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.

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').

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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.

3 Like

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.

2 Like

February 3, 2021 at 12:05am

#1003529

Feed A Cold

But first, a quick brag:

Thanks to all the judges and other participants in "30-Day Blogging Challenge ON HIATUS"

[13+]

You, too, can win a Merit Badge, and with a lot less effort -- details below!

Today we reach into the Wayback Machine for an article from the innocent Before Time, two years ago. A time when we actually worried about being around other people because we might get a cold.

Quick-fix cold and flu remedies do nothing but make you poorer
Emergen-C, zinc, detox baths, vitamins, echinacea: it’s all garbage. Chicken soup, though, works.

I once worked in an office that, I’m convinced, was actually a refrigerator. The people who worked inside that open-plan frozen concrete box were often sick, probably because a large contingent of them never washed their hands after they used the dead-silent restrooms.

Shudder.

But we were not the healthiest people.

This is my shocked face:

We continue to cling to so-called “old-wives’ tales” when it comes to preventing and treating coughs, colds, and flu; in fact, interest in complementary and alternative medicine, such as vitamins, herbal treatments, and acupuncture, has consistently climbed over the past few decades. But ultimately, we’re spending energy and money on prevention methods and treatments that, at best, don’t work, and in rare cases, could actually make you sicker.

On the contrary, these "medicines" have been proven to be very effective... at allowing the people who peddle them to eat.

Freed recently conducted a poll of over 2,000 parents, representative of the general United States population, and found that 70 percent of them employ folk strategies to help prevent their child from catching a cold. Such strategies include disallowing the child from going outside with wet hair, limiting outdoor time generally to avoid getting sick, or conversely, encouraging more time outside to prevent the child from getting sick.

My takeaway from this: Better not go outside at all. Safer that way.

One of the most popular herbal supplements for immune-support, echinacea, is mixed at the very best; while a handful of studies show it can reduce your risk of catching a cold, several others, like a 2004 randomized controlled trial, were unable to replicate those earlier studies’ results; a 2018 review (with a very helpful infographic!) takes that language even further, stating that echinacea, as well as other popular supplements and herbal remedies like zinc, garlic, ginseng, eucalyptus oil, and honey, shows “no evidence of effect” for treating the common cold.

You'd probably be well-protected against vampires, though. I mean, I eat garlic all the time and I have yet to be bitten by a vampire. It really works!

“Detox baths,” in which the patient bathes in hot water with epsom salt, baking soda, ginger, essential oils, apple cider vinegar, and/or ground mustard seed are another cold treatment popular with wellness bloggers.

"Detox" is one of those words that, if you use it unironically, is extremely effective at keeping me away.

These baths supposedly conduct their magic by “opening the pores” and “ridding the body of toxins we pick up from pollution and processed foods.”

Sniff... sniff... ah, the familiar scent of bovine excrement.

So what can you do if there’s a cold going around and you don’t want to catch it, or if you already have one and just want the misery to end?

Stay home and drink booze?

Sadly, not much.

Damn.

The most important thing for prevention is practicing good hygiene, like washing your hands frequently, cleaning surfaces in your house, not touching your face, and “not being in the face of people who have colds themselves, like Uncle Edward or Aunt Freda who want to hug and kiss you,” said Freed.

A year before Trump Mumps, folks.

Some over-the-counter medications, like antihistamines, decongestants, or painkillers like Ibuprofen or Aspirin, can temporarily relieve symptoms...

Ibuprofen is my go-to pain reliever. However, I have never yet found an antihistamine or decongestant that does a goddamn thing for me. Well. Some of them make me feel woozy on top of whatever cold I have at the time, but they don't relieve sinus problems in the slightest. And yes, I've had prescription ones.

One inconclusive (but promising and honestly delightful) treatment method is eating chicken soup.

See? My people were right all along! Just not about bacon.

Until the common cold is eradicated, or at least until we have more, better research on the subject, the best thing to do is wash your hands and hope no one sneezes on you.

Or in your chicken soup.

In spite of these findings, which have been apparent for some time, people continue to waste money on nostrums and snake oil in hopes that, I don't know, something will work. And there's always the person who swears by one treatment or another. "It works for me!" Well, there are several possible reasons for that, including a) placebo effect; b) you're getting better anyway and it just happens to be right after you swallow some pill or other; or, I'm willing to admit, c) you're weird and the thing actually works. Problem with (c) is that unless it passes scientific testing, it's clear that it's not going to work for everyone or even most people.

I'll finish by noting that back in college, I participated in a study on zinc as a potential cold remedy. Participating in studies in college is a time-honored tradition, and kept students in beer money back when more undergrads could drink beer. Of course, I still don't know if I was in the control group or not. Point is, this was in the mid-80s, and here it is over 30 years later and it seems they're still arguing about zinc. Which tells me that it's pretty damn worthless, or there would have been something conclusive by now.

About that Merit Badge, though, it's time for another

Merit Badge Mini-Contest!

We'll make this one easy. Comment below with what you do to prevent and/or treat a cold when you get one (or brag a lot if you never get colds). I'll pick one of the relevant responses at random and give the commenter a Merit Badge tomorrow. As usual, you have until midnight WDC time tonight, Wednesday.

And just to be fair, I'll tell you here what I do: I buy three or four boxes of lotion-infused tissues, take Advil for the headache, drink tea, sit up in bed and blow my nose every 5-10 seconds. Sometimes, rum is involved -- it doesn't make the cold better, but it helps me give less of a damn about it. And then I bitch a lot about how fucking miserable I am to whoever will listen, and maybe a few people who won't.

Why no chicken soup? I can't be arsed to make it when I'm sick (even if it involves nothing more than opening a can of Campbell's), and besides, it couldn't possibly be as good as my mom's -- it was the one thing she could cook well.

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