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 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.

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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.

April 13, 2021 at 12:05am

#1008426

Ancient Life

More science.

Fossil Discoveries Challenge Ideas About Earth’s Start
A series of fossil finds suggests that life on Earth started earlier than anyone thought, calling into question a widely held theory of the solar system’s beginnings.

There are some compelling hypotheses about how life got its start. There are also not so compelling ones, like "it came from somewhere else in space." That one just kicks the can down the road and doesn't explain how life first arose out of non-life; it just postulates that it didn't begin on this planet. Regardless, the question of "when" might shed some light on "how."

In the arid, sun-soaked northwest corner of Australia, along the Tropic of Capricorn, the oldest face of Earth is exposed to the sky. Drive through the northern outback for a while, south of Port Hedland on the coast, and you will come upon hills softened by time. They are part of a region called the Pilbara Craton, which formed about 3.5 billion years ago, when Earth was in its youth.

That's if you can avoid the drop bears and everything else that's trying to kill you in Australia. Really, it's a wonder anyone can live there at all. Maybe the reputation is exaggerated and promoted by the locals to keep tourons away.

In the past year, separate teams of researchers have dug up, pulverized and laser-blasted pieces of rock that may contain life dating to 3.7, 3.95 and maybe even 4.28 billion years ago.

Those are all a Really Damn Long Time Ago. For reference, the Sun is probably about 5 billion years old, a little more than 1/3 the age of the Universe itself. Also for reference, human civilization is maybe 10,000 years old - not even as big as the error bars in whatever they used to date those (possible) fossils.

Taken together, the latest evidence from the ancient Earth and from the moon is painting a picture of a very different Hadean Earth: a stoutly solid, temperate, meteorite-clear and watery world, an Eden from the very beginning.

When you consider that all early life, up until (they think) about half a billion years ago, was aquatic, the water could have provided some protection from asteroidal catastrophes.

The article includes a timeline for early Earth/Moon formation, with graphics that for some reason amuse me.

Anyway, the rest of the story lays out the evidence and discusses some of the debate surrounding it. No need to reproduce most of it here.

“Are there other explanations than life? Yeah, there are,” Bell said. “But this is what I would consider the most secure evidence for some sort of fossil or biogenic structure.”

Thing is, just as it's important to be really, really sure when we find evidence of extraterrestrial life (by which I mean microbes, not Vulcans), it's also important to be skeptical of this sort of finding. Likely there will need to be additional evidence.

Far from fussy and delicate, life may have taken hold in the worst conditions imaginable.

I don't know about "worst." Venus is way worse. (Before anyone starts quoting the bit about finding signs of life in Venus' atmosphere, that finding was conclusively discredited.) But worse than we imagined, sure. We already know that simple life can withstand conditions that more complex life-forms cannot, so it's at least believable.

There's also a video worth watching in the article, if you're interested in this sort of thing. With bonus explosion graphics. (You just have to suspend disbelief with all the sound effects about things going boom and thunk and whoosh in the near-vacuum of space.)

The other reason this sort of thing is interesting is the implication for finding (simple) life elsewhere -- something the article doesn't get into until the very end:

If there was no mass sterilization at 3.9 billion years ago, or if a few massive asteroid strikes confined the destruction to a single hemisphere, then Earth’s oldest ancestors may have been here from the haziest days of the planet’s own birth. And that, in turn, makes the notion of life elsewhere in the cosmos seem less implausible. Life might be able to withstand horrendous conditions much more readily than we thought. It might not need much time at all to take hold.

And again, as a caution, I wouldn't make the leap to the kind of sophisticated life it would take to, say, make radio stations or flying saucers. Nothing in evolution requires that result. But it makes ideas like finding microbes or their equivalent on, say, Mars, Europa, or Titan that much more likely.

Which would be cool.

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