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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Write_Mikey_Write!

about 8 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.

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Adherennium Plotting something

about 10 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

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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.

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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.

June 21, 2021 at 12:01am

#1012247

Count Me In

Entry #6 of 8 for "Journalistic Intentions"

[18+].

In three words I can sum up everything I learned about life: Counting is easier than it looks. -Pumpkin Spice Sox

We don't have just one number system.

You probably learned, long ago, about the difference between cardinal and ordinal numbers. If you need a refresher, I found this

Basically, cardinal numbers are for quantity; ordinals are for position.

From that article: "We use ordinal numbers for dates and the order of something..."

Well, I'm pretty sure if I used ordinal numbers for dates, the dates wouldn't last very long "You're the sixth woman I've seen this week" tends to be a "Check, please!" moment. Especially if I say it on a Tuesday morning.

Okay, fine, that's a pun and one that's probably beneath me, since I haven't been on a date in mumble mumble mumble.

That's English, though. French has a really weird counting system. Well, it's weird to me; I'm sure our way of counting is weird to Francophones. For example, we say today is June twenty-first. In French, you'd say le vingt-et-un juin, which, literally, would mean "the twenty and one June," which sounds more like a cardinal number; their ordinals are more of the form première, seconde, troisième, quatrième, cinquième, etc.

This is, of course, not the weirdest thing about French numbers. They don't have a special word for seventy like we do. Instead it's soixante-dix or, literally, "sixty-ten." Then there's sixty-eleven, sixty-twelve... all the way up to four twenties (quatre vingts) for eighty.

People who are afraid of math would hate learning French.

And also probably reading the rest of this blog entry.

See, numbers get even weirder when you realize that our number system doesn't actually start at 1 but at 0 (yes, I'm ignoring negative numbers and a whole slew of things like rationals, irrationals, transcedentals and *ahem* complex numbers). Programmers know this; most older programming languages and some newer ones start array counting at the zeroth element. The array element indexed at 9 is therefore the 10th element.

I was thinking about numbers just the other day, which sometimes happens when I don't have distractions, like, in this case, when I was driving on a mostly-empty highway. A couple of weeks ago, my housemate showed me a package of food things (I don't remember what they were; let's say "mozzarella poppers" for the hell of it) that was labeled for two servings and contained... five mozzarella poppers.

Why five? If you're going to package something in two servings, freaking make it an even number. Three servings? Multiple of three. It's like trying to evenly split a pizza three ways; you want to cut it into six slices (or twelve, or nine which would be a nightmare). Above all, unless you're putting the food things into a single-serving package, don't make it a prime number. For fuck's sake.

This is almost as maddening as how hot dogs are sold in packs of ten, while hot dog buns are sold in packs of eight. To make it all come out even, you'd need five packs of buns and four packs of weenies. I hope you're hungry or have lots of friends.

In an attempt to solve that age-old conundrum, I found this site

which explains it in great detail, but for the love of sausage, do not click on the embedded video there that shows how franks are made. It's almost enough to turn me into a vegetarian. Well, no, not really.

Still, it's important to know these things, as well as knowing the sound of one hand clapping, and the mystery surrounding trees falling in a forest (I can't demonstrate the former online, but the answer to the latter question is: sound is a pressure wave through air, and it's independent of whether there's someone around to hear it or not, so yes, a tree falling in the forest indeed makes a noise).

As an aside, I also know why we park in driveways and drive on parkways, but that's beyond our scope here because we were talking about numbers.

I was saying that we don't have just one number system. There are, potentially, an infinite number of them. Computers, as you know, do everything in binary, or base-2. So you have 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010 and so on. Often those are collected into groups to make base-8 or base-16 (but for some reason, base-4 isn't a thing in computer science, though it's just as legitimate a base as anything else). Some people have proposed trinary computers; since the word for "binary digit" is "bit," I think I'll leave the word for "trinary digit" up to your imagination. (Just don't give it a byte, unless, you know, she's into that sort of thing.) Ancient Sumerian numbering systems were essentially base-12, not base-10, which is why our clocks aren't decimal even if they're digital. The point is, you can take any positive integer and use it as a base for counting, because all it does is tell you when to tack on another digit place, like when an odometer rolls over from 99,999 to 100,000.

How you'd symbolize such numbers once you run out of 0-9 and all 26 letters (base-16, for example, contains the numbers A for decimal 10, B for decimal 11, and so on up to F for decimal 15, at which point you finally get to hex 10 which is decimal 16), I'll leave up to more creative people. I'd suggest starting with those symbols above the decimal numerals on a computer keyboard, so a lot of numbers could look like cartoon cuss words, like #%&@!.

The point being that base 10 is completely arbitrary, and it may seem natural because most of us (at least those of us who haven't played with fireworks or joined the Yakuza) have ten fingers, tell that to the Sumerians, or to the people who run around barefoot and use base-20.

Incidentally, I long ago learned a trick for counting up to 31 by using just the fingers of one hand. It involves treating each finger as a different binary digit.

What can I tell you? Such things help me get to sleep (those sheep won't count themselves) and it's useful for counting cards in blackjack. Not, of course, that I would ever do such a thing, because I would like to keep said fingers. My system can, of course, be extended up to 1023 if one were to use both hands, but blackjack card counting never goes that high.

And yet, the only number system wherein 3 is equal to 6 is in the Comedy Number System, where pretty much anything goes as long as it elicits at least a chuckle.

What three words (not six) can sum up everything I learned about life? "Comedy breaks everything."

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