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:

Elycia Lee ☮

about 5 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 18 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 21 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 23 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

March 21, 2020 at 12:03am

#978676

'Murica!

This one has been hanging out in my queue for quite some time. Sure, it's Cracked, but it's David Wong from when Wong was still with Cracked.

https://www.cracked.com/blog/20-facts-that-will-make-you-understand-america-or-n...

20 Facts That Will Make You Understand America (Or Not)

There's an old joke: What's the difference between America and yogurt? Give it enough time, and yogurt develops a culture.

Also, I'd dispute the word "facts" in the headline.

20. 4 In 10 Americans Would Save The Life Of Their Dog Over A Foreign Tourist

Applied Trolley Problem.

To be fair, I'd save my cat over... well, anyone.

I mean, I'd probably save the tourist myself, but I wouldn't tell the pollster that on the phone. It'd make things too awkward between me and my dog if she overheard.

"Messing with pollsters" is a time-honored American tradition. Still not culture.

19. Most Americans Think Political Correctness Is A Problem

I'm just going to leave this part here without comment:

What annoys me is that usually we only aim the "PC" complaint in one direction. "I'm offended by the misuse of gender pronouns" is mocked as being "PC," but "I'm offended because you didn't stand for the national anthem" is not, somehow. I'd love to see the poll results when people are given both of those examples before answering. Actually, I probably would not love those results at all.

18. Empathy Among Young People Has Fallen 40%

Yeah, but fuck young people and their ilk.

As the author of the study explains, empathy for strangers has been replaced with intense loyalty to one's own group (which in turn creates hatred for strangers). But you already know that if you have ever used the internet, or even just walked past someone who has.

17. Churchgoing Americans Are More Likely To Support Torture

Well? Their whole theology depends on people being tortured for all eternity if they have sex with the "wrong" person, and by "wrong" I don't mean "underage."

"Hold on," you might be saying, "isn't these Evangelicals' entire religion based on a Middle Eastern man being wrongly tortured to death by the state?"

16. Nearly 8 In 10 Say American Morality Is Getting Worse

Well, duh. See #17.

15. About 40% Support Discouraging The "Wrong" People From Having Kids

I try to discourage everyone from having kids. But not very adamantly.

14. 33% Still Say Homosexuality Is Immoral

Again, see #17.

13. Only 43% Say Pornography Is Morally OK

The US was founded by Puritans and still maintains a lot of their values, if only in an undercurrent. This means our attitudes about sex tend to be whacked in general.

12. Most Think Men And Women Have Equally Difficult Lives

My only issue with this, really, is that it overgeneralizes, like saying something like "men tend to have more upper-body strength than women." Technically true on average, but we're talking about two bell curves that overlap; you can always find exceptions. And since most Americans can't grasp the subtleties of probability theory, making claims like this is problematic. Show me someone with a difficult life, and I can show you someone of a different gender with a more difficult one. Unless of course the one person you show me is the one out of 330 million of us who happens to have The Most Difficult Life.

Surprisingly, the difference in responses between men and women isn't nearly as dramatic as the difference between political parties. Sorry, I realize now that's actually not surprising at all.

11. We Tend To Only Befriend People Exactly Like Ourselves

I don't find this shocking at all. And Wong should know better; he's the one who elucidated the whole Monkeysphere concept. People always gravitate toward people with more stuff in common.

The issue is that many people don't even try to branch out.

10. 11% Of Americans Have Never Left The State They Were Born In

As Wong points out here, it's harder to travel if you're poor.

9. Around 10% Are Anti-Vaxxers

While that number is appallingly large, and should be 0% (okay, I'll adjust that to 1% to account for stubbornness), it's much less than my internet browsing leads me to believe.

I bet it goes down soon, though. "I'm against vaccination." "A shitload of people have COVID-19." "When will the vaccine be ready?"

8. 90% Of Americans Believe In God

... or some kind of higher spiritual power

Yeah, that last bit is kinda important.

This one, together with the last one (and a few others on this list) illustrate the basic problem with democracy: the "majority" is not always right, and public policy can seem contradictory as a result.

7. 1 In 4 Haven't Read A Book In The Last Year

There are still panicked articles insisting that Americans are reading less than ever, with some data to back them up, but even they seem to dismiss the idea that "browsing the internet" can count as "reading." I mean, look how much smarter this article is making you.

I'm pretty sure we don't have that problem here.

6. 65% Believe They Are Of Above-Average Intelligence

We also believe we're above-average drivers.

5. The Number Of Vegetarians/Vegans Is Tiny, And Hasn't Grown In Decades

I'm okay with this.

But this is why you'll continually hear "They're coming for your hamburgers!" as a talking point against environmentalists.

You know, I actually kinda like veggie burgers. And great strides have been made toward vat-grown, cruelty-free meat. I'd switch to that in a clogged heartbeat.

4. Americans Spend Less Time Cooking And Eating Than Almost Anyone

Don't get me started on The Causes of Obesity. But cooking and eating are, in general, big time-sucks. I have more interesting things to do than cook, and I like to do other stuff (like read or watch shows) while I eat.

3. Nearly 80% Of Americans Pee In The Shower

Okay, look, confession time: I can't say I haven't done it. But I certainly don't make a habit of it.

Also, 20% of us bring a drink into the shower. Or at least, that's how many of us thought it would be funny to tell that to a pollster.

There's a local brewery whose flagship product is called Shower Beer. It's a light Pilsener. I don't like it, myself, and I can't say I've ever brought anything into the shower with me except soap and a razor (hey, shaving in the sink is a pain in the ass).

2. 43% Have Assigned Seating In Their Living Room

I don't, but I live with a housemate who has basically taken over the living room. This is fine; I have my own space, and I prefer to watch things on my laptop, which could sit anywhere.

1. 1 In 5 Are In Chronic Pain

And how many are addicted to painkillers? And how much of that addiction is actually pain-avoidance? I don't know; just throwing it out there.

One out of 12 people you pass on the street are in absolute agony.

That's something to keep in mind the next time you run into a rude driver, or a raging customer, or an antagonistic person on the internet. Pain messes with your mind, drains your energy, ruins your concentration.

I do try to be aware that someone else might be having a bad day, week, year, life. Sometimes I fail at that.

Society is, at the end of the day, just a big pile of other people's coping mechanisms. I guess we should try to remember that.

Yeah, especially now.

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