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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Write_Mikey_Write!

about 10 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 12 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').

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

February 21, 2024 at 9:33am

#1064595

That's Not Life

Periodically, astronomers announce the discovery of an exoplanet in its star's "habitable zone," and, inevitably, like the old game of Telephone, this gets filtered down to us garbled as something like "Scientists Discover Evidence of Alien Life."

Is K2-18b an inhabited ocean world? Don’t bet on it
Some fascinating observations of K2-18b have come along with horrendous, speculative communications. There's no evidence for oceans or life.

Worse, this gets understood as "sentient alien life," because we've all grown up with Star Trek and Doctor Who, both of which put sentience on top of an evolutionary pyramid, like it's inevitable once you posit life. Much as I love those franchises, I understand that they're not documentaries.

I’m sorry, everyone, but we need to talk about Hycean worlds and dimethyl sulfide.

Okay, I have some idea of what dimethyl sulfide is (in that it's gotta involve two CH₃ groups and sulfur), but I had to follow their link to "Hycean worlds." The link goes to an earlier Big Think article (same source, different author). Basically, they've got a lot of hydrogen. "Hycean" is a portmanteau of hydrogen and ocean (where "ocean" is understood to be a water ocean, and 2/3 of the atoms in water are, of course, hydrogen. But there may also be methane oceans, and 4/5 of the atoms in a methane molecule are hydrogen, but... whatever, I digress.)

I also checked up on dimethyl sulfide and, yep, a sulfur atom connected to two methyl groups. It's a bit like water, structurally, with sulfur instead of oxygen and methyl ions taking the place of hydrogen. Except that it's not like water at all. Among other differences, water doesn't usually burn, unless you live in Cleveland.

This planet, K2-18b, was indeed observed by the JWST, and did have a fantastic spectrum taken of its atmosphere, revealing many fascinating details about it.

The fact that we can do this for a planet 120 light-years away is wondrous enough in itself.

However, there is no evidence that K2-18b is a Hycean world at all; no water was detected. There’s only dubious evidence for dimethyl sulfide, and even if it does exist in the atmosphere, assigning a biological cause to it is an incredibly dubious proposition.

You said "dubious" twice. Now I'm dubious.

Yes, that word can describe both unreliable observations, and our reaction to them. Love English.

Yet if you’ve read headlines from around the internet, it isn’t just the usual suspects like the New York Post or the Daily Mail with outrageous, alien life-driven headlines, but normally reliable places like National Geographic, the BBC, and right here on Big Think.

NatGeo hasn't been reliable since Fox bought them (though Disney later acquired that property along with lots of other Foxy things, though not the "news" arm); I've been starting to wonder about the BBC; and while I like Big Think as a source, it's still subject to many of the problems all internet-only sources have.

Let’s take a look at what’s really going on with exoplanet K2-18b.

The article proceeds to do just that, but I won't reproduce it all here. But, in summary:

None of these possibilities describe K2-18b, because it’s massive, puffy, and more Neptune-like than Earth-like.

Does this mean it definitively doesn't harbor life? Of course not. But let's not get ahead of ourselves.

K2-18b is about 2.6 times the radius of Earth and 8.6 times the mass of Earth. This means its density is less than half of Earth’s density, implying that it has a large envelope of volatile gases surrounding it.

Let's not be in too big a hurry to compare it to Neptune, either. Neptune is classified as an ice giant, because it's cold. K2-18b probably isn't. But I'll run with the Neptune analogy, because a) the article does, and b) I won't mention the name of the other ice giant in our solar system.

There's a bunch of technical details that follow, which I certainly won't reproduce here, but the article is an easy read and the link is right there.

While a water-covered Earth-sized world would be an incredibly interesting place to look for life, and in particular to look for the biosignatures associated with the processes that occur in ocean waters, it’s an enormous stretch to apply those same criteria to a gas giant world like K2-18b.

Why?

Because there was no water detected on K2-18b.

Again, this doesn't mean "no life." But it's not a good candidate for water-based life as we understand it. And we absolutely cannot make the jump from "possible simple life" to "Klingons."

In other words, it’s not entirely implausible that maybe, just maybe, this is a mini-Neptune version of a water-rich Hycean world, and maybe there really is some sort of extremely exotic form of life that exists on a world like this. After all, the JWST spectrum shows a (weak) indication of dimethyl sulfide, which we know here on Earth is produced biologically. Could that truly be what’s happening here?

Short answer: no. Longer answer: probably not.

Look, finding extraterrestrial life, even if it's just an archaeum or its equivalent, would be a Big Fucking Deal. I genuinely hope it happens in my lifetime (so hurry it up, already). But jumping to conclusions helps no one, and is especially difficult on a high-gravity planet.

Let's also not forget that "habitable zone" isn't the last word on where life might exist. Venus and Mars are (barely) within what's considered the Sun's habitable zone, and despite some flurries a few years back, neither has shown any definitive signs of harboring life. On the other hand, we're looking at moons of Jupiter and Saturn as possible places for life to have gotten a foothold, and those moons are absolutely outside the habitable zone.

Meanwhile, though, K2-18b would make a good name for a Star Wars droid.

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