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Friday, Jun. 28, 2024

10:46am EDT

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Adherennium Dr of Phoolishness

about 23 hours ago
In response to "Wave Rider"

Horrifying fact, I once painted a reproduction of the Hokusai wave on our bathroom wall. Eventually of course it was painted over, but I like to fantasise that in some distant future it will be archaeologically excavated and held up as an example of early cultural appropriation, (badly executed too.)

1 Like

Annette

a day ago
In response to "The Oz Hypothesis"

The nekked wimmin look like something that I might have drawn in first grade.

1 Like

Aiva Raine

Jun 25, 2024 at 10:16pm
In response to "Can't Spell Success without Sucs"

Ok- this made me truly LOL- "sipping a gin and tonic made with low-paid laborer tears"

Thank you. Totally agree with the rest of your article.

Money doesn't solve all problems but it sure as hell eliminates many and makes it easier to go places or buy things that can create some happiness. IE- a property somewhere quiet and peaceful and a trailer with which to chill on said property.

Rich people are typically clueless assholes.

1 Like

Anni Pon

Jun 25, 2024 at 9:30am
In response to "Can't Spell Success without Sucs"

I've never heard anyone say, "That guy is so successful, but he doesn't really make much money." Or anything similar. Sure, you can have a positive impact on the world without making lots of money, but people won't call it success.

2 Like

Annette

Jun 23, 2024 at 1:28pm
In response to "Revisited: "Experience""

This is how I coped:

1 Like

Elisa: Middle Aged Stik

Jun 23, 2024 at 9:51am
In response to "Revisited: "Experience""

Not gonna lie I kinda miss Zoom parties, at least for this community since people on WDC are so spread out. Meeting in person is pretty tricky and/or expensive. It's certainly more socializing than I've been doing regularly since, oh, 2022 or so.

As for server migration, well, I work in the IT world. I've been through that. And honestly, server migration is a bit nerve wracking, as some stuff can go missing. It's still more manageable than a complete redesign. Complete redesigns are awful. You can test in different environments that the public can't see till kingdom come, but things always go wrong when the public version is rolled out. A local grocery store recently redid their website, and the old site held thousands of recipes. When the new site was unveiled, none of those recipes made it over. I ended up reaching out on Messenger and was able to get a copy of a recipe I use somewhat often, but I'm kind of annoyed that the text recipes are currently MIA because somehow that data didn't transfer with the site rebuild. Absolute madness.

1 Like

Wordsmitty ✍️

Jun 22, 2024 at 5:56pm
In response to "Discomfort Zone"

Just so everyone is clear:

- There are credit cards and debit cards and even cash cards. They are treated differently by different places as some prefer debit and cash over credit (which usually chrages the merchant more).

- As Annette mentioned, using text and wi-fi is best and usually cheap or free included most US phone plans. However, 'data' outside N. America is not. If you want to wander streets and go hiking and check maps or GPS in Europe, make sure you have a data plan for Europe. That's where the thousands of dollar charges come in if you don't have that. Voice also is usually extra. So if you plan on a lot of talking, get a plan for that, too. (As I recall without an additional plan for Europe, standard charges were 25 or 50 cents for each text and and a dollar minute for voice.)

- On phones, you may have to turn on or unlock the ability to use European system as it communicates differently than the one used in the US. T-Mobile does have good info explanaing that and data plans once you find it on their site. I think there was a search to input your phone make & model to show you what will make it work outside N. America.

1 Like

Elisa: Middle Aged Stik

Jun 22, 2024 at 5:08pm
In response to "Discomfort Zone"

Funny thing is, everyone mentions day to day spending in cash, but I don't recall seeing a single person in the Netherlands use cash in the grocery store. I was in a couple different grocery stores and a convenience store outside normal tourist areas, and cards were preferred. Even in museums there were readers to get donations from cards instead of having boxes out to drop off cash donations. This may be less of a thing in smaller communities, but I do feel like some of the larger (non-tourist) communities are going cash free. The pandemic's legacy of contactless payments sure has been a weird one.

1 Like

Annette

Jun 22, 2024 at 4:51pm
In response to "Discomfort Zone"

I think that SIM card thing is over now. Check it out though. I didn't change my SIM card and I my bill was the exact same as always. I did only use texting and wi-fi though.

1 Like

Robert Waltz

Jun 22, 2024 at 4:46pm
In response to "Discomfort Zone"

Annette

, merci, tu as raison.

I have a pretty recent Samsung phone, and T-Mobile, but I seem to remember something about a different SIM card or activation or something. Easy enough to find out. I know there were horror stories for a while about people traveling and getting hit by surprise roaming fees in the thousands of dollars, so that would be something to avoid.

Thinking about going this autumn, because like you said, it's not summer. Also, hopefully fewer free-range tourist kids. One of the many great things about being me is that I have some flexibility, so I don't yet know exactly when.

1 Like

June 24, 2024 at 10:38am

#1073116

Sadistics

Thank you to everyone who commented here on Saturday about my travel plans. It clarified a few things for me.

