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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QPdoll

about 15 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 18 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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August 5, 2023 at 9:33am

#1053674

Not What You'd Expect

You're going to die. That (along with taxes) is one of the great certainties of life.

5 Paradoxes About How You’ll Die

Cracked is here to make jokes about it.

If you think too much about your own mortality, you will wind up on the floor, spinning on your back and screaming at the sky.

Floor, yeah. Spinning? It's the room doing that.

You should live forever; anything else is unfair.

Until you realize that "forever" is a null concept, and you'll still expire from energy depletion at some point before the heat death of the universe but long after you began suffering billions of years of pure boredom.

Side note: Nothing we’re going to talk about today is a true paradox.

I'm really glad they pointed this out, so that I don't have to.

In fact, there are no true paradoxes, anywhere. Paradoxes cannot exist. Anytime someone comes up with a paradox, it’s either a trick of language or they invented imaginary, contradictory premises and are now acting surprised that they contradict each other.

Which is what I've been saying, but no one listens to me.

5. If You’re Drunk, You’re Safer Driving Home Than Walking

Drunk driving laws aren't about keeping you safe.

The article does a fine job explaining the basis of this conclusion, which, again, involves no paradox; it's merely counter to common sense. Because common sense is generally bullshit, which is why we have science. In any case, a few moments' thought should be enough to realize that, whether you're drunk or not, driving should be safer than walking (assuming walking along streets and roads); when driving, we're at least protected in a metal cage.

Especially when the person in that cage is drunk.

In reality, you don’t have to choose between driving drunk and walking drunk. The easiest and safest option is to just get a ride, with someone else driving.

Even easier: drink at home.

4. Those Who Drive Slightly Above the Average Speed Get in Fewer Accidents

This earned another "duh" from me, but just because I expected it doesn't mean there was data to support it.

Here’s a helpful chart that notes you’ll be perfectly safe if everyone's stationary, but things get a little riskier from that point on.

There is no such thing as "perfectly safe." (You'll have to go to the article to see the chart.)

One landmark study from 1964 tried to find the relationship between a car’s speed relative to others and the chance of getting into a collision. This look at 10,000 crashes determined that, yes, if you drive faster than the median speed, that raises your chance of a collision.

Oh. That explains minimum speed limits that exist on some roads. Two semesters of traffic engineering classes, and I never did get a good justification for that, or how many tickets cops gleefully write when there's a jam and everyone's below the minimum speed limit.

For decades, attempts to replicate the study produced similar results. More recent studies haven’t, so the latest advice is to err on the side of going too slow if you can’t keep pace with everyone else.

You know what really burns my butt from both a driving and traffic engineering perspective? It's when they give trucks a lower speed limit on the same stretch of road. That never made sense to me.

In any case, automobile (and to a lesser extent, road) technology has advanced in the last 60 years since that study, so it's not too surprising that it hasn't been replicated recently.

3. People Who Take Multivitamins Die Sooner

You know who else supposedly dies sooner? People who eat only raw fruits and vegetables. Well... allegedly. I was trying to find a link to the story about the influenza who promoted such a diet and croaked at the age of 30, but all I found were tabloids. Not linking those here. Oh, well; I knew it was too good to be true.

Unless you’re vitamin-deficient, popping vitamin supplements will offer you no benefit whatsoever.

No, but it does benefit the chemical engineers who make the vitamin supplements. Won't anyone think of their well-being?

Nutrition scientists found themselves with a lot of vitamin data to examine thanks to the Iowa Women's Health Study, which surveyed some 40,000 women as they aged through 20 years. Comparing people who took vitamin and mineral supplements to those who didn’t, the researchers found that the supplemented humans were 6 percent more likely to die each year than their unenhanced counterparts.

Ugh. "Nutrition scientists:" I don't trust that lot. "Women:" While it's good that there's more focus on women's health, it's not exactly an exhaustive cross-section of society. About the only positive thing about that study is the sample size.

This is why it's important to not succumb to confirmation bias.

Also, consider this possibility (in addition to the caveats in the linked article): When you do one thing that's "good," it's possible that you slack off on other things that are "good." Like maybe you think it's enough to eat a balanced, low-calorie diet, and don't bother with exercise, or vice-versa. Or, like me, you figure that since you didn't reproduce, you're free to emit as much carbon into the burning atmosphere as you like.

So "I'm taking vitamins, so I can slack off on other health habits" might very well be a thing. I don't know.

2. A Century Ago, Rich Women Died Giving Birth More Often Than Poor Women

This one's interesting enough, but I don't have much to say about it. This final one, though, that's what really lit up all the LEDs on my confirmation bias:

1. The Stockdale Paradox: Optimism Kills

Vice Admiral James Stockdale spent over seven years as a prisoner of war in Vietnam.

You already know that situation is what's commonly known as "fucked."

For his own amusement, he left fake notes for his captors to find, and to further pass the time, he did logarithmic math by writing in the dust.

Conclusion: math can keep you alive and relatively sane.

Years later, he was asked about the hundred or so American POWs in the Hỏa Lò Prison who died before the prisoners were released in 1973. Stockdale said the ones who didn’t make it were “the optimists.”

As I've been saying: expect the worst, and you can only be pleasantly surprised.

This is the least scientific of the phenomena we’ve talked about today. But we wanted to leave you with it because we began this post by saying that reflecting on your mortality will leave you delirious and feeble. That’s not really true.

It's also, again, not a paradox. It's a perfectly reasonable conclusion. No, it's not scientific. Designing a study to back this up would itself be a war crime. Even interviewing other former POWs is ethically questionable, as it brings up trauma. Still, I'll continue my pessimist ways, to wit: we're all going to die.

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