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 15 hours 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 ✍️

about 18 hours 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

about 21 hours ago
In response to "Gray Matters"

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

2 Like

tj-Merry Mischief Maker

a day ago
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

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

May 27, 2024 at 10:22am

#1071760

Distrust the Process

One reason I think science confuses people is that nutrition science, in particular, is a hot mess that can't seem to make up its mind about anything. Case in point, from Inverse (reprinted from The Conversation):

The Ultraprocessed Food Hype Is Masking This Other Major Health Predictor, 30 Years of Data Reveal
Much of the recent evidence related to ultra-processed foods tell us what we already knew.

Even I gave up on trusting its results after several iterations of eggs being good, then bad, then good, then bad, then good, then bad, then good, etc. Not to mention all the questionable studies funded by people who push for a particular result. Nowadays, I mostly just eat for enjoyment; let other people get neurotic about what's healthy or not this week.

I recognize that part of the problem here is that biology is fiendishly complex, and has all sorts of mechanisms to return to equilibrium after a push away from it, some of which we don't fully understand. Another part of the problem is that it can be extraordinarily difficult to remove confounding variables from a nutrition experiment. And a big part of the problem is breathless reporting, eager to announce the Next Big Bad Thing or explain why the Last Big Bad Thing wasn't so bad after all—the latter of which seems to be the case for today's article.

In recent years, there’s been increasing hype about the potential health risks associated with so-called “ultra-processed” foods.

There's always a bogeyman, because we can't just let people enjoy their meals in peace. First it was fats (turns out we need some), then carbs (turns out we need some of those too), then gluten (still a source of grave misunderstandings) or whatever. Now, apparently it's ultra-processing, which I'm sure is defined somewhere.

But new evidence published this week found not all “ultra-processed” foods are linked to poor health. That includes the mass-produced wholegrain bread you buy from the supermarket.

Like I said, carbs went from good to bad to maybe good. The "maybe" is probably a matter of high vs. low glycemic index. Meaning, for example, whole wheat bread is probably better for you than white bread, which I did a whole entry about a couple of weeks ago.

Ultra-processed foods are industrially produced using a variety of processing techniques. They typically include ingredients that can’t be found in a home kitchen, such as preservatives, emulsifiers, sweeteners, and/or artificial colors.

I told you it was defined somewhere. Well, sort of. I'm pretty sure we all have things in those categories in our home kitchens. Salt, for example, is a preservative. Egg protein is an emulsifier (a thing that keeps oil and water from separating as nature intended). And so on.

Common examples of ultra-processed foods include packaged chips, flavored yogurts, soft drinks, sausages, and mass-produced packaged wholegrain bread.

In other words, anything that actually tastes good.

The new paper just published used 30 years of data from two large US cohort studies to evaluate the relationship between ultra-processed food consumption and long-term health. The study tried to disentangle the effects of the manufacturing process itself from the nutrient profile of foods.

I'm not reading that, so I'm not going to comment on the validity of the study. I'll just point out that this is just throwing another osis into our collective neuroses.

The study found a small increase in the risk of early death with higher ultra-processed food consumption.

The obvious question here is: correlation or causation?

Existing national dietary guidelines have been developed and refined based on decades of nutrition evidence.

But mostly based on pressure from lobbyists who work for manufacturers of ultra-processed foods, so that sentence makes me laugh. Nutrition science is convoluted enough as it is; throw in government bullshit, and you can see why I've given up.

Which doesn't mean I'm going to snack on Doritos all day. Just that I'm done worrying about every little thing that goes into my food and drink.

6 Like • Comment • View 2 comments

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