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:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

October 29, 2024 at 9:35am

#1079108

The Numbers Don't Lie

From Big Think, an article about how the numbers sometimes lie:

The “McNamara fallacy”: When data leads to the worst decision
Don’t make the mistake of blindly following quantitative metrics — whether you’re helping clients or looking for lunch.

Or do they?

In 2014, I took a weekend break to York.

You know what sometimes lies? Anecdotes.

York is a lovely city in the north of the UK, with an ancient cathedral, quaint cobbled roads...

Oh, hell no.

...and an interactive Viking experience.

Is that where you get hung by the ankles and forced to watch your clan murdered and village pillaged while you're upside-down?

The reason for the anecdote is quickly revealed: the author made the mistake of trusting online reviews whilst choosing a restaurant, Skewers, which turned out to have high ratings due to drunks lovingly appreciating their late-night-early-morning fare.

Let's be clear, here: This is not a failure of TripAdvisor. It's not a failure of the food establishment. It's certainly not a failure of the well-meaning drunks. The blame lies entirely on the author. I travel quite a bit, and I prefer to form my own opinions of places I visit, rather than relying on others' (sometimes real, sometimes not) experiences. Sometimes, it sucks, but usually, I'm pleasantly surprised. Either way, I get to write reviews of my own, which may or may not factor into someone else choosing to visit.

The story of Skewers is an example of the McNamara fallacy, and learning about it can help us all (especially the underprepared tourists among us).

In spite of my misgivings about the opening anecdote, I'm willing to read on.

Fortunately, the author again gets quickly to the point, which is how I know I'm reading Big Think and not The New Yorker:

The McNamara fallacy is what occurs when decision-makers rely solely on quantitative metrics while ignoring qualitative factors.

So, like, when you rely only on the studies that claim booze is bad for you, and ignore how good it makes you feel.

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 where considering certain human and contextual elements would have been successful.

I'm no expert on war history, but claiming that different decisions "would have been successful" strikes me as arrogant. I'd weasel out with "might have been successful," instead.

The McNamara fallacy is not saying that using data is bad or that collecting as much information as you can is wasted time.

I'm just leaving this here lest someone be thinking, "What's the point in data, then?"

For example, it’s not uncommon for someone to deeply love a book that has few or no reviews on Goodreads. It’s possible to enjoy a restaurant or a movie in spite of what others say. Data is a great starting point, and a great many idiotic and dangerous things are done when we ignore data, but it doesn’t always make for the best decisions.

I'm very familiar with that assessment, anyway. People are different, and you might love something others hate, or vice-versa. Reviews and the like are aggregations, and blindly following them is roughly the same thing as blindly following the crowd. You can do that if you want, but I don't, because that way, I miss out on the joy of making my own discoveries.

If everyone liked the same things, there'd only be one beer, and that would be a boring world indeed.

I have a much bigger problem with the explicit comparison of "finding a place to eat" and "running a war where people die."

Still, I suspect the basic ideas of the fallacy are sound, and the article goes into other examples and applications thereof, which I don't feel the need to reproduce here.

Fear not, though; it's short, and there are very few actual numbers. So, a solid four stars out of five from me. Your experience may vary.

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