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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Write_Mikey_Write!

about 5 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 7 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.

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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.

April 29, 2024 at 9:25am

#1070095

Supplementary

Snake oil gonna snake.

A natural deception: 3 marketing myths the supplement industry wants you to swallow

Though snake oil doesn't deserve all of its reputation for fakery. It had a use in traditional medicine

(whether it was actually efficacious, I don't know). The problem was, hucksters started promoting it as a magical cure-all, so now "snake oil salesman" is a synonym for fraudster.

The business of supplements is booming, and with all the hype around them, it’s easy to forget what they actually are: substances that can powerfully affect the body and your health, yet aren’t regulated like drugs are. They’re regulated more like food.

Okay, but let's be clear, here: food can "powerfully affect the body and your health." The line between food and drug isn't always sharp. Hence you get a long string of "superfood" fads.

It’s important to consider why so many people believe supplements can help them lead a healthier life. While there are many reasons, how supplements are marketed is undeniably an important one. In my years following the industry, I’ve found that three mistaken assumptions appear over and over in supplement marketing.

There follows a discussion of those "three mistaken assumptions" (I'd have phrased that differently, using words like misleading, false, or lie, but I'm not getting paid to mince words to avoid lawsuits).

1. The appeal to nature fallacy

The appeal to nature fallacy occurs when you assume that because something is “natural” it must be good. The word natural is used a lot in the marketing of supplements.

Leaving aside for the moment the inherent ambiguity of "natural," I've long wondered how they can take, say, a plant, extract its juices, distill out the desired chemicals (everything is chemicals), stuff it into a pill, and still call it natural.

To be clear, “natural” does not equate to “better,” but that’s what the marketing wants you to think.

Remember, poison ivy is natural. So are those mushrooms that'll kill you horribly and painfully. And the pollen which, even though I'm not technically allergic, is playing havoc with my sinuses right now (and for which I'll happily take a manufactured antihistamine).

2. The belief that more of a good thing is always better

There’s another assumption that piggybacks on the appeal to nature fallacy: If something is natural, it must be good, and more of it must also always be better.

This seems to be pervasive in human psychology, for whatever reason. I have to guard against it, myself, even though I know it's not the case. Not just with food, but activities as well. You can overdose on anything, even water.

3. The action bias

Finally, the supplement industry likes to capitalize on the idea that doing something is better than doing nothing. This is the action bias. Taking action makes people feel like they have more control of a situation, which is especially powerful when it comes to health.

This one's pervasive, too. It doesn't help that, in some cases, it seems to be true. Getting five minutes of exercise a day is said to be better than getting no exercise at all, though again, more is better... up to a point.

But in this particular case, taking a supplement might not be better than not taking one. Okay, that's a convoluted sentence with a lot of negations. Given the risks, it might be best to avoid supplements. As with most things, I feel like a person should consult with their doctor as to whether a supplement is useful or necessary, and safe. Doctors don't know everything, either, but they're more likely to have the relevant information, based on your own health profile, than good old Doctor Google.

On the flip side, if very few people bought these supplements, they wouldn't continue to be manufactured. This might have adverse effects on the people who, for whatever reason, do have to take supplements. It's like how I get snarky about "gluten-free," but the fad helped make available more products that people with legitimate gluten allergies can eat, improving their quality of life.

So, if you're a supplement fanatic, maybe rest easy knowing that you're making life better for other people... if not necessarily yourself.

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