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.

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').

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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

November 14, 2022 at 12:02am

#1040628

Crabbit Hole

Today's article, another one from Cracked, is a fun (and sometimes funny) exercise in speculation.

Cracked Goes Galaxy Brain: 4 Bizarre Ways Aliens May Or May Not Look

And I do mean speculation. Which is okay; you gotta start somewhere, and speculating is better than being incurious.

Humanity's most important question, excluding pop media relationship statuses, Incognito rash inquiries, and whether or how various animals would wear pants, is this: are we alone?

Well, some of us definitely are.

Oh, you mean "we" as a species. Or maybe "we" as a biosphere.

Are there any intelligent aliens we could have a brew and watch the game with, or are they all crabs?

There was an idea floating around a while back that came out as "everything eventually evolves into a crab." This is, of course, crabshit, as a moment's thought of how evolution works to create and fill environmental niches will conclude. The original idea is that a lot of, specifically, crustaceans, eventually take on the morphology of crabs, and there may be something to that. But it does represent one important leap in popular speculation about extraterrestrial intelligence: the idea that the human form isn't necessarily what evolution works toward (it actually doesn't "work toward" anything).

Speaking of which, this article uses "intelligence." I've beaten that dead crab a few times; basically, let's not conflate intelligence with technological capability. And please, please, stow the tired old "but we're not intelligent either" jokes. This article has enough of them.

Chillingly, we may be the gleaming example of advanced life in the entire universe.

Maybe. The Universe is a big place, though. So I doubt it.

In its usual style, the list is in reverse numerical order. Look, it's just their brand.

4. It's Not Just What Other Life Looks Like, But How We See It

Plants are green because they reflect green light. But the chlorophyll that powers their photosynthetic planty prowesses is extra reflective in near-infrared. Sadly, we're limited to seeing the visible spectrum of light, which is a tiny portion of the entire spectrum.

This is, essentially, true. But there's a decent reason for why we see the sliver of spectrum that we do, and not way out in other wavelengths: it's the relative transparency of water (where our distant ancestors evolved) and air (where our more recent ancestors evolved) to those particular frequencies. Now, some species see higher or lower wavelengths, but our red-to-violet vision is more than acceptable for what evolution produced vision for in the first place: seeing predators coming, and seeing prey.

The rest of this section goes deep into the speculation bit, and it has helpful images designed to be seen by our puny-sliver-of-spectrum-seeing eyes.

3. Aliens Could Look And Maybe Even Communicate Like Us, Dawg

Regarding convergent evolution, maybe nature isn't as creative as we thought and survival problems "only have a few good solutions."

Again, not borne out by evidence right here on our own planet. Every single living thing right now has been subject to evolution just as long as humans (and crabs) have, and this includes such varying survival techniques as nonskeletal molluscs (octopuses), opposable thumbs (primates), claws (crabs), bills (ducks), mushiness (jellyfish), ants, trees, and many other wildly varying features.

Photosynthesis necessitated loads of tweaks to many cell types, so plants produce oxygen and not, perhaps, farts. So, such intelligence as ours may occur on only 1 in 100 trillion habitable worlds. But while there may not be any civilizations in our galaxy, it's quite possible that the Milky Way still harbors tens of billions of planets covered in prokaryotic purple slime.

This aligns with what I've been saying all along. But remember, we have a sample size of exactly one when it comes to "examples of worlds with life on them." It could be that technology (again, not using "intelligence") happens on 1 in 10. It could be 1 in a googolplex. Personally, I suspect it's closer to the latter than the former. We don't know.

2. They might be robots, or robotic brains the size of your city

The problem with organic "wet" brains is that they're limited by size and processing power. Similar challenges are faced by the organic “wet” under-parts that get so many of us in trouble today. But inorganic brains theoretically have no limits of perception or conception, and robotification may be the ultimate destiny of all lifeforms that don't nuke themselves into glowing dust.

This is certainly not a new idea. Our own history of space exploration is "send the robot first." It's entirely likely that if another tech civilization exists, we'll meet their robot probes first. Or they'll meet ours; whatever. The logical extension of that would be consciousness transfer to robotic forms, which isn't remotely possible with our current technology (not to mention we don't really know what consciousness is), but hey, we're speculating here.

1. We May Be All Alone

Or maybe advanced aliens don't look like anything because they don't exist. We may be the only intelligent (ish) life in the universe. Based on statistical models, Oxford researchers say "average evolutionary transition times likely exceed the lifetime of Earth." And the universe is only a few Earth-ages old.

Oh, it's worse than that, though. Life as we know it depends on certain heavier elements. Not just the molybdenum mentioned in a recent blog entry, but something as seemingly basic as oxygen, or the iron that makes our blood work. And such elements just aren't found in the early universe. No, they have to be forged in stars, supernovae, and things like neutron star collisions. This takes time. It's not like life as we know it could have begun early on. "Sure, but what about life not as we know it?" Sure, we can just make stuff up.

Still, let's not put too much stock in statistical models, even if they do come from a reputable place like Oxford. Again, we have one data point.

My favorite theory? The one with the greatest potential for mindscrewiness: that aliens may, or may have initially, looked like us.

That's not a theory. That's more speculation. I also want to take this opportunity to reiterate what I've said in the past: there is no universal law of evolution that requires the emergence of a technology-using species. Plenty of species get along just fine without building computers or rockets.

They may even address us in English, which isn't crazy as it sounds. If they can traverse space, why couldn't they learn our language by jamming to Spotify in Moon orbit?

More likely they'll be speaking Mandarin Chinese or Spanish; more people speak those. As for what they'd listen to, I'll just note that the radio waves with the greatest chance of punching through Earth's atmosphere are in the FM band.

So I really, really hope they're listening to NPR and not some morning show shock jock.

When they do show up, just stay out of range of their pincers and you should be fine.

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