Complex Numbers (Book)

Blog Calendar

◄ August ►
S	M	T	W	T	F	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31
Archive				RSS

About This Author

Contact Me

Come closer.

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/

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:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

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.

2 Like • Comment • View 1 comment

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.