplicably varies across the face of the sun and throughout the 11-year solar cycle.

I mean, sure, we have lots of evidence to suggest that sunshine is ultimately caused by the fusion of hydrogen to helium, a process that releases a metric shitton of energy. That's easy enough to understand. It's the details, and how the energy transfers from the core of the sun to its surface, and what happens above the surface, that we're still working on.

And Seckel, Stanev and Gaisser’s model said nothing about any dip. According to Seckel, it’s difficult to imagine how you would end up with a deep, narrow dip in the gamma-ray spectrum by starting with cosmic rays, which have a smooth spectrum of energies. It’s hard to get dips in general, he said: “It’s much easier to get bumps than dips. If I have something that comes out of the sun, OK, that’s an extra channel. How do I make a negative channel out of that?”

Well, that sounds like an absorption spectrum to me. But I find it highly unlikely that they didn't consider that possibility. Still, you'd think the article would at least mention it, if only to rule it out for nerds like me.

Besides, what exactly is doing the absorbing?

They’ve long suspected that the sun’s core might harbor dark matter — and that the dark matter particles, after being drawn in and trapped by gravity, might be dense enough there to annihilate each other. But how could gamma rays produced by annihilating dark matter in the core avoid scattering before escaping the sun? Attempts to link the gamma-ray signal to dark matter “seem like a Rube Goldberg-type thing,” Seckel said.

I'd imagine that this is because we don't know what dark matter really is. That's why it's called "dark matter" and not "stuff we understand."

I've been saying for a while now that the whole dark matter and dark energy thing reads like the luminiferous ether that was proposed as an interstellar medium before Einstein et al. showed that entirely new physics was involved. Not in the details, of course, but that it's a placeholder for, again, "stuff we don't understand."

The sun is the most extensively studied star, yet its magnetic field — generated by the churning maelstrom of charged particles inside it — remains poorly understood, leaving us with a blurry picture of how stars operate.

I would hope it was the most extensively studied star. It's not like we can go haring off to study other stars up close. Yet. But it's also notoriously difficult to study. Want to know what Mars is made of? Send a robot with sampling tools. Want to know what the moon is made of? Send astronauts, bring crackers and wine. Want to know what the sun is made of? Don't get too close. Figure it out from out here where our instruments won't melt. Much.

A solar panel malfunction kept the Fermi Telescope mostly pointed away from the sun for the last year...

Irony, thy name is Science.

Scientists are also eager to see whether the spatial pattern of gamma rays changes relative to 11 years ago, since cosmic rays remain positively charged but the sun’s north and south poles have reversed.

Misleading, thy name is Science Reporting. Of course cosmic rays are positively charged; they're basically protons moving at relativistic speeds. While electricity and magnetism are of course related, the charge on a hydrogen ion (proton) is independent of the direction of a magnetic field.

Since NASA is publicly funded, “anybody can download it if they want to glance through,” said Linden, who downloads Fermi’s new data almost every day.

No, thanks. I'll rely on science reporting, with all of its flaws. I'm smart but I'm not that smart.

“The worst that can happen here is that we find out that the sun is stranger and more beautiful than we ever imagined,” Beacom said. “And the best that could happen is we discover some kind of new physics.”

Oh, I can think of much worse things that could happen "Sorry, guys, the sun's going to blow up in 10 years instead of 5 billion. Our bad."

But no, I find that unlikely in the extreme. It's exciting to learn new stuff, especially about the thing that, I reluctantly admit, enables life to thrive on Earth.

I suppose I should explain the title of this entry for any younger readers...

Blinded by the light
Mama always told me not to look into the sights of the sun
Woah, but mama that's where the fun is

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