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 24 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

a day 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.

December 2, 2019 at 12:12am

#970837

Pure Imagination

https://www.vox.com/science-and-health/2017/10/18/15995478/chocolate-health-bene...

Dark chocolate is now a health food. Here’s how that happened.

Let me guess - Willy Wonka paid for "scientific" studies.

The Mars company has sponsored hundreds of scientific studies to show cocoa is good for you.

See? Cynics are usually right. Shunned and ignored, but right.

A year after James Cadbury, the 30-something great-great-great-grandson of the British chocolatier John Cadbury, launched his luxury cocoa startup in 2016, he introduced an avocado chocolate bar.

Way to destroy both avocado and chocolate, Jim.

The company promised to deliver the nutrition of avocados — in a chocolate bar. Journalists were dazzled.

Journalists shouldn't be dazzled. Journalists should be even more cynical - and skeptical - than I am. But we live in the 21st Century and, clearly, in Bizarro World.

Big Chocolate’s investment in health science was a marketing masterstroke, catapulting dark chocolate into the superfood realm along with red wine, blueberries, and avocados — and helping to sell more candy.

I'm a big fan of science. I'm also a big fan of chocolate. But science mixed with chocolate is like... M&Ms mixed with Skittles. I had a chaotic evil Dungeon Master who did that once. Once.

Point is, science, especially when it comes to food and nutrition, is largely statistical. And you know what Twain said about statistics.

Actually, Twain said (or, more accurately, wrote), "There are three kinds of lies: lies, damned lies, and statistics," but he didn't come up with it. So that's a lie too. It's lies all the way down!

Which brings me back to healthy chocolate.

Such overwhelmingly positive findings suggest this area of industry-funded nutrition science may be biased.

YA THINK?!

Fawning cocoa coverage is so pervasive, one journalist even created a bad chocolate study, suggesting the candy promotes weight loss. His goal was to fool the media into picking it up and make a point about how easy it is to do so. (It worked.)

I find that hilarious. And sad. Sadlarious.

“I don’t want to be cynical — a lot of their science is good; it’s put in peer-reviewed journals,” Coe added. “But just keep in mind that too much of anything is not really good. If you’re hooked on chocolate, you’re hooked on sugar.”

Funny you should mention sugar. Turns out a whole bunch of studies have been done on sugar as well, and it appears that added sugars are Really Fucking Bad For You. There might well be bias in that science as well, but you don't hear as much about it. Now, me, I prefer chocolate that's as bitter as my ex-wife and as dark as her lawyer's heart, but the vast majority of people? "Oh, I hear chocolate is good for you so I'mma eat this whole bag of M&Ms." Like one guy I knew who heard that red wine was good for you so he chugged a gallon of it every day. Pretty sure he's dead now. Liver and pancreas got into a fight and he exploded.

Still, most nutritional studies that you hear about are touting the health benefits of foods that taste like ass. Kale, for example. I was shocked to learn recently, by the way, that there was supposedly a time when no one had heard of kale. Shocked, because my family grew that by the bushel in the garden and sold it at farmers' markets when I was a kid. Always hated that stuff. It's only recently that I figured out how to make it palatable.

And that's one problem with "health" foods - the Puritan undertones of American society steer us away from believing that anything can be good for you if it's not like doing penance. Which is probably why so many people are eager to believe the chocolate stories - they reverse the narrative.

My personal opinion? Eat chocolate. Don't make it a "guilty pleasure." Don't pretend it's a health food. Just enjoy it for what it is - limit yourself if you have to (I certainly do), but god DAMN I'm tired of all the anxiety and neuroses surrounding food and drink these days.

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