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 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

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Jayngle Bells

Dec 23, 2024 at 10:03am
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.

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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.

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Annette

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

Comments work.

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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.

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August 6, 2024 at 10:37am

#1074837

Hall of Fame

The article I'm sharing today, from SciAm, is a bit esoteric. To me, it illustrates why common sense is neither, and why probability can get really weird.

See Why Everyone Gets the Monty Hall Puzzle Wrong
How to finally wrap your mind around the uniquely counterintuitive Monty Hall dilemma

In high school statistics class, my teacher presented a probability problem that haunts me to this day. It was a puzzle inspired by the TV game show Let’s Make a Deal and named after its longtime host, the late Monty Hall.

Several generations of people would never have heard of this guy if it weren't for this puzzle. And it's not like he came up with it; he's just part of the setup. As for the details of the puzzle itself, the article provides a fun illustration of it, but, in short:

There are three doors in front of you.
Behind one of the doors is a new car. Behind the other two doors are goats.
The host invites you to pick a door, any door.
The host throws open one of the doors you did not select, revealing a goat.
Then the host gives you the option of switching your [initial selecton].

Let's leave aside for the moment the relative value of cars and goats. I don't want a new car. I'd be responsible for taxes and insurance on the thing, though I could maybe just sell it for money. But right now, I could really use a goat; you wouldn't believe the condition of my back yard.

Most people think it doesn’t matter whether they stick with their original choice or switch to the other unopened door because the odds are 50–50—that it’s nothing more than a coin toss.

Yeah, that's just common sense. (Which is why I hate "common sense.")

But you should always switch doors. You win two thirds of the time if you switch and one third of the time if you stay. In other words, switching doors doubles your chance of winning.

I have a quibble about "doubles your chance of winning:" If you buy two lottery tickets instead of one (assuming different numbers), that doubles your chance of winning from, I don't know, 1 in a billion to 1 in half a billion. Wow. But, okay, in this particular formulation, it's meaningful.

This counterintuitive problem, first described in relation to the TV show in 1975, is taught in introductory math and statistics classes across the world. But it was widely popularized in 1990 in Parade magazine. After writer Marilyn vos Savant wrote about the puzzle in her Ask Marilyn column, she received an estimated 10,000 furious letters declaring the answer she gave was wrong, including 1,000 or so that were signed by people with a Ph.D. in their title.

It's entirely possible to be smart and wrong. Even if you have a Ph.D. Even if your name is Marilyn vos Savant.

How are the odds not 50–50? Since that fateful column, mathematicians, psychologists and philosophers have been trying to understand what makes this answer uniquely hard to grasp. They’ve found that some of the most common cognitive biases may be to blame, along with a core misunderstanding of how probability works.

The article, of course, goes into the details, and I won't rehash them here. But I will point out another glaring problem we often have with probability, which is almost always illustrated for me when I'm at a blackjack table.

Put as simply as I can, you have the choice in blackjack to stand (keep the cards you have) or hit (dealer gives you another card). The actual probabilities vary, of course, and there can be other complications, but in most cases, you want to either stand or hit, and one of those two has better odds of not losing the game.

But a probability is just that; it's not a certainty. Even if it pays off most of the time (with "most" being anywhere from 50.1% to 99.9% or thereabouts), it won't pay off all the time.

In blackjack, if your strategy fails, you just pony up another bet and play again (or give up in disgust and visit the buffet). But the Monty Hall problem is a one-off. You'll never again be faced with maybe getting a shiny new car or maybe getting a poison-ivy-eating goat, so you only get that one shot. And that one shot is, according to the math, about a 2/3 chance: better than 50-50, but in the end, you might still "lose."

My point being that even the best strategies can fail, and sometimes, we have a problem confusing "probably" with "certainly." And that one data point doesn't mean anything; if you lost on an 80% probability win, that doesn't mean that the next one doesn't have an 80% probability.

Which is why I quit giving probability lessons to other blackjack players.

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