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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 11 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 13 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.

March 25, 2022 at 12:02am

#1029446

Bite Me

Here's an article about a thing I don't think I've ever tried.

How Bagel Bites Took a (Tiny) Bite Out of Snacking History
What happens when a simple, innovative idea meets two hardworking families who will do anything to follow their dreams? A miniature, delicious pizza-bagel, of course

I don't know why I've never tried it. I love pizza, and of course bagels are part of my culture. I think, probably, I always assumed that they aren't real bagels (which they're not) or real pizza (which they're not), so I didn't want to be turned off from either or both by consuming this horrid abomination.

I have no problem eating pizza toppings on naan, though.

“You won’t believe this,” Stanley Garczynski, the co-creator of Bagel Bites, says with a pause, “but I hate pizza.”

I have a cousin who worked at a pizza place in high school. He couldn't eat the stuff for years.

Still, the article may be a puff piece (that's a joke, as bagels are anything but puffy), but it's still an interesting look at the business.

It all started sometime around 1982, when 30-year-old Garczynski moved to Fort Myers, Florida in search of a fresh start. Along with that, he needed a new tennis partner, which is how he met Mosher, a caterer at the time. The two became fast friends, and they and their respective girlfriends would get together often to party, drink and shoot the breeze about everything from their dreams for the future to humankind’s near-universal love for pizza. It was during these casual get-togethers a simple, yet innovative idea was born, one that would not only make Garczynski a millionaire, but change American snacking habits forever.

Now, that's a bit of a stretch. One of the reasons I've never tried bagel bites is simply that no one I know has ever produced them at a party I was at. To be fair, though, few people ever invite me to parties.

In 1982, I had been a teacher for seven years in Londonderry, New Hampshire. I loved teaching and I loved the kids, but there was no money in it. I was getting restless living in New Hampshire, so I moved to Florida and began working in marketing and sales.

I hate that "marketing and sales" pays better than teaching. What people earn is supposed to be proportional to their value to society. Hey! Stop laughing!

So we got to talking about coming up with some kind of hors d’oeuvre or appetizer that would be fun for kids — something we could put pizza toppings on, like toast, bread or English muffins. Around this time, bagels had become very popular. In the past, they’d mostly been big in Jewish areas of New York City, but Lender’s Bagels out of Hartford, Connecticut was making them more common everywhere.

I'm sure if the only bagels you ever eat are Lender's, you have a warped idea of what those toroidal carbohydrate modules are. Go to New York. Eat a real bagel. You'll never eat Lender's again.

One of the early challenges was that we had to figure out where to get our ingredients. We already decided on the Lender’s mini-bagels, and for the cheese, we got Sargento. For the sauce, we used Heinz, who would end up buying Bagel Bites some years later.

Still, if you're going for mass-production, you could do worse than those suppliers.

People would see our booth and say, “Bagel Bites? What is a Bagel Bite?” and then see these little pizzas. They’d pick them up, turn them over and see that dimple. “Oh, they’re bagels!” they’d say. “Wow! Pizza on a bagel!” We’d serve thousands of them.

Bagels shouldn't have dimples. Bagels should have holes. The holes in a real bagel would make it somewhat awkward to put pizza toppings on them. More edges for shit to fall off of.

So yeah, a pretty standard success story, and a one-sided one at that. It often amazes me how things that, by all rights, should not exist become popular and profitable. History is littered with them. Chicken wings, for example. Bottled water. I mean, whoever first came up with the idea: "Hey, you know what the world needs? We take the parts of the chicken that usually become dog food, and we market them as sports-watching snacks."

This is why I'd never be a successful businessman. I'd be too focused on what should be to try to sell something that should never be.

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