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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 hours 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.

1 Like

Adherennium Plotting something

about 8 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.

August 15, 2020 at 12:08am

#990810

Wait, Isn't Oxygen A GOOD Thing?

Bit of a sensationalist headline here, but I found the article interesting.

http://nautil.us/issue/15/turbulence/fruits-and-vegetables-are-trying-to-kill-yo...

Fruits and Vegetables Are Trying to Kill You
Antioxidant vitamins don’t stress us like plants do—and don’t have their beneficial effect.

I should point out that this article is from 2014. Since then, I'm sure nutrition science has refuted, confirmed, refuted, confirmed, and refuted again the content by now.

You probably try to exercise regularly and eat right.

I did. For a while. In the Before Time.

Perhaps you steer toward “superfoods,” fruits, nuts, and vegetables advertised as “antioxidant,” which combat the nasty effects of oxidation in our bodies.

Nope.

Maybe you take vitamins to protect against “free radicals,” destructive molecules that arise normally as our cells burn fuel for energy, but which may damage DNA and contribute to cancer, dementia, and the gradual meltdown we call aging.

Here's where the refutations come in. Over-the-counter vitamins seem to be, at best, harmless and ineffective; at worst, harmful. But that's today. Wait a few hours.

But that's really the point of the article here.

Warding off the diseases of aging is certainly a worthwhile pursuit. But evidence has mounted to suggest that antioxidant vitamin supplements, long assumed to improve health, are ineffectual.

"Assumed?" No one did any actual, you know... studies?

Fruits and vegetables are indeed healthful but not necessarily because they shield you from oxidative stress. In fact, they may improve health for quite the opposite reason: They stress you.

Then there's the whole "fiber" thing, which I'm pretty sure they have done studies on.

That stress comes courtesy of trace amounts of naturally occurring pesticides and anti-grazing compounds. You already know these substances as the hot flavors in spices, the mouth-puckering tannins in wines, or the stink of Brussels sprouts. They are the antibacterials, antifungals, and grazing deterrents of the plant world. In the right amount, these slightly noxious substances, which help plants survive, may leave you stronger.

I do love me some hot pepper. Brussels sprouts, though? Thanks to science, they don't stink as much as they used to. And do I need to sing the praises of wine here?

Parallel studies, meanwhile, have undercut decades-old assumptions about the dangers of free radicals. Rather than killing us, these volatile molecules, in the right amount, may improve our health. Our quest to neutralize them with antioxidant supplements may be doing more harm than good.

This actually tracks with some other stuff I've read. But that doesn't mean much. As you know by now, when it comes to nutritional science, I don't believe a damn thing.

Oddly, our mitochondria, the energy factories of our cells, emitted ROS naturally. So degenerative disease seemed to stem in part from our own metabolic function: Your mitochondria “burned” fuel, emitted this toxic exhaust, and inadvertently set the limits on your existence. That was the working hypothesis, at any rate.

This is what happens when you take a working hypothesis and try to defend it rather than disprove it.

Observational studies, meanwhile, suggested that people who regularly ate vitamin-laden fruits and vegetables were healthier. So were people with higher levels of vitamins E and C in their blood.

All together now: Correlation is not causation.

Vitamins were strongly antioxidant in test tubes.

And bleach kills cancer cells in test tubes. It also kills other cells.

But if those ROS were so harmful, some scientists asked—and the basic design of our (eukaryote) cells was over 1 billion years old—why hadn’t evolution solved the ROS problem?

Do I need to go into this again? Evolution simply doesn't care. If you live long enough to reproduce, what happens to you after that is entirely immaterial. A worthy object for study, sure -- but we're talking about effects that usually happen in a human's fifties and beyond, well after peak reproductive years.

According to the ROS model of aging, animals that exercised and fasted should have died younger. But they lived longer.

