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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Show Recent Comments

10 Most Recent Blog Comments:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 22 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
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

Dec 23, 2024 at 4:55pm
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

March 5, 2024 at 11:12am

#1065605

Ignorance is...

Here's an example of finding an article that aligns with my existing worldview but having my doubts.

New Study Links Optimism To Lower Cognitive Abilities

That doesn't mean it's right, of course. But it also doesn't mean it's wrong.

Optimistic thinking, often celebrated in self-help literature as a pathway to happiness, health, and longer life, can also lead to poor decision-making.

This has been clear to me for a while.

Research from the University of Bath shows that excessive optimism is actually associated with lower cognitive skills such as verbal fluency, fluid reasoning, numerical reasoning, and memory. Whereas those high in cognitive ability tend to be both more realistic and pessimistic in their expectations about the future.

Basically: ignorance is bliss.

Well, no, not really, because ignorance is simply not knowing something, which has nothing to do with how smart you are. Everybody's ignorant about lots of things. But stubborn, willful ignorance is another matter entirely... also not necessarily correlated with intelligence.

In any case, I'll note that the passage I just quoted seems to be very careful not to imply causation. Does excessive optimism make a person less clever, or does being an idiot lead to optimism? Or, alternatively, is there something else causing both effects? Much as I want to believe the article, that seems like an important thing to find out.

“This points to the idea that whilst humans may be primed by evolution to expect the best, those high in cognitive ability are more able to override this automatic response when it comes to important decisions. Plans based on overly optimistic beliefs make for poor decisions and are bound to deliver worse outcomes than would realistic beliefs,” Dr Dawson added.

Okay, I'm going to need a reference on that "primed by evolution" thing, hopefully not from bogus evolutionary psychology. Also, why isn't this "override" attributed to evolution?

“Unrealistically optimistic financial expectations can lead to excessive levels of consumption and debt, as well as insufficient savings. It can also lead to excessive business entries and subsequent failures. The chances of starting a successful business are tiny, but optimists always think they have a shot and will start businesses destined to fail,” Dr Dawson said.

But, clearly, they're not all destined to fail (unless you take the really long view that every business will fail eventually, even if it takes a few centuries). If we didn't have people taking a shot at these things, we wouldn't have businesses at all. It's like the baby turtles' march to the sea.

The study took data from a UK survey of over 36,000 households and looked at people’s expectations of their financial well-being and compared them with their actual financial outcomes. The research found that those highest on cognitive ability experienced a 22% increase in the probability of “realism” and a 35 percent decrease in the probability of “extreme optimism”.

For once, I can't complain about the sample size. But there are other obvious yellow flags here. UK only, for starters. How cognitive ability was determined from a mere survey: did they basically ask trivia questions? Even if not, there are major issues concerning the standard IQ tests. And how do we know that they're studying truly independent variables?

“Unrealistic optimism is one of the most pervasive human traits and research has shown people consistently underestimate the negative and accentuate the positive. The concept of ‘positive thinking’ is almost unquestioningly embedded in our culture – and it would be healthy to revisit that belief,” Dr Dawson added

Sounds to me like someone hated the idea of positive thinking and set out to debunk it. I can understand that impulse, as I share it. But, first, if science goes into an investigation wanting a certain outcome, that outcome is more likely (even excluding fraud or other shenanigans). And second, I have serious questions about the methodology.

So I put this in the category of "stuff I want to believe, and which tracks with stuff I've been saying, but can't trust the science."

3 Like • Comment • View 1 comment

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