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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 5, 2021 at 12:02am

#1018723

Lyin's Mane

Entry #2 for "Journalistic Intentions"

[18+]

Lion's Mane

Probably for as long as there have been people, it's been in our nature to classify things. I'm sure someone could make up a bullshit evolutionary biology reason for it ("it was important for our ancestors to quickly tell a lion from an antelope" or something), but whatever the reason, we have a need to sort things into groups of like objects.

No one is more obsessed with classification than scientists. Thing is, though, some things aren't so easy to group. Take rocks for example. You have boulders, stones, rocks, pebbles, and sand, among other names (those being generally in descending order of size). But reality doesn't always provide neat, well-defined boundaries. At what size does sand become pebbles? When do pebbles become stones? This is further complicated by the irregular shape of some hunks of mineral. Usually, it doesn't matter for everyday purposes, but rock sizes exist on a continuum. The difference between a stone and a boulder probably has a defined cutoff, but that cutoff is purely arbitrary.

People are still sore about the "demotion" of Pluto from planet to dwarf planet. (Get over it already; whatever we call it, it's still there.) Orbiting bodies, too, exist on a continuum, and if we're going to define them, there has to be some minimum size or other criteria for when to call something a "planet" or "dwarf planet" or "moon" or "asteroid" or "dust." Size alone won't do it, because there are moons in our solar system bigger than Mercury, which is defined as a planet.

But we were talking about Lion's Mane. Lion's Mane is, by all classification systems I'm aware of, a mushroom. All mushrooms are fungi; not all fungi are mushrooms. Biology is absolutely fucking intent on classification, so you get the practice of taxonomy. I don't pretend to know everything about it, but in the early days of taxonomy, before we were able to do cool things like sequence genomes, and even before Darwin changed the way we looked at life on Earth, every known species (which has a particular definition that's itself kind of fuzzy around the edges) got lumped into groups, subgroups, sub-subgroups, etc. At the top of that list were, at first, plants and animals -- two "kingdoms" that were pretty easy to tell apart: if it moves, it's an animal; if it doesn't, it's a plant.

It didn't take long to find a) plants that moved b) animals that don't and c) other realms of life. But for a long time, fungi were lumped in with plants because they didn't move.

Turns out (spoiler) that fungi are much closer, genetically, to animals than they are to plants, and are now considered to be a kingdom of their own. Long, long ago in the course of evolution, after animals split off from plants, fungi split off from the line that would lead to animals and, ultimately, us. This has been determined through genetic analysis.

Keep that in mind next time you see a vegan happily chomping on a portabella. "You know, that's closer to animal than plant." "Yeah, whatever, it's not cute. But it is very, very tasty."

And mushrooms can, indeed, be very, very tasty. At least according to me. But it's also well-known that some mushrooms have... other effects besides nutrition. And still others will kill you dead. So, unlike the question of "Is Pluto a planet," the field of mycology (the word comes from the Greek word for fungus, which itself is Latin, so there's another classification thing going on here) has a direct impact on everyday life, at least if you're going to eat shrooms.

But, as with most classification systems, things aren't always black-and-white, live-or-die. There's a good bit in between. Some mushrooms are definitely safe, like creminis. Others, like the death cap... well, it's called death cap for a reason. Others fall somewhere in the middle; as they say, the dose makes the poison.

Incidentally, I've grown to dislike the word "edible." Not only has it been hijacked by the cannabis industry, thus muddying its definition (I have no problem with the cannabis industry; it's just that, like "organic," words are supposed to mean something), but the simple truth is that everything is edible at least once. What I want to know is, will I live to eat it again? Or, will I wish I hadn't because my insides are becoming my outsides for a couple of days?

Where does Lion's Mane fall on this spectrum? Well, for that we have to look at the article

linked in the prompt, or the much better organized Wikipedia page

. Okay, looks safe enough to eat, though I'm not sure about having a hairy thing on my plate. But both sources talk about its medicinal qualities, and that's where my inner skeptic kicks at my peritoneum.

Look at the language used in the first article.

"Research suggests..." weasel

"...including reduced inflammation..." any time I see the word "inflammation" my mind goes, "quack."

"...both the fungus and its extracts appear to be beneficial to health." weasel

"Lion’s mane mushrooms may help..." and I "may" have a beer later.

"Lion’s mane mushrooms may enhance the immune system..." Or it may not.

"Research on mice suggests that lion’s mane mushrooms may boost the activity of the intestinal immune system." in rodents.

"Extracts from lion’s mane mushrooms may be beneficial in the treatment of anxiety and depression." Or they may not.

"In a 2015 study, mice that consumed lion’s mane mushroom extract..." were mice that consumed lion's mane extract.

"In a small Japanese study, women with a variety of health complaints... ate cookies containing lion’s mane extracts or placebo cookies for 4 weeks." Yeah, I looked at the study. 30 women. Four weeks. Pass.

"It is possible that lion’s mane mushrooms might boost cognitive function, but the existing research is mainly on animals." Do I have to spell it out again? Starts with W. Ends with L.

I could go on, but I won't. Just because something is "traditional medicine" doesn't mean it works. It also doesn't mean it doesn't work. People used to chew willow bark for pain relief. Scientists did studies on it, and behold, aspirin was invented. (This "natural" remedy is not without side effects, though.) So it's not outside the realm of possibility, but dammit, study it! Preferably using more than 30 people from a single demographic for more than 4 weeks.

The Wikipedia link sums it up more succinctly: "There is no high-quality evidence from clinical research to indicate that lion's mane mushroom has medicinal properties."

It also notes that the German name for the mushroom translates to "hedgehog goatee," which, I mean, honestly, is a lot funnier than lion's mane.

But whatever. It's apparently nonpoisonous in at least moderate quantities, so if you like it, eat it. Me? I can't get past the hedgehog goatee thing.

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