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 15 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 17 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.

April 10, 2020 at 12:38am

#980657

Snake Oil

Couple days back, I said I linked the last David Wong Cracked article in my queue. Well, I had some other articles from that source. Plus, right after I said that, you know what happened? That's right, Cracked published another one by Wong. Well, I'll get to that one eventually.

For now, what with all the discussion about whether a certain malaria treatment is or isn't effective against covfefe-19 (spoiler: it's probably not), a reminder might be in order that just because someone says something is effective doesn't mean it is. Also, anecdotes aren't proof of anything.

https://www.cracked.com/article_26749_5-hip-health-trends-that-science-says-are-...

5 Hip Health Trends That Science Says Are Just Trash

These days, it seems like every celebrity, fitness model, and lifestyle guru wants to let you in on a revolutionary new essential oil, supplement, or sentient water that'll fix everything wrong with your body. (And lord, there is so very much wrong with it. Who trapped you in that meat prison, and for what crime?)

I'd take this opportunity to rag on Paltrow's snake oil, but other people have done that so much more effectively. I'll just stick to this article.

5. CBD Oil Isn't The Medical Wonder Sellers Are Claiming

Most folks' cannabis comprehension starts and ends with THC, the compound that makes food taste better and freshman philosophy sound interesting, and which turns you into the greatest Mario Kart player of all time.

I'm firmly in the "legalize it" camp. And by "legalize it," I mean treating it similarly to alcohol. This is not because I'm a stoner. I've used it, and sometimes even enjoyed it, but it's not really my thing. I'm not a stoner; I'm a drunk. People don't get this. They seem to think that if I'm pushing for legalization, that means I must want to get stoned all the time. I guess the idea that I think it's okay for other people to like something I'm not a huge fan of is alien to them, which worries me about people.

Right now if you're thinking any question that starts with the words, "But what about..." feel free to bottle it up inside. Maybe take a bong hit or two to relieve the stress of doing so.

The problem is that most CBD in the U.S. is consumed in the form of supplements, a market that's a few roaming covered wagons away from being the next snake oil industry. Not only do these pills often contain illegal THC runoff without disclosing it, but they also tend to house dangerously random levels of CBD potency and quantities.

That said, I don't care what you call it - drug, supplement, whatever - nothing is perfectly safe, not even water. For instance, I could get liver damage from simply eating grapefruit, because of some medication I take (yes, worse than what I could get from booze). I'm all for freedom, but I also think people should know the actual risks involved before getting into something. You know what kept me from doing drugs all my life? It wasn't a "don't do drugs, drugs are bad" campaign. It was a chart in the nurse's office in elementary school that listed common illegal substances, their short-term effects (positive and negative) and their long-term effects (almost entirely negative). I don't remember everything about it but I do remember reading that list (I don't think it was meant for us kids, but I was precocious) and going, "You know what? All of those substances are extremely dangerous, except marijuana." And that, kids, is why I've only ever drank booze and smoked weed. You'll note that I said I didn't do drugs, and then I said I drank booze and smoked weed. I chose my words carefully.

Anyway, I'm getting off track again. Point is, I think people need to make these decisions for themselves - with the usual "responsibility" caveat. But we can't use things responsibly if we don't know the actual facts.

4. Alkaline "Smart" Water Is No Different Than Dumb Regular Water

If you know someone who's really basic, they may be interested in becoming even more basic by drinking water spiked with alkaline. Millions of health-obsessed people swear by its boosted pH levels, which lower your body's acidity, give you more energy, help you lose weight, and even fight cancer. And whatever else you want it to do, because there's no proof that alkaline water makes your body amazing besides the fact that Beyonce drinks it.

Whoever convinced us that bottled water, in general, was a thing we should be consuming, was a marketing genius. Right up there with whoever invented chicken wings. "Let's take the part of the chicken that usually becomes dog food and turn it into a bar snack." I hate chicken wings. But I drink bottled water. What can I say? I like the convenience of being able to take it with me. If, you know, I ever go anywhere ever again.

Still, it's water. Not some pseudoscientific nonsense.

Conversely, just because it does zilch to your body doesn't mean alkaline won't affect the water it's been swirled into. Experts have warned that increased pH may scrub out many of the minerals that make regular water so good for us. But with plenty of celebrity endorsements and the promise to get thin quick, alkaline water keeps getting more and more popular.

So, not just bullshit, but potentially dangerous bullshit. At least chicken wings are still chicken.

3. Activated Charcoal Toothpaste Can Be Worse Than No Toothpaste

Activated charcoal sounds powerful, as if it comes from a sacred tree set ablaze by ancients druids to awaken the cleansing magic stored inside. So why not slather it directly on your teeth and let its cleaning force absorb all those coffee and cigarette stains, right? That's the logic behind activated charcoal toothpaste. But like most medicinal products boasting to be all-natural and holistically healing, this black paste has little science supporting its claims.

Oddly, even though I've been exposed to the study of chemistry almost my entire life, I've never been real clear on what "activated charcoal" actually is. So finally, with this, I looked it up. Turns out it's charcoal with a greater surface area

. Charcoal in general is mostly carbon, and I'm guessing - just guessing - that the greater surface area ensures more carbon binds with whatever. Carbon binds to a lot of other elements very easily; that's why there's a whole subset of chemistry featuring carbon-based molecules.

Worse, not only do the toothpaste's holistic pushers often leave out unnatural, scientifically sound fluoride, but the charcoal particles can also crawl into tiny tooth crevices, leaving users with a smile like a chimney sweep. So if you don't want any health benefits whatsoever, you don't mind black and yellow teeth, and you just can't get enough of that refreshing charcoal taste, scrub away!

Yeah, when I first heard about "activated charcoal toothpaste," that's literally the first thing I thought of, so I never looked into it further. Also, fluoride is just as natural (or unnatural) as carbon. Want to convince people to stop using activated charcoal as toothpaste? Spread the word that it's a "chemical." Pedants like me will point out that everything is a chemical, but no one listens to us so you're safe.

I have to admit, though, that technically and pedantically, charcoal is organic.

2. Brain Supplements Simply Do Not Work

If they did, we wouldn't need articles like this, now, would we? "I'm stupid, so I'm going to take this pill." Later, "This pill didn't make me smart." Then, "So maybe I shouldn't take it. Hey, maybe it did make me smart."

1. Cryotherapy Chambers Are Nothing But Really Expensive Cold Packs

The logic behind the treatment is simple. First you get put into a cryo locker that looks like a cannibal's pressure cooker, then liquid nitrogen makes the chamber colder than temperatures that naturally occur on Earth. This "hyper-cooling" purportedly sends your body into survival mode, which will then increase your fitness by hyper-oxygenating the blood so you'll have a better chance at surviving this sudden Ice Age.

You have got to be fucking kidding me. I spend my life in the pursuit of not being cold. This just sounds like hell to me. I don't care if it does work; it's a ton of nope.

Really, my only problem with this article is that there are only five examples. I'm sure we can all think of more.

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