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:

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.

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.

1 Like

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').

1 Like

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.

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

1 Like

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 18, 2020 at 12:02am

#996159

Animal, Vegetable, Mineral

Cribbing from Cracked again today. Because I haven't done one in a while, let's make this a Merit Badge Mini-Contest! (Details below.)

5 Crazypants Scientific Discoveries (Happening Right Now)

Now, keeping in mind that Cracked is, first and foremost, comedy, I wouldn't take any of these as the last word on the subject. Those of us who try to be funny would sacrifice the truth for a good laugh at the drop of a hat. Do people still drop hats? Hell, do people still wear hats? Other than those stupid-looking backwards baseball caps, which don't count.

Chances are you're not keeping up with science news these days, what with the world being four different kinds of on fire.

I'm keeping up with science news because the world is fourfive different kinds of on fire. After all, the only way we're getting out of any of this is science. Maybe also voting. But mostly science.

And even if you tried, 90% of it now boils down to headlines saying: "There's still no vaccine, put your goddamn mask back on."

Still, this bears repeating.

5. Dinosaurs Had So Much Cancer

Despite Spielberg trying to drill it into our brains every few years, it's hard to fathom dinosaurs really existed. That they were actual flesh and blood critters instead of the mythic tar-breathing dragons we like to imagine them as.

Interesting you should mention dragons. My pet theory, which is mine and isn't actually a theory but a hypothesis I'd look further into if I weren't so damn lazy, is that dinosaurs were dragons.

Well, not exactly. Okay, bear with me here: you're a pre-scientific cave-dwelling humanoid, wandering through a wilderness of everything unknown, inexplicable, and potentially carnivorous. Kind of like Australia today. One day, you kick over a clod of dirt and uncover a bone.

Not just any bone, mind you. A big bone. You've seen bones before, of course; you've probably cracked them for the marrow. Some of them are quite large, like the ones the woolly mammoths leave behind after you've munched on their soft parts. But this particular bone is not only bigger than any that comes from a mammoth, but bigger than your shaggy frame.

How do you explain this, as you've never seen a creature that could leave a bone so large? Well, you come up with the idea of a secretive, giant creature, and you take this story home with you, along with the bone to provide incontrovertible evidence that such a creature must exist. You have no idea of evolution, or deep time, or maybe even the entire concept of the possibility of a species going extinct everywhere.

Over time, the legend grows, takes wing, maybe spits fire, and becomes what we now know as dragons.

While this is not inconceivable as an origin story, keep in mind that I have absolutely no evidence for it. Like I said: pet hypothesis. It's just as likely that it's entirely a figment of our messy collective imagination, which doesn't explain to my satisfaction why people as disparate as East Asians and Western Europeans both had dragons in their mythology, albeit of entirely different mien.

Anyway.

The Cretaceous cancer patient in question was an unfortunate Ceratops (an ancestor of the Triceratops named after the deathless ancient that is Michael Cera). While the beast likely drowned with its entire family before the cancer could kill it (a mixed blessing) it suffered from such aggressive osteosarcoma that this bone cancer had visibly warped its lower leg bone -- which is how it came to the attention of the oncologist researchers.

While it's good, in a sense, to have confirmation of these things, it should really come as no surprise to anyone. Cancer, in my admittedly very rudimentary understanding of the subject, is the result of errors in cellular replication, errors that exceed the body's natural powers to fix. Thus, any living thing has the potential to contract cancer. If this happens after the organism has already reproduced, there's no evolutionary tendency to wash it out of the gene pool.

What's amazing isn't that dinosaurs had cancer, but that some of them lived long enough, what with all the predators, tar pits, time travelers, asteroids and whatever, to develop it in the first place.

4. Gardens Are Being Overrun By Superpowered Poison Ivy

Thanks to climate change, poison ivy has become the poster girl of (post)apocalyptic prosperity, having doubled in size since the 1950s. And much like the world's shittiest Pokemon, Dr. Jacqueline Mohan of the University of Georgia warns, this poison ivy has also evolved to be "significantly more poisonous."

You know, once, just once, I want to hear of a good thing that comes from climate change. Like, maybe, no more ice storms in Virginia. That would be nice. Well, up to a point, anyway.

Anyway, what, exactly, has doubled in size? Its range? Leaf size? Individual plant size? The link at the article doesn't make it much more clear than the Cracked article itself.

I guess it doesn't matter. Poison ivy sucks. (Harley Quinn is a much more interesting villain.)

Over the past decades, the weed has been given a CO2 boost of 149%, making them bigger, stronger, and faster growing.

