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:

Write_Mikey_Write!

about 9 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 11 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.

February 28, 2024 at 8:49am

#1065075

That Was Easy

Not every invention is great, but the article I'm linking today, from Cracked, is about great inventions that were unappreciated at the time. The article's a bit on the lengthy side, so I'm not going to mention all of them. Just a few things I want to comment on.

5 Innocent Inventions That Sparked Uproars When They Were New

5. Push Buttons

At the end of the 19th century, a few different electric devices like the lightbulb were set to change the world.

Meanwhile, I'm sure candlemakers and whale oil suppliers were freaking out about their impending loss of revenue.

Still, people resisted electricity entering their homes.

I truly hope that was an intended pun.

Then came a new ancillary invention that made electricity a lot less scary: the push button.

The actual definition of "easy," at least according to one well-known marketing campaign.

But the push button received unexpected pushback from the scientific community itself. While marketers realized the button would convert people to the church of electric power, educators already had their own plan for managing this: education. They wanted to bring people closer to the inner workings of electricity, not farther. In schools, they were teaching boys and girls about how to put together motors and batteries, not as part of vocational training but just standard learning. Understanding electricity demystified the process.

I kind of get it. I liked the internet a lot better when you had to have some level of technical proficiency and the desire to use it. And also when it was less commercialized. Okay, mostly the latter. Still, I kind of get it. Teach people, instead of dumbing things down for them.

Problem is, some people refuse to be taught, and some simply cannot be taught. They already know everything they need to. Just ask them, and they'll tell you.

4. ZIP Codes

Everyone was being assigned new numbers? That was pointless — and dehumanizing. It was (theorized some people) surely a communist plot, with an uncertain goal. Some random comments from disgruntled customers were preserved so we can marvel at them, generations later. “Dear Sir, Zip Code is a complete boo-boo and you just don’t want to admit it,” wrote one woman. “It has set our mail delivery back 100 years.” Another message claimed, “The Pony Express would be more efficient.”

Sound familiar? It should. 60 years later, we're still getting comments from the same kinds of novelty-resistant people, only now with a lot more abbreviations, LOLs, OMGs, emoji, and maybe a few cutting gifs. Which, I suppose, satisfies the definition of "irony."

Today, you use them without complaint, but how often do you use the full ZIP code, with the initial five digits as well as the four digits that come after them? Do you even know your own full ZIP code?

No, but I can look it up. And therein lies the problem: Anyone can type in an address, anywhere in the US (which is the only place ZIP codes apply; places like Canada and the UK use similar but different systems), and find their ZIP code, complete with the rarely-used +4 suffix.

Which means that now, ZIP codes are kinda anachronistic in general. Hardly matters, though, at least for me: I can't remember the last time I had to address an envelope. It's been a long, long time.

2. The Cheese Slicer

If you try cutting a block of cheese into slices, you need a steady hand, lots of concentration and also a high tolerance for failure because the result will come out terrible no matter what. You’ll wind up with a bunch of awkward wedges instead of slices. Then, in 1925, a hero named Thor Bjørklund forged a new tool, which would be called the ostehøvel.

I suspect it would be very, very difficult to find a more Norwegian name than Thor Bjørklund.

Everyone who cut food at home loved the ostehøvel. Professional cheese men did not. If cheese cutting was going to be so easy going forward, why had they wasted all those years getting a degree from Colby College (and then a master’s, from Stilton)?

And that should sound familiar, too. Many new inventions threaten to displace old industries. It's only when the industry is powerful enough to have a lobbying group that laws get passed against the new invention. At least, that's how it works in the US. Not sure about Norway.

1. Toilet Paper

Look, if you value your mental health, never, ever look up "what did people wipe their asses with before toilet paper?" This article doesn't even go into the real details. For which you should be ever grateful.

4 Like • Comment • View 1 comment

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