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 3 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 4 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.

April 10, 2021 at 12:05am

#1008142

One Weird Trick

This one's been kicking around for nearly five years, so you've probably seen some version of it by now. Still, as is often the case, I have something to say.

How non-English speakers are taught this crazy English grammar rule you know but have never heard of

Don't be too put off by the clickbaity headline.

English grammar, beloved by sticklers, is also feared by non-native speakers.

Even sticklers get it wrong sometimes. Waltz's First Law of the Internet is that any post criticizing someone's spelling, grammar, or punctuation will itself contain at least one spelling, grammar, or punctuation error.

As with many Laws, it is possible to find exceptions. But it happens often enough to make me chuckle every time I see it. "You missed a comma there. Your an idiot." That sort of thing.

Many of its idiosyncrasies can turn into traps even for the most confident users.

I have heard that English is one of the hardest languages to master. I'm not sure any of us ever really do so.

This week, for example, the BBC’s Matthew Anderson pointed out a ”rule” about the order in which adjectives have to be put in front of a noun.

"This week," of course, was five years ago now.

Adjectives, writes the author, professional stickler Mark Forsyth, “absolutely have to be in this order: opinion-size-age-shape-colour-origin-material-purpose Noun. So you can have a lovely little old rectangular green French silver whittling knife. But if you mess with that order in the slightest you’ll sound like a maniac.”

And that is where I'm going to quibble.

First of all, let's get "knife" out of that example phrase, because it's too easy to "sound like a maniac" when describing a knife. Let's stick with utensils, though, and change it to "spoon."

Since there's no such thing as a whittling spoon (except insofar as language allows me to put those two words together), let's further change the "purpose" slot to "soup." The only other questionable adjective, then, is "rectangular," because I've never heard of a rectangular spoon, but for the sake of keeping the phrase as close as possible to the original, imagine a rectangular spoon. Maybe it has rounded corners, and "rectangular" is less than precise. Doesn't matter. The point is, it's harder to sound like a crazy person when describing a spoon than it is when describing a knife.

So we have a lovely little old rectangular green French silver soup spoon. The quote above asserts that if you mess with the order, it sounds wrong. And yes, if you mix it all up at random, it probably will. But consider: You can easily have a little old rectangular green silver French soup spoon. There, I swapped "French" with "silver." Instead of sounding like you need to be put away, you've emphasized that the spoon is made of green silver (not that I've ever seen green silver, either, but it's easy enough to imagine it).

If you're having trouble with "green silver," though, pretend it's made of wood instead. "green wood French soup spoon" might actually be a better description than "green French wood soup spoon," depending on what you want to emphasize. The former emphasizes "green wood," while the latter implies that it's green and made of French wood.

My point being that, like all rules, there are exceptions.

It makes me wonder how English became so ubiquitous in the world.

Still, at least in English, we know that, in very nearly every case, adjectives precede nouns. The only exceptions I can think of are done for effect, such as "arts dramatic" instead of the more natural "dramatic arts." Point is, off the top of my head, I can't think of any situation where you'd be wrong to put every single damn stupid adjective before the noun it modifies.

From what little I've learned about French, some adjectives come before the noun, and some after. More, there are some that precede certain nouns and follow other nouns. Like, "last month" would be "le mois dernier" (literally "the month last"), while "the last book" would be "le dernier livre." (that one's easy; it's the same word order as English).

To me, French adjective positioning is harder to remember, but then, I've been speaking and reading English for most of my life, so I suppose a Francophone person would find the English adjective thing weird. Speaking of weird:

Learning rules doesn’t always work, however. Forsyth also takes issue with the rules we think we know, but which don’t actually hold true. In a lecture about grammar, he dismantles the commonly held English spelling mantra ”I before E except after C.” It’s used to help people remember how to spell words like “piece,” but, Forsyth says, there are only 44 words that follow the rule, and 923 that don’t. His prime examples? “Their,” “being,” and “eight.”

Whereas my favorite example of that rule being broken is "weird."

I suppose when it comes to spelling, the best thing to do is to memorize each word individually. I've always been fairly good at spelling, though there are words that trip me up all the time, like "commitment" and "embarrassing."

Either way ("either" being another one, and I dare you to pronounce it correctly), that's why I almost never correct someone's grammar in an Internet post. I don't want to fall victim to my own First Law.

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