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 2 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 4 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.

December 6, 2022 at 12:02am

#1041452

In the Habit

Science is cool.

A Dream of Discovering Alien Life Finds New Hope
For Lisa Kaltenegger and her generation of exoplanet astronomers, decades of planning have set the stage for an epochal detection.

Just to get this out of the way: we're not talking about finding Vulcans, here. But even finding so much as a microbe somewhere other than Earth would be a Big Deal.

Kepler had glimpsed its first two Earth-size exoplanets with a decent chance of having liquid water on their surfaces. These were the sort of strange new worlds that everyone at the conference — and possibly most of the human race — had imagined at least once. Would Kaltenegger confirm that the planets might be habitable?

Another disclaimer: "habitable" doesn't necessarily mean "by humans." Water, in its liquid form, is essential (but not by itself sufficient) for life as we know it. Sure, the possibility of life not-as-we-know-it is something to consider, but if you're going to be looking for alien life, it makes sense to start with the conditions we know can produce and support life.

Kaltenegger, at the time an astrophysicist at the Max Planck Institute for Astronomy in Heidelberg, Germany, started running new climate models before the conference was over, incorporating basic facts like the planets’ diameters and the lukewarm glow of their star. Her ultimate answer: a qualified yes. The planets might be suitable for life, or at least for liquid water; they could even be water worlds, encased in endless oceans without a single rocky outcrop poking above the waves. The caveat was that she would need more advanced observations to be sure.

I'll also note that there's at least one other place in our own solar system that is potentially habitable (if not by humans): Europa, a moon of Jupiter. It's not in what astronomers would call the habitable zone of our system, but other factors make liquid water there a possibility. From what I understand, they're sending a probe to Jupiter to get more data. So we're not even sure about that, yet, but that's not stopping astronomers from searching exoplanets as well.

Her overarching quest — the search for alien life — is entering an unprecedented phase. Barring the bolt-from-the-blue arrival of something like an extraterrestrial radio broadcast, most astronomers believe that our best near-term chance of encountering other life in the cosmos is to detect biosignature gases — gases that could only have come from life — floating in exoplanets’ atmospheres.

When life first arose on Earth, it sucked in carbon dioxide and produced oxygen as waste. Plenty of life still does that here. So yes, our entire biology is about using other life's waste products (not just the alcohol that beer yeast produces). Oxygen is very reactive, so it probably doesn't stick around in an atmosphere for very long unless it's continually being produced by some process, including what we call life. I've known for some time that finding oxygen in a planet's atmosphere would be a promising sign that there could be some sort of life there. There are other telltales, but oxygen is a pretty obvious choice to look for. The trick, however, is to detect it.

The sort of remote measurement necessary to make that kind of detection has strained the capabilities of even humanity’s most advanced observatories. But with the James Webb Space Telescope (JWST) now in its first few months of observations, such a discovery has become possible.

Remarkable that the same telescope that can tell us stuff about the early universe could also tell us about things relatively close by.

Most optimistically — if biospheres bloom easily from Earth-like worlds — the telescope may detect odd ratios of, say, carbon dioxide, oxygen and methane on one of these planets. Astronomers may then be sorely tempted to attribute the concoction to the presence of an extraterrestrial ecosystem.

Astronomers, tempted or not, probably wouldn't jump to that conclusion. It would be a real "giant leap," and not in a good way. The best you're going to get from them is "maybe." Science reporters, on the other hand, will confuse the issue in the public's collective mind by hopping from "oxygen in the atmosphere" to "life" to "OMG ALIENS."

I've said this before and I'll say it again: life is probably out there, but there is nothing about evolution that requires the emergence of sentient life, the kind that might build spaceships or write science fiction.

It would still be a big damn deal if we could remove "probably" from that sentence.

Not only will the atmospheric signals they’re looking for be weak, but she and her colleagues must model a planet’s possible interplay of starlight, rock and air accurately enough to be sure that nothing besides life could explain the presence of a particular atmospheric gas. Any such analysis must navigate between a Scylla and Charybdis, avoiding both false negatives — life was there but you missed it — and false positives that find life where there is none.

Points to this reporter for the Odyssey reference.

Unlike most scientific endeavors, the search for signs of extraterrestrial life happens under an unavoidable spotlight, and in a turbocharged information ecosystem where any scientist crying “Life!” warps the fabric of funding, attention and public trust.

Because immediately, people are going to expect an alien invasion, or we will prepare to stage one of our own.

The goal at the time was to compare spectra from rocky, temperate planets to what Earth’s spectrum would look like from far away, seeking conspicuous signals like a surplus of oxygen due to widespread photosynthesis. Kaltenegger’s objection was that, for the first 2 billion years of Earth’s existence, its atmosphere had no oxygen. Then it took another billion years for oxygen to build up to high levels.

That's another thing I hadn't considered: that for more than half the time we know life existed here on Earth, it wouldn't have produced enough oxygen to be detectable. But then, I'm just a spectator; I don't get paid to think about these things.

The most pertinent case study rocked the astronomy world in the fall of 2020. A team including Seager announced that they had spotted an unusual compound called phosphine in the upper atmosphere of Venus, a sweltering, acid-washed planet typically dismissed as sterile. On Earth, phosphine is commonly produced by microbes. While some abiotic processes can also make the compound under certain conditions, the team’s analysis suggested those processes weren’t likely to occur on Venus. In their view, that left tiny floating Venusian organisms as a plausible explanation. “Life on Venus?” the New York Times headline wondered.

I remember that. You probably do, too, because, as with the Martian meteorite from even further back, any time someone says "maybe life," it gets blown up to "OMG ALIENS."

Spoiler: no aliens on Venus. Probably not even microbes. Well, let me qualify that: no definitive evidence; it's remarkably hard to prove a negative, and the possibility remains, however unlikely.

Anyway, the article is long but thorough, and I barely scratched the surface here—much as our Martian robots have barely scratched that planet's surface in their quest for even the slightest hint that life might once have had a foothold on the Red Planet.

Nothing yet. But we'll keep looking, because that's what we do.

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