And then there's my philosophy: what's the point of living longer if you have to spend all that extra time exercising, and avoiding one of the few pleasures (eating) that makes life worth living? (Yes, I've reduced my caloric intake and, at least when I could still go to the gym, exercised every day. But not to live longer; just to feel better. Now that exercise isn't as easy for me, I'm not doing it as much, and I find myself sinking once more into depression.)

He had 39 male volunteers exercise regularly over several weeks; half took vitamin supplements before working out. The results, published in 2009, continue to reverberate throughout the field of exercise physiology, and beyond. Volunteers who took large doses of vitamins C and E before training failed to benefit from the workout. Their muscles didn’t become stronger; insulin sensitivity, a measure of metabolic health, didn’t improve; and increases in native antioxidants, such as glutathione, didn’t occur.

Small, non-diverse sample.

But the primary role of vitamins in our body, according to Ristow and others, may not be antioxidant. And the antioxidant content of fruits and veggies does not, he thinks, explain their benefits to our health. So what does?

I think we're about to find out.

Mark Mattson, Chief of the Laboratory of Neurosciences at the National Institute on Aging, has studied how plant chemicals, or phytochemicals, affect our cells (in test tubes) for years. The assumption in the field has long been that, like vitamins, phytochemicals are directly antioxidant. But Mattson and others think they work indirectly. Much like exercise, he’s found, phytochemicals stress our bodies in a way that leaves us stronger.

Again: test tube science, while the body is incredibly complex. Also, again: "assumption."

Plants, Mattson explains, live a stationary life. They cannot respond to pathogens, parasites, and grazers as we might—by moving. To manage the many threats posed by mobile life, as well as heat, drought, and other environmental stresses, they’ve evolved a remarkable number of defensive chemicals.

As I noted recently, every living thing on Earth has been evolving exactly as long as we have. It's silly to assume that, because plants don't have opposable thumbs, mobility, and a central nervous system, that they're not fit for their niche.

Mattson and his colleagues say these plant “biopesticides” work on us like hormetic stressors. Our bodies recognize them as slightly toxic, and we respond with an ancient detoxification process aimed at breaking them down and flushing them out.

Sounds good, but as I said, we're going to need more than test tube science here.

In a study on Type 2 diabetics, broccoli-sprout powder lowered triglyceride levels. High triglycerides, a lipid, are associated with an increased risk of heart disease and stroke. Lowering abnormally elevated triglycerides may lessen the risk of these disorders.

"Powder?" Why not just... you know... eat a fucking vegetable? I don't fully buy in to the whole "eat non-processed foods" fad, nor am I vegetarian, but even I have to admit that an actual goddamned vegetable is probably better for you than powder.

Implicit in the research is a new indictment of the Western diet.

Of course it is. Can't leave well enough alone. Gotta convince people they're Doing It Wrong, so you can sell them the "solution."

Sinclair studies another class of native proteins, called sirtuins, associated with health. They’re triggered by exercise and also, Sinclair contends, a molecule called resveratrol, found in grape skins and other plants. “It’s too coincidental that time and time again these molecules come out of nature that have the surprising multifactorial benefit of tweaking the body just the right way,” Sinclair says.

Coincidental? Pfeh. Again: we evolved together. But to be fair, the article goes on to explain that, albeit in a way unconvincing to me.

In the dance between animals and plants, says Hooper, “I think there’s true mutualism. We’re in this together, the plants and us.”

This, however, rings true.

While xenohormesis is a compelling idea, it remains unproven.

As I noted above, this is from 2014. I suppose more studies have been done since then. I'd be curious to see the results, but I can't be arsed to look for scientific papers that I'm certain will contradict each other.

That critique speaks to a larger problem: It’s often unclear how lab research on simple organisms or cell cultures will translate, if at all, into recommendations or therapies for genetically complex, free-living humans.

One thing I'll say in favor of this article: it's pretty clear what's hypothesis and what's established. The whole reason I linked this is to point out that just because articles like this exist, doesn't mean we should take them at face value. Still, it's an interesting article... so here it is.

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