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 20 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 22 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.

February 16, 2020 at 12:11am

#975619

Critical

Today, we learn "how to spot a bogus health fad." Sadly, "you see it on the internet" is not a complete answer, nor necessarily a factual one. Sigh. That would make life a lot easier for critical thinkers.

https://www.self.com/story/spot-bogus-health-claim

8 Simple Ways to Spot a Bogus Health Fad
Opinion: The plural of anecdote is not "data."

By Yvette d'Entremont

Hey, that looks French. I think it means "Yvette d'Entremont."

In 2018 the global weight-management market was valued at $212 billion, and the wellness industry was worth a staggering $4.2 trillion in 2017.

I have this vision of enterprising capitalists selling junk food to us with one hand and gym memberships with the other. Good capitalism. Bad ethics. But it might explain why there's a Krispy Kreme right across from my gym.

You may be wondering: In an internet landscape offering cures of suspect origins for all that may be ailing you, is it even possible to sort out what’s real from what will inevitably just become a pile of dust-gathering items in your collection of dubious supplements and accessories?

Surprisingly, yes.

It's called "science." Failing that, not jumping on every promise that some random influencer makes would go a long way.

1. Does it promise too much?

Ever seen a product or a diet that claims it can do everything? It won’t just make you lose a few pounds, but it’s going to clear up your acne, make your nipples perkier, and earn you thousands of dollars a week working from home in your Rainbow Cat Unicorn Underoos.

Now I'm going to buy stock in the company that makes Rainbow Cat Unicorn Underoos. They can't fail.

This crap is age-old. The term "snake oil" comes from the products hucksters would sell from their carriages, all of which promised everything.

2. Does it blame all your health woes on a single thing?

I had a friend who's a chiropractor, and she was absolutely convinced that subluxations were the cause of all disease.

3. Is the evidence confined to anecdotes?

What this section leaves out is the nonequivalence of correlation and causation. Most diseases either a) eventually kill you or b) get better. If the disease was going to do (b) anyway, it doesn't matter how many vitamin pills you took. And if it was going to do (a) anyway, then you're not around to push the "cure." See also: Steve Jobs.

4. Who’s promoting it?

Don’t take this as a hard-and-fast rule that something isn’t reputable if it’s being promoted on Instagram.

Nevertheless, if it's being promoted on any sort of social media, I consider it immediately suspect until proven otherwise.

5. Which scientific or medical boards or bodies have evaluated it?

You know those labels that let you know, give or take, “these claims haven’t been evaluated by real doctors or scientists, may contain up to 30% unicorn dander”?

I'm the first to admit that science doesn't know everything. Hell, that's the entire foundation of science: "we don't know everything, but we'll find out what we can." So not everything has been tested - nor can it legitimately be. It's possible that your grandmother's herbal remedy for whatever actually works. As we all know, aspirin started out as a natural herbal (sort of) remedy. Then it comes down to "if I use this method, am I closing off other avenues of possible relief?"

6. Are you drawn to it for reasons other than its efficacy? Like, say, desperation to find a solution?

I feel for people with no recourse. Having lived through the whole Laetrile debacle (feel free to Google it if you're too young or don't remember), it's clear to me that desperate people will hare around looking for desperate measures. But sometimes, as they say, the cure is worse than the disease - though I'd be hard-pressed to think of anything worse than cancer. I understand that if you're actually dying, you'll grasp at whatever straws life, or snake oil salesmen, throw your way. The problem is that these things often substitute for actual care that could actually work.

7. Is there a way to measure evidence that the product works?

If something is claiming to tinker with your health, assess the study quality (if there even was a study), and don’t let yourself be the lab rat for a product that’s probably just going to be a wallet cleanse.

"Wallet cleanse" is what I'm going to call all claims of "cleanses" from now on.

8. And if it seems like bullshit but you just can’t put your finger on it, ask around.

I like to think I have a pretty good bullshit detector, but I could be deluding myself. We've gotten to where we put equal stock in what some Internet huckster says as we do in what doctors say, or even get swayed more by the former. Yes, doctors are people and people are fallible, but so are "alternative medicine" promoters. Me, I don't always follow a doctor's advice, but that doesn't mean I don't respect it; it just means that I think the doctor doesn't understand my basic philosophy: that there's no point in living longer if you have to give up the things that make life worth living.

So, don't believe everything you hear. More importantly, don't believe everything you think.

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