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!

4 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.

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.

May 1, 2024 at 7:51am

#1070355

Discovery

Despite my well-known appreciation for Star Trek, the title of this entry has nothing to do with the series with that name, or its namesake starship. Nor does it have anything to do with the cable channel or Land Rover model. It is, in fact, misleading, as today's article, from Cracked, is all about...

5 Renowned Pioneers Who Didn’t Discover Shit

The problem with history is that someone’s gotta write it. At least before recording technology and a way to preserve those recordings, getting data down required going through the natural bias of at least one human brain.

Even with recording technology, one often wonders. Like how cops' bodycams always seem to malfunction right before a suspect punches himself in the kidneys.

Here are five pioneers who get credit for things they didn’t really discover…

As usual, I'm just going to touch on some highlights.

5. Christopher Columbus

Perhaps the most obvious and well-known individual in this category.

The premier poster child of false pioneers, and one that got the capital city of a world superpower named after him in the process: Christopher Columbus.

And then there's the city in Ohio, which, because of issues with the dude's legacy, some people are trying to get renamed. To Flavortown, in honor of native son Guy Fieri.

This is not an improvement. Don't they have an astronaut or something they can rename to?

Immediately and obviously, it’s difficult to be the first person to discover a country that’s already fully occupied by a culture and civilization.

Unless, of course, your definition of "culture and civilization" is exclusively "15th century Europe," which was apparently the attitude of 15th century Europeans.

Even if you do want to give credit for crashing into the right rock, you’d still have to hand it to Leif Erikson, who landed in North America all the way back around the year 1000 A.D. He even named it, calling it “Vinland” or the “land of wine,” a name that seemingly, nobody bothered to put down on any maps, at least the ones that Columbus would later use.

I'd totally live in the USV. Hell, these days, I think I do, what with the proliferation of wineries in every state (yes, even Alaska). And every province of Canada. I didn't check on Mexico, but who needs wine when you have tequila?

It's entirely possible that the French heard about this "Vinland" thing and did their best to erase it from history in an attempt to keep their monopoly on spoiled grape juice.

4. Captain James Cook

Speaking of white sailors that hog credit, let’s take a look at the man given popular recognition for the “discovery” of Australia, Captain James Cook.

His name always bugged me, anyway. Are you a captain, or a cook? Only Christopher Pike gets to be both.

(I had to get a Star Trek reference in somewhere, right?)

Obviously, again, Australia was already populated by humans, as the article points out. Then, something I wasn't aware of:

Unlike Columbus, this is much wider than a 1v1, two-explorer debate. By the time of Cook’s trip in 1770, fully double digit European landings on Australia were already recorded. Beyond that, there may well have been visits by Portuguese sailors and Chinese fleets in the preceding centuries.

There is one continent that we know was "discovered" in recorded history and that's Antarctica. Even this is disputed, as the Maori claim to have landed there in the more distant past. This claim is unsupported by evidence, as I understand it. But apparently, Cook almost discovered the place, but turned away at the slightest mention of icebergs, making him at least smarter than a certain other captain on the other side of the world a couple of centuries later.

Incidentally, Australia also makes some kick-ass wine.

3. Frank Benford

Let’s flip to a different textbook: mathematics.

And we both just lost half our readers.

There is a law, known as Benford’s law, which states that the first digit in any real-world number, regardless of size, is disproportionately likely to be a low number like 1 or 2.

It's also not really a "law," in either commonly used sense of the word.

Why is this true? I have no earthly idea.

As I understand it, it's because real-world numbers tend to count objects, like sheep or dollars. These don't tend to show up with equal probability. For instance, you're more likely to have 100-199 sheep than you are to have 900 of them. You're more likely to have a net worth of $100,000 than $900,000 (or $-10,000 than $-50,000). (You can't have negative sheep, even in Australia.) Lots of things cost in the $100 range; fewer things cost in the $200 range, even fewer in the $300s, etc.

To be fair, neither did Frank Benford, the man whose name graces this law. So if he didn’t explain it, he must have received the honor for discovering it, right? Weirdly, also no. It was discovered, and written about, by an astronomer named Simon Newcomb in 1881.

To be even more fair, astronomers tend to be recognized for discovering, I don't know, stars or galaxies or whatever, and not number theory statistics.

Incidentally, this "law" is used in forensic accounting. People who cook (pun intended; see #4 above) their books tend to put in semi-random numbers, and you get enough of these and they violate the... whoever's "law" it is.

2. Elvis Presley

Him and Buddy Holly are often plopped onto Rock’s Mount Rushmore, and sure, they’re significant artists in the genre, but pretending they came up with it themselves is a whitewashing job Tom Sawyer would be proud of.

"He and Buddy Holly." Sheesh, editors.

Unfortunately, the general populace preferred to ignore the obvious and clear Black influence on his music.

Still, I have to question: is a musical genre "discovered" or "invented?" If one takes the Platonic view of the world, everything is discovered, and nothing is invented. Plato invented, I mean discovered, that philosophy. Probably.

1. Edmund Halley

Hey, look, an actual astronomer on the list.

The fact is, the comet we know as Halley’s was observed and recorded long before he was around. His great discovery was that the comet orbited the sun, making its return consistent and predictable. People had seen Halley’s comet before, they just didn’t realize it was the same one over and over.

Splitting hairs, if you ask me. It's still a worthy discovery. That's kind of like saying that whoever figured out that the sun gets its sun-like qualities from the fusion of hydrogen into helium didn't do shit, because we kind of suspected the sun was bright all along. (Okay, I looked it up; it was actually a bunch of people working off Einstein's inventions. I mean discoveries.)

In any case, while Halley didn't "discover" the comet named after him, he did make discoveries, like how stars are actually moving around. Which was a bigger deal than it sounds like now, because it was a watershed moment between "the stars are fixed for all time" and "holy shit, those suckers are moving." And thus, he contributed to our better understanding of the universe.

Unlike, say, Columbus who, like Blackadder II, discovered bugger-all.

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