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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

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.

5 Like • Comment • View 1 comment

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