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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:

tj: Tenebrous Jekyll

about 7 hours ago
In response to "Phases"

Perhaps, if the full moon drives one to madness, the half moon will only make one half-crazy.

1 Like

Battywyn🎶

Oct 29, 2024 at 12:18pm
In response to "The Numbers Don't Lie"

I hear ya. I like to think I'm discerning enough to detect fakes, but that might just be my ego talking. When it comes to local businesses, I also go to social media and request recommendations. With both reviews from strangers and recommendations from friends, I give more credence when more (anecdotal, heh) details are provided. It's more an art than science, but liars either have such flawless details that their story sounds fabricated, or they get embarrassed when they trip themselves up. Truthtellers make errors, because memories are fallible, but they get more confused than embarrassed. Ultimately, the only way to really know is to try the product/service yourself. Which is why you'll hear about my Cancun vacation next month, but you don't have to believe me.

1 Like

Robert Waltz

Oct 29, 2024 at 11:07am
In response to "The Numbers Don't Lie"

Battywyn🎶

- I, too, look at the 1-star reviews, though again, reviews aren't my sole criteria. But sometimes it's blatant fake reviewing from the competition: "Don't stay here! It has rats, bedbugs, and roaches! Stay at this other place instead!" where "this other place" is clearly owned by the reviewer.

On the other paw, five-star reviews are often salted by the establishment in question.

Used to be, you could look at the four- and two- star reviews to get more honest, nuanced opinions, but the establishments, and their competition, have figured out that trick.

Which is one reason why reviews aren't my sole criteria.

(None of this applies to WDC reviews, or to the artificially inflated Uber rating system)

Battywyn🎶

Oct 29, 2024 at 11:00am
In response to "The Numbers Don't Lie"

And I was going to conclude by saying, when evaluating Caribbean vacation destinations, even if 5,000 people gave it a 4.8 rating, if the one star reviews were from seven people who encountered rats in their rooms, I'm very likely to not choose this destination.

2 Like

Battywyn🎶

Oct 29, 2024 at 10:58am
In response to "The Numbers Don't Lie"

Well, that was annoying. I was still in the process of editing, so please excuse the multitude of errors. I also intended to italicize the quotes from your post. Stupid f****** Parkinson's.

1 Like

Battywyn🎶

Oct 29, 2024 at 10:56am
In response to "The Numbers Don't Lie"

You know what sometimes lies? Anecdotes.

Also, dogs, I've heard.

Oh, hell no.

🤣 I love a good inside joke, but my marketing guru would tell you to cross link here

The fallacy is named after Robert McNamara, the U.S. Secretary of Defense during the Vietnam War, where his over-reliance on measurable data led to several misguided strategies

Bet he's proud of his claim to fame.

What's the point in data, then?

The point in data is a decimal separator, unless here in non-english speaking countries, in which case it's more likely a thousands separator.

So, whenever I'm evaluating online reviews for pretty much anything, and for the record I check them for almost everything - I always consider three things: 1. The average rating (eg, number of stars); 2. The The number of contributors to the rating; and three. What the one star ratings say.

1 Like

Annette

Oct 28, 2024 at 2:47pm
In response to "Waste Not"

In Southern California, every single family residence has a green bin. Every trash day, three different trucks make the rounds.

1 for composting
1 for recycling
1 for garbage

Trader Joe's sells almost exclusively packaged foods, even perishables. They have contracts with college food pantries. In many colleges, there are now pantries that are open to the students 4 or 5 days a week plus 1 day when people from the community can drive up and receive a bag with a mix of fresh fruits and vegetables along with non-perishable foods and donated sundries.

Churches are huge food redistribution centers also.

The "too hard to set up" systems have created good jobs here. Instead of seeing food donations as a liability, they have created a supervised and refrigerated second market.

The cost for employees is small because, of course, many of the positions are volunteers. But the supervisors and some of the staff are full time employees with benefits.

The article has thrown up obstacles that are easy to overcome.

3 Like

AnotherDreamer

Oct 28, 2024 at 11:33am
In response to "Waste Not"

https://cdn.prod.website-files.com/6156cebc4c605e281289a232/6156cebc4c605eb8bc89... e%20for%20beginners%2024%2009%2018.pdf

1 Like

Robert Waltz

Oct 28, 2024 at 11:33am
In response to "Waste Not"

AnotherDreamer

- Hey, it's a replicator from Star Trek!