My backyard is a testament to that. Oh well. Maybe it'll keep the burglars away. It certainly keeps me from doing yardwork.

So the next time some climate change denier starts their bullshit about nature finding a way to cope, feel free to use that as an invitation to rub a gloved handful of super poison ivy into their dumb faces.

They're probably not wearing a mask, so this should be easy.

3. Florida Fights Its Mosquito Problem By Releasing 750 Million More Mosquitos

I know we here can be a tad harsh on Florida, that unnoticed meth-head booger hanging from America.

I have... more colorful, pus-filled anatomical metaphors.

Since we're still decades away from creating a continent-sized bug zapper, many scientists have accepted that we are unable to wipe out the mosquito population. But you know who's even better at wiping out populations? Mosquitos. So under the maxim "if you can't kill 'em, breed 'em" biotech corporations have been creating clouds of genetically modified male mosquitoes.

At least one semester of reading, analyzing, and writing papers about science fiction/horror books and/or movies should be required for anyone pursuing so much as a bachelor of science degree. Preferably two semesters. Otherwise, those things that we ordinary people take for granted as a Bad Idea, like unearthing mummies during a pandemic, or freakin' genetically engineering freakin' mosquitoes, will just keep happening.

2. Using Mindfulness Meditation To Cure Depression Can Cause Depression

DUH. "When I think about how I'm in an existential void, I'm sure it will help to concentrate on how void-y my existence is."

Mindfulness is the latest mental health fad that loses a lot of its medicinal cred if you just add "whoa" and "like" to its description. According to Psychology Today, mindfulness is, like, to live in the moment and, like, reawaken oneself to the present, whoah, rather than dwelling on the past or, like, anticipating the future *mind blown gesture.*

In other words, try to shed the third-most important thing that makes us human (after comedy and Oreos). Yeah, I'm not a fan of mindfulness. I've mentioned this before.

That kind of ongoing pose-posing is downright dangerous for these one out of twelve meditation practitioners. When they start experiencing the supposedly non-existent side-effects of meditation, they have no idea what's going on. This can make their suffering even worse, both by them now clueless on how to address mental issues that clearly require medical treatment but also by making them feel like they're somehow doing sitting down and not falling asleep wrong.

Fortunately, I haven't seen too many "you're doing it wrong" articles on the internet lately. They used to crop up in my article-searching on a regular basis. Maybe there's finally been enough backlash? Or maybe there's just as many, but the Great Algorithm has finally realized I won't read them. At its nadir, I would not have been one bit surprised to see a "you're falling asleep wrong!" headline, which I would have proceeded to shred here in this blog. (The sitting down thing is... well. Lots of us sit wrong.)

1. We Finally Know Where Stonehenge Got Its Stones

And, sorry, History Channel fans, it's not Procyon 5.

The mystery of the rolling stones was finally uncovered by a team of historians from the University of Brighton. Using a portable X-ray gun (how's that for living in the future?) they managed to analyze the sandstone's unique chemical composition.

Nope, not really living in the future until I have a ray gun, a jet pack, and a flying car. And maybe a robot maid. Preferably a sexy one.

And while the team never dreamed of matching them to a nearby quarry, they did exactly that, tracing 50 out of 52 boulders to a site in the nearby West Woods of Wiltshire.

The article goes on to quote both 25 miles and 35 kilometers as the distance the stones were carted. I'm not completely unfamiliar with either unit of measurement, though, and I'd tend to think that the units were simply interchanged, but 25 miles is more like 40km, so... well, like I said, don't fully trust comedy sites for accurate science. If you care, pull up Google Maps. They have a neat-o linear measuring function now.

But remember, this was 3000 BC Britain. A time when the known world ended forty feet past the nearest tree line and a place where traveling straight for 5 miles meant you no longer understood people's dialect (not unlike contemporary Britain).

Just quoting this because I laughed my ass off. Since it's true.

But the distance doesn't matter. To pull forward monoliths the size and weight of the moon (you assume, since you're from the Bronze Age) alone took ingenious levels of engineering.

Nah, you just get a dragon to carry it for you.

Merit Badge Mini-Contest!

What's your favorite cool (or, as with the genetically engineered mutant skeeters or world-devouring poison ivy, potentially apocalyptic) scientific discovery, breakthrough, achievement, or whatever? I don't care if it's a recent one, as with the above, or an older one (though, fair warning, "harnessing fire" ain't gonna cut it). As always, you have until midnight WDC time. Comment here below, please. The one I like best will earn its writer an appropriate Merit Badge.

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