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.

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

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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

October 18, 2024 at 11:02am

#1078516

Lithium

It's science, not magic; and it's not secret anymore, thanks to Noemi. Fair warning, this is a long read:

The Secret, Magical Life Of Lithium
One of the oldest, scarcest elements in the universe has given us treatments for mental illness, ovenproof casserole dishes and electric cars. But how much do we really know about lithium?

It may surprise some people to know that I don't have the Periodic Table memorized. I go, "Hydrogen, helium, lithium..." and then I have to look it up. Fortunately, I live in the 21st century, and it's easy to look things up rather than commit them to long-term memory.

(Beryllium's next, by the way, followed by boron and carbon.)

The universe was born small, unimaginably dense and furiously hot. At first, it was all energy contained in a volume of space that exploded in size by a factor of 100 septillion in a fraction of a second.

This view of universal origin may be in need of revision, thanks to JWST observations. It does not change the material (pun intended) facts of the piece.

Quarks and gluons had congealed to make the first protons and neutrons, which collided over the course of a few minutes and stuck in different configurations, forming the nuclei of the first three elements: two gases and one light metal.

Those would be the ones I mentioned above.

For the next 100 million years or so, these would be the only elements in the vast, unblemished fabric of space before the first stars ignited like furnaces in the dark to forge all other matter.

"Unblemished" is surely poetic license. Also, again, details subject to change pending study of recent observations.

Almost 14 billion years later, on the third rocky planet orbiting a young star in a distal arm of a spiral galaxy, intelligent lifeforms would give names to those first three elements.

"More than" 14 billion, and they need to be careful about calling us "intelligent," as doing so invites self-contradictory jokes about intelligence. (Self-contradictory, because one needs to be intelligent to make such a joke, but it's still stupid.)

“Lithium has one of the most complex stories,” Fields said. “The oxygen you’re breathing, the carbon in your DNA, the iron in your blood — that came later, out of stars. But lithium comes straight out of the Big Bang.”

This is misleading, as lithium can also be produced post-Bang, but the article does mention that later.

But the sources of lithium in the universe have, since the late 1980s, presented astronomers with an odd problem. As Fields told me, astronomers studying the abundance of elements after the Big Bang have created complex calculations that account for the expansion of the universe, nuclear reactions and the behavior of subatomic particles like photons and neutrinos. The math pencils out for hydrogen and helium — the measurements match the predictions. Not so with lithium — only a third or less of the expected amount of lithium is observable in the universe.

This sort of thing is exactly why we do science.

The story of how humans discovered lithium goes back to the late 18th century and a Brazilian scientist, statesman and poet named José Bonifácio de Andrada e Silva who was hopscotching around Europe on a sort of early study abroad program.

I'm betting he studied more than "a" broad.

Because the element was discovered in a mineral, Arfwedson and Berzelius named it “lithia,” after the Greek word “lithos,” for stone.

It should surprise no one that I find the origins of element names fascinating. Apart from obvious ones like iron and gold, known from antiquity, their names tend to reflect how we discovered them. Hydrogen, made from water. Helium, first detected in the sun. Beryllium, found in the mineral beryl (emeralds, e.g.) That sort of thing.

Finally, in 1855, two chemists — Robert Bunsen and Augustus Matthiessen — were able to isolate lithium in a quantity large enough to study its properties.

Yes, that Bunsen, namesake of every chemistry student's favorite flame source.

The article goes on to describe the beginnings of what might be lithium's most famous application: treating mental disorders. Don't read it if you like guinea pigs.

Research in Texas, Greece, Lithuania and elsewhere has found that naturally occurring lithium in drinking water is associated with lower rates of suicide and violent crime, prompting some scientists to advocate adding it to municipal water supplies, much like fluoride is sometimes added to strengthen teeth.

1) I want to take this opportunity to point out that Lithuania and lithium are not etymologically related.
2) Oh, adding elements to drinking water is sure to be uncontroversial.

The article then turns to what's probably the second most famous application of lithium (and might have already claimed the #1 spot): batteries.

Thomas Edison died in 1931 with a tin of lithium on his desk. One of the final projects in the great inventor’s life was experimenting with new chemistries in what he called “storage batteries.”

Yeah, right. "Great inventor," my fat ass. Edison was a hack with a flair for marketing and melodrama, who coasted to most of his patents on the writhing backs of his underpaid servants. Could be he invented a thing or two by himself along the way, but he was a massive dick. Muskmelon reminds me of him.

Anyway, the article ends with a section highlighting lithium's ubiquitous role in today's technological life. As I expected, nothing magical about it, but it is interesting to read about.

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