Today, I'm going to discuss an article about numbers: this 2018 Ars Technica discussion of statistics. Trigger warning: math ahead.

“Fixed mindsets” might be why we don’t understand statistics
Study finds people prefer complicated methods because that's what they're used to.

Just reading the subhead there triggered my skepticism. People prefer complicated methods? Not in my experience, piped up my inner skeptic. People like things simple.

And then I remembered that we're reading and writing in English, with all of its confusing and complicated grammar and spelling rules, and how every attempt to simplify the language has met with indignant resistance.

Sure, I'm used to it. I've spent a lifetime trying to improve my command of the language. If we suddenly decided that everything should be spelled consistently, I'd have to learn a whole new set of rules, and even if those rules are simpler, it's more work. Besides, shouldn't everyone else have to put in the same work I did to learn something?

Anyway, that has nothing to do with statistics (well, not directly). It was the first analogy that popped into my head.

A new study in Frontiers in Psychology examined why people struggle so much to solve statistical problems, particularly why we show a marked preference for complicated solutions over simpler, more intuitive ones. Chalk it up to our resistance to change. The study concluded that fixed mindsets are to blame: we tend to stick with the familiar methods we learned in school, blinding us to the existence of a simpler solution.

As I noted above, it wouldn't be a "new study" anymore. I mostly mention the time frame because the article's publication was before the entire world got hit by a working example of statistics.

Now, I know I keep harping on this, but one study isn't sufficient to establish fact. I have no idea if any follow-ups have been done in the 5+ years since this article was published. Fortunately, this is handled by weasel words like "might be."

Roughly 96 percent of the general population struggles with solving problems relating to statistics and probability.

67.56% of statistics are made up on the spot.

Okay, no, I just made up that number, but if you're going to assert something like "96% of the general population" does anything, shouldn't you back that up with some sort of citation?

Assuming it's true, though, would it surprise you to know that I'm one of the 96%? I necessarily encountered statistics and probability in my education, but that doesn't mean I don't "struggle." I'm not an expert, and it doesn't come naturally to me. It's one reason I like blackjack at casinos: it exercises that part of my brain. Sure, it can be an expensive education, but so is college.

I think the bigger problem is that many people don't even try.

Recent studies have shown that performance rates on many statistical tasks increased from four percent to 24 percent when the problems were presented using the natural frequency format.

The "natural frequency" thing is explained fully in the article (basically, saying something like 1 in 4 instead of giving it a 25% probability, the two of which are mathematically equivalent). But I had to chuckle at the way they phrased this... using percentages.

What I finally figured out they meant by "four percent to 24 percent" was: the 4% is the complement of the 96% above, and "natural frequency format" lowered the "struggle" rate from 96% to 76%. But the way that was presented was somewhat ambiguous.

I encounter this sort of thing sometimes in video games. Say you have a 30% chance of scoring a critical hit on an opponent. You find some item whose description includes "raises the crit hit chance by 10%." Now: does that mean it's additive, raising the crit hit chance to 40%? Or does it increase your chance of scoring a critical hit by 10% of the current probability, raising it to (30+0.1*30=) 33%? This matters because the latter might not be enough of an edge to justify the opportunity cost of using that item rather than, say, one that makes you prettier.

And these games are generally designed, at least in part, by nerds who oughta know better than to be that ambiguous about numbers.

The article goes on to describe a Baysean probability problem (in brief, Bayes' Theorem takes into account prior knowledge), and the thing that amused me there is that at the end of the section, which involves a bit of math, there's an editorial comment in italics and brackets: [corrected].

The implication is that even the author got it wrong. Which I'm not going to snark on them for, because I get it wrong sometimes, too (there's one particularly egregious math error in an old blog entry that I'm really hoping doesn't come up in one of my Revisited posts, because I'd then be honor-bound to publicly kick myself for getting it wrong). But if you're specifically writing an article about how people get statistics wrong, and in it you get the statistics wrong, well, that's funny.

The students had to show their work, so it would be easier to follow their thought processes. Weber and his colleagues were surprised to find that even when presented with problems in the natural frequency format, half the participants didn't use the simpler method to solve them. Rather, they "translated" the problem into the more challenging probability format with all the extra steps, because it was the more familiar approach.

A while back, I encountered an article about second-order polynomials. The quadratic equation is traditionally taught as a rigorous method for finding their solutions. In my experience, I had to memorize the quadratic equation long before anyone really explained to me why it worked, which I find to be essential for my true understanding of something. The QE is somewhat complicated, involving square roots, addition, subtraction, multiplication, and division; not to mention that it's got that pesky "plus or minus" sign, because the square root of any positive number can be either positive or negative (sqrt(4), for example, can be 2 or -2).

Anyway, this article presented a faster, simpler method for deconstructing polynomials. As I recall, it's indeed simpler. And yet, on the rare occasion these days when I want to think about this sort of thing, I still go to my hard-fought memorization of the QE.

So I can believe the conclusion reached in the study.

But that just means I have to be more skeptical of it.

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