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:

tj-Merry Mischief Maker

about 9 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

about 14 hours 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

about 16 hours 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 ✍️

a day ago
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

a day ago
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

🐕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

December 27, 2024 at 8:29am

#1081585

Sensational

From SciAm, an opinion piece that seems to align with my own opinions:

A Science Breakthrough Too Good to Be True? It Probably Isn’t
The more exciting, transformative and revolutionary a science result appears, especially if it comes out of nowhere, the more likely it is to be dead wrong. So approach science headlines with a healthy amount of skepticism and patience

Regular readers might have noticed that this is what I try to do.

I'd also add: pay attention to retractions. There are still people stubborn enough to believe vaccines cause autism.

In 2014 astronomers announced a whopper of a discovery: primordial waves from the earliest moments of the big bang.

You know, I have no memory of that one, or its retraction.

Or remember Tabby’s Star? In 2015 astronomers speculated that its strange light pattern might be the product of alien megastructures. Cue media circus, high-profile talks, the works. Further analysis revealed that it was ... dust, again.

That one, I recall fairly well. I wasn't blogging at the time, but I remember reading about it and going, yeah, let's do some more research before jumping to "aliens."

Further analysis revealed that it was ... dust, again.

Or... that's what They want us to think.

More recently, a group of astronomers claimed to find phosphine in abundance in the Venusian atmosphere, proposing that there might be some form of exotic life floating in the cloud tops.

That one's a bit more complicated. In brief, further studies are being done, and some are contradictory. That's okay. That's how we figure things out. Until there's something definitive, though, I think it's completely safe to assume "no life on Venus."

It’s not just astronomy. Neutrinos can travel faster than light. Mozart makes your kids smarter. Dyeing your hair gives you cancer. Smartphones make us stupid.

Very, very stunning. But very, very wrong.

Some of that is wishful thinking or deliberate misinformation. "Mozart makes your kids smarter," for example, sounds like something that music producers might push.

First, much, if not most, scientific research is wrong. That’s why it’s research; if we knew the answers ahead of time, we wouldn’t need to do science.

That's... well, it could be phrased better, I think. Not that scientific research is "wrong," which can easily imply a moral judgement, but that a) some hypotheses turn out to be falsified and b) sometimes scientists reach the wrong conclusion, which is later caught through peer review and replication attempts.

Second, scientists endure perverse incentives to publish as much as possible—to “publish or perish”—and to get their results in top-tier journals as much as possible.

That's a problem, but I wouldn't have the first idea how to fix it.

Lastly, there’s the modern-day hype machine. While many journalists respect scientists and want to faithfully represent the results of scientific research, publishers face their own incentives to capture eyeballs and clicks and downloads. The more sensational the story, the better.

This bit is what I mainly focus on in here, because I'm not involved in science or science publication, but I do try to recognize when a headline or link is deliberately sensationalized.

The more times that the public sees science contradict itself, the less likely people are to believe the next result that makes headlines. And the more times that scientists are loudly, publicly wrong, the more ammunition antiscience groups have in their fight against trusted experts.

This is absolutely a problem. I've used this example before, but it's like how eggs have gone from good to bad to good to bad to maybe okay to maybe not too many to good to bad (and then to way too expensive anyway), all just within my lifetime. Just because nutrition science has major flaws, however, doesn't mean astronomy and physics do.

And let me be clear here. I’m sure you find most, if not all, scientific research absolutely fascinating—as do I. But the more interesting a result is to the wider community, with more headlines, chatter, attention and raised eyebrows, the more likely it is to be worthy of a bit of healthy skepticism.

I think that's a pretty sound generality. Like any generality, it's not always true. One day, perhaps, someone will find unequivocal evidence of life on another world: microbes on Mars or eukaryotes on Europa, or something. That will be a Big Fucking Deal, one of the most important discoveries in human history. However, if the evidence is not unequivocal—as with the Venus phosphine, or, before that, the Antarctic meteorite from Mars—it's all sensation with no confirmation.

The best approach to take with science results, news, and headlines is the same approach scientists use themselves: healthy skepticism.

To be clear, "skepticism" doesn't mean "reflexive disbelief." It's more like looking at it critically while keeping an open mind. Seeing a science headline and immediately disbelieving it on principle is just as bad as immediately believing it because you want it to be true.

Beware big headlines; don’t believe everything you see. But when study after study comes out, building up an interconnected latticework of theory and experiment, allow your beliefs to shift, because that’s when the process of science has likely led to an interesting and useful conclusion.

That, to me, is of great importance. Being stubborn, clinging to what has been disproved or debunked, is not a positive character trait. But then, neither is naïve acceptance of every claim you see.

3 Like • Comment • View 3 comments

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