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.

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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.

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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.

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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.

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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.

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October 4, 2021 at 12:02am

#1018650

Inventive

People can be clever sometimes. While I think the most important invention of all time was the snooze button, I will admit that some others deserve mention too. Here are some of them.

The Stories Behind 20 Inventions That Changed the World

I'm not going to go over all of them; that's what the above link is for. Still, some highlights:

You might find it impossible to imagine a world without your smartphone, or have trouble remembering a time when Wi-Fi wasn’t everywhere, but many of today’s most relied-upon technologies would not have been possible—or even dreamed of—if it weren’t for the game-changing inventions that came before them.

Yes, I remember well when fire was invented. That led to all sorts of problems, but in the end it turned out to be a fairly good deal for us.

Dry and flush toilets have been around for thousands of years, and while many of us take these pieces of porcelain hardware for granted these days, there’s no doubt that life would look much different—and much worse—without them. “Toilets are the key to a thriving, healthy society,” Kimberly Worsham, sanitation expert and founder of FLUSH (Facilitated Learning for Universal Sanitation and Hygiene), tells Mental Floss.

Okay, you see that? That acronym right there? Someone thought they were clever, coming up with that, torturing the language to find vaguely appropriate words whose first letters make another word. Someone else approved it. Even more people perpetuate it. Stop it. It's on the same level as Calvin's Club G.R.O.S.S (Get Rid Of Slimy girlS)

Though many of today’s kids didn’t know what a Walkman was until they saw Chris Pratt’s Peter Quill flaunt one in 2014's Guardians of the Galaxy, they pay unofficial homage to the device every time they play a song on their smartphone. Transistor radios had been around since the 1950s, but it was Sony co-founder Masaru Ibuka who really revolutionized the idea of playing whatever you want wherever you are (provided that you had the cassette tape on hand).

I read somewhere else that the prop used in Guardians of the Galaxy had to be made from scratch, because genuine Walkmans (Walkmen? I never did figure that one out) were being sold for thousands of dollars. Why it was cheaper to pay union scale to fake one than to buy a real one off eBay, especially for a movie with a multi-million-dollar budget, is beyond me.

When Lyman Spitzer proposed the invention of a space telescope in the 1940s, humans could look at our universe only through land-based instruments. Earth’s atmosphere acted like a veil between the land-based telescopes and space, blurring images and hindering detection of far-off celestial phenomena. Spitzer’s research paved the way for the Hubble Space Telescope, the first space-based major optical telescope, launched in 1990 and named for the American astronomer Edwin P. Hubble.

I gotta admit, space-based telescopes are almost right up there with the snooze button in terms of great inventions. Of course, in the 1940s, we'd put nothing into orbit - not until Sputnik at the end of the 50s. But the vision was there (pun intended). This was at about the same time that Arthur C. Clarke figured out that geosynchronous orbit would be an awesome parking spot for broadcast satellites, but still decades after Tsiolkovsky proposed building an elevator to space.

I'll also point out another invention for space viewing: adaptive optics. That shit's cool as fuck. Basically, the problem with ground-based telescopes is they have to look through a soup of air molecules, and that soup is in constant motion. Density waves refract an object's light as it passes through the atmosphere. Adaptive optics compensate for the density waves by actually moving the mirrors by a tiny bit, steadying the image so you can get a longer exposure. It's still not as crisp as an image made by a space telescope, but there aren't that many space telescopes. There will soon be one more, as the article notes, and I for one can't wait to see what it'll show us.

The pizza industry has undergone numerous innovations in recent decades, but one element that has remained largely the same is the box your pie comes in. Domino's Pizza founder Tom Monaghan changed the game in the early 1960s when he worked with Triad Containers in Detroit to develop the modern pizza box. Prior to this, pizzas were delivered in bags or paperboard bakery boxes. These containers were flimsy and often crumpled under the intense heat of the pie before they reached their destinations. Domino’s corrugated cardboard containers were much more durable.

As a bonus, they tasted better than the "pizza" inside them.

Duct tape was the brainchild of Vesta Stoudt, an Illinois mom whose two sons were in the Navy. Stoudt worked at Green River Ordnance Plant packing and inspecting boxes of ammunition. The boxes were sealed with paper tape, dipped in wax, and had a tab to open them. Stoudt noticed that the boxes had a flaw: The tape was flimsy and tabs often tore off, which meant that soldiers couldn’t quickly open the boxes when they were under fire. Why not create a cloth-based waterproof tape to seal the boxes?

Okay, I can admit when I'm wrong. Duct tape is maybe above the snooze button in the list of awesome inventions.

Anyway, there are many more at the link. Twenty, in fact, as promised in the headline.

As you can see from the list, the inventions range from the dead simple (the pizza box, e.g.) to the incredibly sophisticated (like the space telescope). One should not discount the value of simplicity -- when you want to prop a door open, you use the ancient low-tech wedge; you don't stick your computer monitor in front of the door. Unless it's broken, in which case, go for it; that's about all it's good for at that point anyway.

But the fascinating thing about inventions, to me, is how they build on each other. This is what misanthropes miss when they smugly proclaim that other animals use tools, too, so we're obviously not that special. Yes, some have been observed using simple tools, sure, but I haven't heard of a single case of a nonhuman animal using a tool to build other tools. That requires a level of planning beyond "I see a piece of food and I'm hungry."

Which brings me back to how awesome it was to learn how to make fire.

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