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 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.

1 Like

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.

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.

July 2, 2020 at 1:35am

#987000

East is West

The 30DBC prompt hasn't been posted yet [Edit: it's at the bottom now], but I'm on a blogging roll, so...

https://aeon.co/essays/the-logic-of-buddhist-philosophy-goes-beyond-simple-truth...

Beyond true and false
Buddhist philosophy is full of contradictions. Now modern logic is learning why that might be a good thing

The sentence you're reading right now is false.

When Western philosophers look East, they find things they do not understand – not least the fact that the Asian traditions seem to accept, and even endorse, contradictions.

Common mistake. No one understands philosophy, east or west. Anyone who says they do is just trying to sound impressive.

An abhorrence of contradiction has been high orthodoxy in the West for more than 2,000 years.

Hence the prevalence of statements of paradox, most of which are a result of semantics.

Let’s start by turning back the clock. It is India in the fifth century BCE, the age of the historical Buddha, and a rather peculiar principle of reasoning appears to be in general use.

If the Buddha provided the foundations for Eastern philosophy, Pythagoras can be said to do the same for Western. I found out once that they were roughly contemporaries. I've already used this as a basis for a novel.

At around the same time, 5,000km to the west in Ancient Athens, Aristotle was laying the foundations of Western logic along very different lines.

Aristotle actually came along a bit later, and his "contributions" to Western thought can best be described as holding it back for nearly 2000 years.

Unfortunately, Aristotle’s own arguments are somewhat tortured – to put it mildly – and modern scholars find it difficult even to say what they are supposed to be.

Which is one reason science didn't progress as it could have.

The article goes on to express, in formal terms, how to describe something as "both true and false" and "neither true nor false," but I'll skip that part.

Even so, you might be wondering how on earth something could be both true and false, or neither true nor false. In fact, the idea that some claims are neither true nor false is a very old one in Western philosophy. None other than Aristotle himself argued for one kind of example.

And so, after arguing against the possibility of contradiction, Aristotle contradicts himself.

The notion that some things might be both true and false is much more unorthodox. But here, too, we can find some plausible examples. Take the notorious ‘paradoxes of self-reference’, the oldest of which, reputedly discovered by Eubulides in the fourth century BCE, is called the Liar Paradox. Here’s its commonest expression:

This statement is false.

Where’s the paradox? If the statement is true, then it is indeed false. But if it is false, well, then it is true. So it seems to be both true and false.

This is the one I cribbed to start this commentary. As I mentioned above, most paradoxes are actually just semantic problems. That's one of them. It's only a paradox if you believe that everything is either true, or it's false. But the very fact that we can construct such sentences is evidence that some things are both, or neither.

Nagarjuna’s writings defined the new version of Buddhism that was emerging at the time: Mahayana. Central to his teachings is the view that things are ‘empty’ (sunya). This does not mean that they are non-existent; only that they are what they are because of how they relate to other things.

I was thinking along those lines myself, a few months ago, before I found this article. To take a very mundane example: what is a hole? A hole isn't a "thing," as we expect most nouns to be. It is, by definition, a not-thing. A hole in the ground is not-ground. The hole in a donut is not-donut. (Mmmmm... donut.) I can't just point at a spot in space and say "this is a hole in a bagel."

If something is ineffable, i, it is certainly neither true nor false.

To which I reply: Eff the ineffable.

Now, you can’t explain why something is ineffable without talking about it. That’s a plain contradiction: talking of the ineffable.

Again: semantics.

The similarities between this and our Buddhist paradox of ineffability are, you must admit, pretty unnerving. But those who developed plurivalent logic were entirely unaware of any Buddhist connections. (I say this with authority, since I was one of them.) Once again, the strange claims of our Buddhist philosophers fall into precise mathematical place.

Can't be arsed to look up who said it: "East is East, and West is West, and never the twain shall meet." Demonstrably false.

I'll add to this entry later, after the prompt shows up. Probably after I sleep, too. Yeah, I slept earlier, for five hours, but that was more of a passing-out than actual sleeping. Gin has that effect. Worth it.

PROMPT July 2nd

We all know the mood-ring fad of the 70’s only predicted body temperature and not necessarily mood, but what if people could actually see your emotions, like an aura of color surrounding you. Would you try to mask it, display it proudly, or something in between?

I tried to find a way to relate this response to the prompt to the title of the entry, but I'm not at my best at 10:30 in the morning. Which is why I post much, much earlier. So this will be kind of a nonsequitur.

I'd find or invent some technology to fake it. You know, kind of like people do when they post on Instagram. "Look at my shiny happy life! I'm so lucky! Here's a pic of me in Belize! Here's another one kitesurfing the Mediterranean!" Reality: "I'm broke after renting these photoshoot backgrounds, I'm about to get a divorce, and my kids hate me."

Also imagine playing poker. It'd be like dogs playing poker. Not the classic epic velvet painting, but, you know, actual dogs playing poker. "Thump thump thump." "I fold. Larry's tail says he's got a full house."

I mean, in reality, people are pretty good at seeing my emotions anyway, which is why I prefer blackjack over poker. And it gets really annoying really fast. "You don't look happy! Are you okay?" "Really, I'm fine." "Let me cheer you up!" "No, really, I'm fine." "Are you sure you're okay? Really sure? Like, sure-sure?" "WELL NOW I'M ANNOYED AND DEPRESSED, THANKS!"

And then I slink off alone to listen to Brandi Carlile and/or Leonard Cohen until I feel better, which I suppose is a bit like cutting yourself deeper when you have an injury, but for whatever reason that works for me.

And the only thing worse than that are the people who, when I am happy, decide that it's the perfect time to deliver bad news. "You look happy, Waltz." "I am happy!" "Great. By the way, I'm going to date other men now. Bye!"

You know what else was a big fad in the 70s?

Pet rocks.

Some guy in the 70s had the brilliant idea to find rocks (cost: free), nest them in shredded paper (cost: $0.01, or close to it), put each in a cardboard box (cost: I dunno? Five cents?), and sell them for four bucks. He became a millionaire doing this, which was a much bigger deal in the 70s than it is now, and ended up opening a bar called Carry Nation, for which I do have to admire him, as it's a punch in the face to the most evil woman in history (and considering that history includes Margaret Thatcher and Elizabeth Bathory, that's saying a lot).

How did he pull that off? Marketing. You can sell anything with the proper marketing. Well, not you, and not me, but someone can. Smart cars. Hello Kitty. Bottled water. Chicken wings. "Hey, let's take the part of the chicken that we've been using for dog food, dip it in spices, and sell it for a 1000% markup!"

Mood rings at least had the distinction of having fashionable settings, at least by the questionable standards of the 1970s. But hey, fifty years from now, the survivors in the wasteland will find our Instagram posts and laugh at our hairstyles and what we were wearing.

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