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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Write_Mikey_Write!

6 minutes ago
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.

Adherennium Plotting something

about 2 hours ago
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.

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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').

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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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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.

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