Now let's invent the holodeck.

2 Like

AnotherDreamer

Oct 28, 2024 at 11:27am
In response to "Waste Not"

We the people of the world could use those products before they are compost as free sources for 3D printed foods that could be used for emergency food, or if tasty enough, re sold.https://builtin.com/articles/3d-printed-food

2 Like

December 4, 2019 at 12:27am

#970931

Science Is Almost Always Wrong, And That's A Good Thing

Today, I'm going to snark on physics. Don't worry; it's an article for the uninitiated.

https://www.quantamagazine.org/einstein-symmetry-and-the-future-of-physics-20190...

The Simple Idea Behind Einstein’s Greatest Discoveries

Lurking behind Einstein’s theory of gravity and our modern understanding of particle physics is the deceptively simple idea of symmetry. But physicists are beginning to question whether focusing on symmetry is still as productive as it once was.

The flashier fruits of Albert Einstein’s century-old insights are by now deeply embedded in the popular imagination: Black holes, time warps and wormholes show up regularly as plot points in movies, books, TV shows.

Yes, and most of them get it wrong. I mean, seriously, how hard is it to get the damn science right? I personally know a guy who works as a science consultant to certain TV shows, and the problem is they don't listen to him.

Perhaps ironically, though, what is arguably the most revolutionary part of Einstein’s legacy rarely gets attention. It has none of the splash of gravitational waves, the pull of black holes or even the charm of quarks.

You know, probably about 5% of your readers got that last pun. I hope it was worth it.

The key insight came to Einstein in one of his famous thought experiments. He imagined a man falling off a building. The man would be floating as happily as an astronaut in space, until the ground got in his way.

1. There weren't any such things as "astronauts" then; 2. Newton figured shit out with falling apples, not humans. WTF, Albert?

It took a while for him to pin down the mathematical details of general relativity, but the enigma of gravity was solved once he showed that gravity is the curvature of space-time itself, created by massive objects like the Earth.

For the record, this is a little bit misleading. Yes, he solved one "enigma" of gravity (the article specifically mentions the one; it's how heavy and light objects fall at the same speed). But physicists are still ultimately puzzled by gravity. Maybe if they find (or fail to find) the hypothetical "graviton," some of that will clear up, but science is science, meaning that'll just lead to more puzzles.

So the one fundamental force that we all feel all the time - with the notable exception of a select few astronauts, for brief periods - is the one we have the least understanding about. Life is funny that way.

Nearby “falling” objects like Einstein’s imaginary man or Galileo’s balls simply follow the space-time path carved out for them.

That's going to be my new curse. "Hey, Waltz, someone keyed your car." "Galileo's balls!"

At the same time, symmetry-based reasoning predicted a slew of things that haven’t shown up in any experiments, including the “supersymmetric” particles that could have served as the cosmos’s missing dark matter and explained why gravity is so weak compared to electromagnetism and all the other forces.

1. Dark matter isn't "missing;" they're all but certain it's there; it's just that they have no idea what it is. 2) The relative weakness of gravity starts to make sense if you hypothesize invisible dimensions. ... which I suppose doesn't help much.

Periodically - and fortunately, this particular article doesn't do this - you see someone going "Einstein was WRONG!" This plays to the masses, who yearn to see some genius or authority taken down a peg or two. But it displays an ignorance about the way science works. Was Einstein wrong, ever? Most definitely. For all his vaunted intelligence and insight, he was human, and humans are wrong from time to time (some more than others, granted).

Einstein was no more "wrong" than Newton was wrong; he just didn't have all the data. Newton's theories about gravity work to a really high degree of accuracy in both our everyday lives and the things we observe. Einstein extended them. It's analogous to how we think of the Earth itself: it's basically a sphere, and can be modeled as such for all kinds of purposes, such as charting airline flight paths or drawing maps. But to get greater precision you have to take into account the difference between the equatorial diameter and the polar diameter, which is (roughly) 40 miles, for most purposes a rounding error in the average diameter. Discovering the equatorial bulge doesn't make "the Earth is round" wrong; it just adds precision. And of course it's also lumpy, but not 40 miles worth of lumpy; Everest, for example, tops out at only about 5 or 6 miles above sea level.

Point is, though, the theories Einstein came up with have withstood numerous experimental tests. So no, he wasn't "wrong" in that sense. But we keep searching for more refinements to the theories. And I find that sort of thing fascinating.

3 Like • Comment

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