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 6 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.

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Adherennium Plotting something

about 8 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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

July 23, 2021 at 12:03am

#1014161

Numbers, Not Even Complex Ones

Warning: I've been drinking. But I still wanted to do another prompt from "JAFBG"

[XGC].

There's a difference between ignorance and willful ignorance. What reality do you wish people would face?

Ignorance is our natural state.

It is also our natural state to combat this ignorance by learning shit.

People who refuse to learn are therefore fighting against nature.

As we are fallible minds, the only effective way to approach reality is through systematic trial and error. Making judgements based on singular events leads to misconceptions. It's like... say you've just learned how to ride a bicycle. You're out on the street enjoying this newfound skill, when someone in a car cuts a corner and knocks you off your bike. That's your one and only data point, so you stop there. You might decide, all bloody and bruised and shit, "Fuck this bike-riding noise; I'm going to stay in a car where it's safe."

But that ignores the large number of people who ride bikes without ever being hit by a car, or the traffic accident statistics. Your experience is an outlier, but you don't know that, because you're living it and it's the only experience you know. Sure, cycling has its dangers... but so does driving a car, or sitting at home, or doing literally anything. Yes, some things are more hazardous than others, but again, if your only data point is your own experience, you'll end up with a warped view of reality. You might spend years taking showers every day, secure in the knowledge that it's perfectly safe, until one day you slip in the tub and crack your head open. Surprise, it wasn't perfectly safe after all.

At first glance, everything is binary. Either you get hit by a car, or you don't. Either you contract a dreaded disease, or you don't. Either you die in the shower, or you don't. Either the jump off the bridge kills you, or it doesn't.

But life doesn't work like that. There are always outliers, but by definition, they're not representative. To use a current hot-button issue, sure, some people die after getting the covfefe vaccine. Statistically, this is inevitable, just as it's inevitable that some people die from being in a car or an airplane. This ignores that more people die from NOT getting the vaccine, or that the death might not have had the vaccine as proximal cause, or that there are far, far worse outcomes from illness than mere death. There's always nuance, and it often contradicts one's lived experience. This is why people like me rail against "anecdata," the stories people tell that inevitably go against the majority experience. (I'm aware that my own situation is similar, which is why I'm going to suck it up and get the cataract surgery; the numbers are on my side on this.)

Vaccines (ALL vaccines), by the way, are real-life trolley problems. Can we ethically justify taking actions that will have negative effects on a small minority, in order to avoid negative effects on a larger population?

My answer is yes. Qualified, depending on the actual numbers involved.

It's a similar situation with self-driving cars. Right now, the technology is in its infancy, I'll grant. But also right now, every year in the US, about 35,000 people die in auto accidents. Many more are injured. That's obviously not an exact number, but whatever; the exact number is irrelevant to the point I'm trying to make, which is that we have decided, as a society, that the benefits of being able to drive cars and trucks outweigh the costs in life, property, and injury from driving cars and trucks. I'm not contradicting this. But what if we come up with a method to decrease that fatality statistic by 1,000? 3,000? 10,000? What is the threshold number that will make us, as a society, say "You know what? This is worth it to do?"

To hear some people talk, any major change such as turning over control to an algorithm would require a reduction in transportation fatalities to zero -- which is a completely unreasonable number. Sure, it would be nice, but wouldn't it also be nice to, say, just reduce it by one standard deviation? To have, instead of 35,000 fatalities in a year, 31,500?

Yes, I know, these numbers are really only meaningful as compared to total population in a particular country or whatever. I'm just throwing out figures here to make an argument, not to be absolutist.

The perfect cannot be the enemy of the good.

When I started writing this, I thought my answer to "What reality do you wish people would face?" would be "That science is imperfect but it is still the best means we've ever come up with to understand our universe." But I think I want to be a little more focused than that. Not just science, but statistics and probability.

But that would require people being comfortable with math, and also with accepting the idea that their lived experience may not be representative, and I don't see that happening anytime soon.

Yes, I could have just seen it on Netflix. But then I wouldn't have gotten the theater experience, or the drafthouse experience.

One-sentence moviecomedy special review: Bo Burnham: Inside

Funny and relatable as all hell, this guy's genius is all over the place, and it's totally worth seeing in the theater or on streaming -- but some comedy is best shared with a large group of like-minded strangers.

Rating: 4/5

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