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:

Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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

December 8, 2018 at 12:33am

#947045

Biased

A couple of days ago I mentioned the survivorship fallacy. Well, here's a bit on that and a few other cognitive biases we tend to have.

https://getpocket.com/explore/item/5-common-mental-errors-that-sway-you-from-mak...

1. Survivorship Bias.

I won't go into this further. I think the article is clear enough, and I already talked about it.

2. Loss Aversion.

Loss aversion refers to our tendency to strongly prefer avoiding losses over acquiring gains. Research has shown that if someone gives you $10 you will experience a small boost in satisfaction, but if you lose $10 you will experience a dramatically higher loss in satisfaction. Yes, the responses are opposite, but they are not equal in magnitude.

I've tried to train myself to reverse this. That's helpful for me because I'm a gambler. I also have money in the stock market. No, those are not the same thing, but they do have one commonality: you win sometimes, and you lose sometimes. But I see loss aversion catered to in stock market analysis all the time; lately, the markets have been on a bear tear and people are freaking the fuck out. Any downturn is exacerbated by panic selling.

Don't get me wrong; losing money never feels good. But it can feel less bad if you're aware of this kind of bias.

3. The Availability Heuristic.

The Availability Heuristic refers to a common mistake that our brains make by assuming that the examples which come to mind easily are also the most important or prevalent things.

For example, research by Steven Pinker at Harvard University has shown that we are currently living in the least violent time in history. There are more people living in peace right now than ever before. The rates of homicide, rape, sexual assault, and child abuse are all falling.

Okay, we're going to use fancy words like "heuristic?" That strikes me as being obfuscatory. What this bit boils down to is this: While anecdotes can make compelling narratives, intelligent decisions can only be made on hard data. I'm sure you've already heard that "data" isn't the plural form of "anecdote."

For example, many people are afraid of flying. While some fear is understandable because you're putting your life into someone else's hands, much of the fear - of being killed in a plane crash, of getting hijacked, etc. - is almost groundless (see what I did there?) You have a much greater chance of being killed driving to and from an airport than in transit between airports. But what happens is, all but the worst highway incidents don't make international news, while every incident involving an airplane gets reported all over the world, so we start to think of flying as a hazardous proposition.

Make no mistake, there are dangers - life is inherently risky - but really, you're more likely to kick it by slipping in the shower than in a plane crash.

Or hey, maybe I'm just trying to psych myself up; I have a plane to catch in about 8 hours.

4. Anchoring.

There is a burger joint close to my hometown that is known for gourmet burgers and cheeses. On the menu, they very boldly state, “LIMIT 6 TYPES OF CHEESE PER BURGER.”

My first thought: This is absurd. Who gets six types of cheese on a burger?

My second thought: Which six am I going to get?

I didn't realize how brilliant the restaurant owners were until I learned about anchoring. You see, normally I would just pick one type of cheese on my burger, but when I read “LIMIT 6 TYPES OF CHEESE” on the menu, my mind was anchored at a much higher number than usual.

I have to admit I fall for this one all the time. But I'm working on it.

5. Confirmation Bias.

The Grandaddy of Them All. Confirmation bias refers to our tendency to search for and favor information that confirms our beliefs while simultaneously ignoring or devaluing information that contradicts our beliefs.

Yeah, this is a big one. It's one of the reasons I try not to stay in an information bubble. But it's also the hardest to deal with, because it works both ways. The article uses the example of climate change. It's a hot-button (see what I did there) topic in many forums on the internet. As far as I've been able to determine, not one single person's mind has changed on the issue based on arguments started on the internet. On the contrary, at some point, any information contrary to what a person believes only causes that person to believe even more deeply. Data is dismissed as falsified. Charts are thrown out as bogus.

Internet arguments are stupid for many reasons, but this is a big one: you're not going to change anyone's mind with any number of anecdotes, any amount of data, any quantity of reason, or any pure charismatic persuasiveness.

I've found this to be the case even with things that have a lot less potentially at stake than climate change. For instance, a lot of people seem to have internalized the definition of "blue moon" as the second full moon in a calendar month. This has been proven to be incorrect from a historical perspective, and yet when I present evidence of that, people go right on believing, promoting, and propagating the incorrect definition. Now, you could say that I'm being stubborn in clinging to the older, true definition, and you'd be right. That doesn't change the fact that the "second in a month" definition is based on an admitted error in a publication from the 1940s. To me, it's a perfect example of how falsehoods get entrenched and then promoted as fact.

If you can't persuade someone of something with such an insignificant outcome, how can you hope to persuade them about, say, vaccinations, climate change, or the nearly-spherical nature of the planet?

As a result of confirmation bias, any internet argument that doesn't dissolve into chaos will always bog down into epistemology. This is Waltz's Second Rule of the Internet, and I haven't found any evidence to contradict it. But then, I'd probably ignore said evidence anyway.

(For anyone interested, Waltz's First Rule of the Internet is that any post attempting to correct someone's spelling, grammar, or punctuation will inevitably contain spelling, grammar or punctuation errors.)

NOTE: I'm going to be traveling over the next 11 days, so updates will be spotty if they happen at all. But if you don't hear from me after that, feel free to enjoy the irony of me dying in a plane crash mere hours or days after insisting how safe air travel is.

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