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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February 13, 2022 at 12:01am

#1026588

The Measure of a Mind

Old article is old, but I just found it recently.

How To Become A Genius
Whether you’d like to admit it or not, you probably have an average IQ. Luckily, there’s a lot you can do to increase your intelligence.

Right. Even if that's true, if everyone does it, we're right back where we started.

You probably think you’re pretty smart.

No, but I think you're pretty smug.

Most people believe they’re smarter than the average American, according to a study from YouGov.

In fairness, most people worldwide are smarter than the average American.

Yet when it comes to IQ, most of us are indeed average, falling in the 80-119 point range. While this number peaks in our late teens to early 20s and remains relatively stable as we age, that doesn’t mean your potential is fixed.

Wasn't the whole point of IQ tests to measure a fixed value? Leaving aside for the moment the controversies over implicit bias in the tests.

“Those who claim that IQ is fixed for life are in fact referring to our IQ test scores, which are relatively stable–not to our intelligence levels, which are constantly increasing.”

Uh... okay, I guess I'm too stupid to know the difference. Apparently Intelligence Quotient is not a measurement of intelligence? Or is the person who is quoted there making the unintelligent but common mistake of conflating knowledge (which of course is usually gained over time) with intelligence? Let's read on to find out.

David Shenk, author of The Genius in All of Us, says it’s virtually impossible to determine any individual’s true intellectual limitations at any age; anyone has the potential for genius or, at the very least, greatness.

Who says greatness is related to genius? In science or philosophy, maybe, sure. But there have been great sportsball players, great actors, etc., many of whom would not score high on an IQ test. Also, intelligence is hardly the truest measure of a person. There have been many highly intelligent people who were also complete dickwads, or who simply have no desire for greatness of any sort.

If you want to be smarter than the average American, it’s not only possible; it’s within reach.

Sure. Just be something other than American.

Intelligence is the ability to acquire and apply knowledge and skills, and includes the ability to reason, solve problems, remember information, and be creative.

Just for the hell of it, I looked up the dictionary definition of intelligence. I normally avoid doing this, because dictionary definitions are descriptive, not prescriptive, and I find resorting to a dictionary in order to support one's point to be the last refuge of a pedant. But guess what my first search result (Oxford) found? "The ability to acquire and apply knowledge and skills." Fucking article quoted directly from the dictionary. And also added some stuff that's only tangentially related to intelligence.

While taking a class and reading a book are two ways to learn something new, here are six surprising tasks that boost your brainpower, make learning easier, and put you on the road to greatness:

Learning new shit doesn't automatically increase one's intelligence; it just means you have a good memory. Intelligence is the ability to make connections between the shit you've learned.

That's not in the dictionary; that's my interpretation.

1. Train Your Memory

While a professor at the University of California, Irvine, Susanne Jaeggi found that an activity known as the n-back task increases fluid intelligence, which is the ability to reason and solve new problems independent of previous knowledge.

Regardless of whether or not you can, or want to, increase your intelligence (however it's defined), an exercise like that couldn't hurt.

2. Open Yourself To New Points of View

Another way to increase your intelligence is to expand your network and consider other people’s points of view. The exercise will open your mind to new opportunities and promote cognitive growth.

Hey look, another good idea not necessarily because of its effect on intelligence. If more people did this, maybe we wouldn't be so dug in to our own thought trenches.

“Open your mind and listen to arguments that make no sense to you–and try to find some sense in them,” writes Roche.

I mean, I try to do that. I don't always succeed.

3. Find Motivation

Uncommon achievement takes a source of motivation, says Shenk. “You have to want it, want it so bad you will never give up, so bad that you are ready to sacrifice time, money, sleep, friendships, even your reputation,” he writes in The Genius in All of Us.

Well, I'm out. I'd rather have time, money, sleep, friends, and a reputation that's not terrible. That's what motivates me.

4. Do Cardiovascular Workouts

Cardiovascular fitness can raise your verbal intelligence and improve long-term memory, according to a study from the University of Gothenburg in Sweden.

I think there are plenty of other reasons to do this, too. And one good reason not to: they take time that I could use to play video games.

5. Play Video Games

Hey, look at that.

While it looks like a good way to waste time, video gaming can actually stimulate the growth of neurons and promote connectivity in the regions of the brain responsible for spatial orientation, memory formation, and strategic planning.

I'm no scientician, but it seems to me that the benefits would vary greatly depending on the game.

6. Meditate

Mindful meditation can increase the neuroplasticity in the brain, according to a study from the University of Oregon and Texas Tech University’s Neuroimaging Institute.

Nope, all it does for me is increase the sleep levels in my brain. But as I said, I do like my sleep, so that's fine.

In summary, I think the main point of the article verges on bullshit, but I can't argue with most of the advice for other reasons.

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