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

June 14, 2023 at 8:39am

#1051050

Mythinformation

Not even two weeks ago, I mentioned the writer Cormac McCarthy in a quoted passage. Now I hear he's passed on. So I've been racking my brain to come up with another name to drop here, to test my newfound powers. Sadly—or not, depending on your perspective—I can't think of anyone I'd actively wish death upon. Not even that politician you're thinking of right now.

As Clarence Darrow (not Mark Twain)

once noted, "I have never killed any one, but I have read some obituary notices with great satisfaction."

To clarify, I didn't hate McCarthy (at least not that McCarthy), but he had little impact on my life. I never read his books or, to my recollection, seen any of the movie adaptations. So I had no reason to wish him dead, or to mourn his passing any more than that of any other individual. But I admit to great satisfaction upon hearing of the death of Pat Robertson last week.

(Segué!) Today's article, from Cracked, has nothing to do with death, but does touch on the subject of commonly misattributed quotes.

5 Famous Stories About People From History That Are Really Just Myths

When you were young, you learned about how George Washington cut down a cherry tree, Ben Franklin flew a kite in a lightning storm and an apple hit Isaac Newton on the head.

True or not (they are not, except maybe, to some extent, the one about Ben), those are "myths" in the original sense: foundational stories, kin to, but much more recent than, stories about Olympians or Asgardians. They reveal more about us than about the original subjects.

You cursed your kindergarten teacher for feeding you those myths and gleefully accepted a new teacher — the internet.

For me, that took a while, as the internet as we know it was invented maybe 20 years after I was in kindergarten. But obviously the joke here is that there's more misinformation on the internet than in a whole school full of urban-legend-spreading children.

5. Myth: Marie Curie’s Books Are Now Stored in Lead

The Story: ...Her notebooks will be radioactive for another 1,000 years, and the French National Library (Bibliothèque Nationale de France, or BNF) stores them in lead boxes.

Setting aside that we know from the setup that this is a myth, it's not like radioactivity happens for a set amount of time and then suddenly disappears. Almost everything is radioactive to some degree, and the ionization fades over time. The question is when it's safe enough to handle, and for how long.

We were unable to visit Paris this week, but we reached out to the Bibliothèque Nationale de France. They asked exactly which publication we represented. The answer must not have impressed them because upon hearing it, they ceased replying.

Probably with a muttered "putains américains."

Vairon found the lead idea improbable, but he dutifully checked in with BNF, who confirmed that, no, the books are not stored in a lead-lined container, just in plastic, and they’re kept with the other precious archives.

Coincidentally, that's exactly how my comic books are stored.

Other people may be getting the fact from Wikipedia, which cites 2005’s A Short History of Nearly Everything by Bill Bryson, who is best known for writing funny travel books. Here, the name does give you a good idea of what it is: It’s a pop-sci book with too great a scope to be an authoritative source on anything. You shouldn’t cite it, for the same reason that you shouldn’t cite Cracked listicles or Wikipedia; you should look to their sources, if they have any.

Which is what I've been saying.

Yes, I do link to Wikipedia on occasion, but it's not like I'm writing scholarly articles here.

As for Bill Bryson, I seem to be the only person on the planet who doesn't find him especially funny.

4. Ernest Hemingway Never Wrote ‘For Sale: Baby Shoes, Never Worn’

In fairness, this does sound like something Hemingway could have written. But as the article notes, he did not (or if he did, he plagiarized it). See also my comment above regarding a quote misattributed to the actual American humorist, Mark Twain.

3. Alfred Nobel’s Inspiration for the Nobel Prize

Well, you see, Nobel lived next to a church and got tired of the bells ringing when he was trying to take a nap. So when someone finally removed the bell's clapper, Alfred awarded that person the first No-Bell Peace Prize.

Yes, I just made that up. Fortunately, we're in no danger of that joke going viral.

Anyway, to summarize this section, people think Nobel founded the Prizes in a fit of remorse after inventing something that blows shit up, but that turns out not to be the case.

2. We Have No Proof Alan Turing Killed Himself

The Story: Alan Turing saved the world by cracking the Nazis’ Enigma code, and we did him dirty in return. After the war, Britain prosecuted him for homosexuality and forced him into chemical castration. That’s a terrible way to live. Turing ended up killing himself, in a fairy-tale manner: He injected cyanide into an apple then took a bite from the poison apple and died.

As the article notes, there is no proof of this. But as apples have figured prominently in many much older myths and stories (see the above bit about Newton, e.g.), it's easy to understand how this story would have staying power.

1. Franz Ferdinand’s Assassin Never Stopped for a Sandwich

This one is maybe a bit obscure unless you're a WWI history buff. Most everyone knows about the assassination itself, but the sandwich detail is the possibly obscure part. Regardless of details, though, we humans love a neat, tidy origin story. WWI started with the assassination of Ferdinand. The Trojan War happened because Helen batted her long, silky eyelashes at some Greek dude. The American Revolution started when some colonists polluted Boston Harbor. WWII was a direct result of the Germans bombing a different Harbor. That sort of thing. In reality, wars usually have complex causes that take actual work to tease out.

The Story: Gavrilo Princip managed to shoot Ferdinand on June 28, 1914, thanks to a sandwich. Earlier plans to assassinate the guy didn’t go so well, so Princip ducked into a café for a quick bite. By amazing coincidence, Ferdinand’s revised motorcade route passed right outside this café. Princip dashed outside and took his shot, resulting in World War I, and all of the world history that followed.

As the article also notes while dispelling this enduring myth:

Nevermind whether, if Princip missed that shot, the group backing him would have pulled off the assassination on some other day, or whether some other event would have sent all the dominoes tumbling regardless.

I just have to say one more thing about this, to debunk another myth:

The sandwich was invented in the 18th century, and by 1914, plenty of people in different parts of the world ate sandwiches, but not in Sarajevo.

The sandwich was not invented in the 18th century. The idea of putting other food into bread and eating it together goes back at least to Roman times, and probably earlier. Hell, the sage Hillel reportedly had the idea back in the first century BCE, and I seriously doubt he invented it. That Earl of Sandwich story? Great story, and maybe it's even true (though I'm somewhat disappointed that it was the Earl of Sandwich and not Lord Penistone, which would have given great humor to the question, "Is a hot dog a penistone?")

Sliced bread, as we know it, wasn't invented until the 20th century, as I noted in "The Greatest Thing Since..."

. Before then, people had to actually use a knife or break the bread with their hands. Saying that Montagu, Earl of Sandwich, invented the food is a bit like saying that Henry Ford invented the automobile—it's more like Ford invented the assembly line that made cars cheaper, thereby popularizing them.

We all have to watch out for misinformation, especially when it sounds like it could be true. Such as the false definition of the Blue Moon, which I'm bringing up again (I hear you groaning) because we're scheduled to have another false Blue Moon in this coming August; the actual next Blue Moon is in August of next year.

I'm not entirely immune to myth, myself. But at least I try to switch to the facts whenever I find them.

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