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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Show Recent Comments

10 Most Recent Blog Comments:

tj wanderlust-words-in-motion

about 2 hours ago
In response to "Revisited: "Unwritten, Expanded""

I seldom use emojis in texts and spell out words, use correct (I think) punctuation; it is, after all, still writing. But, just recently I was shown a paper written by her fifth-grade daughter for English. The entire story was written in "text-language" complete with emojis (a lot even took the place of words) on a computer and printed out. The child was very proud of the B+ she had gotten on the assignment!

Jayne

about 8 hours ago
In response to "Revisited: "Unwritten, Expanded""

As someone who does not have, and will not have, TikTok: isn't that vertical by default? Does that skew the overall tolerance for vertical video? (among the masses, not for you 😂)

Stik: I'd agree those get a pass. I've had to do it in an emergency, and you really don't think about turning your phone, especially if you don't shoot a lot of video.

2 Like

Elisa: Snowman Stik

about 8 hours ago
In response to "Revisited: "Unwritten, Expanded""

"If I see a vertical video, I ignore it."

Does this apply to videos that call out harmful/discriminatory behavior? I'm willing to give those videos a pass if they're vertical.

2 Like

tj wanderlust-words-in-motion

a day ago
In response to "Let It Lie"

I cannot say honesty is always the best policy, I write fiction for Pete's sake, or whoever else reads and enjoys it. However, in questions such as, "Does this dress make my butt look big?"

I can honestly answer with, "No." After all, it's not the dress. As for the question, "Is there a Santa Claus?"

"Hell Yeah, I saw him at the mall."

2 Like

Adherennium - Maybe Writing?

a day ago
In response to "Let It Lie"

I remember telling my youngest that Santa didn't exist. He thought for a moment then said "Does that mean you and Mom bought all those presents? Cool!" But then my youngest is one cool dude.

4 Like

🌖 HuntersMoon

a day ago
In response to "Let It Lie"

Reminds me of an old joke:

What do you call a kid who doesn’t believe in Santa? A rebel without a Claus.

3 Like

tj wanderlust-words-in-motion

Jan 24, 2025 at 4:30pm
In response to "Watch Out"

I'm reminded of my Grandma who would only have a radio in the kitchen which she unplugged when not in use, because she was convinced it was eavesdropping on conversations. She also had a big, thick, homemade quilt she covered the TV with, so no one could watch activities in the living room.

She was already on to something!

2 Like

Graham B.

Jan 24, 2025 at 1:23pm
In response to "Watch Out"

George Orwell wrote books about guys like Larry Ellison.

2 Like

Sum1's Home

Jan 24, 2025 at 11:50am
In response to "Watch Out"

Your statement about being watched is spot on. I'm privy to a bit of knowledge I can't share, but after that mass-killing in the Fort Lauderdale airport baggage claim, cameras were added. Not just ordinary surveillance cameras either. These cameras would monitor a designated area, the computer it fed would 'know' the scene based on provided data. If a bag, piece of baggage, or almost anything was in one place (and unattended) for too long, an alarm would sound. Where? I am not sure, but I would think it would be at a monitored station/computer, much like a dispatcher working in a radio system.

That was quite a few years ago, camera's and monitoring software have been greatly improved. Here's an example of what could happen. Let's say a camera is monitoring a parking lot, sweeping back and forth slowly. It sees your car pulling in as it sweeps. No big deal.... Yet. On it's next pass, the computer it feeds expects to see your car in a parking spot. If not, an alarm might be raised. Furthermore, it may have caught you getting out of the car in one of it's passes, and now expects to find you approaching an entrance. It can zoom in automatically and capture your license plate. It can do the same to your facial features. So yeah, we're being watched almost all the time. Why do I say almost? Those things are expensive, they are not deployed everywhere. When they are, or once they are, we'll be living in the times of the novel 1984. BUT.... you are still safe in your home, unless Alexa or Siri is listening to you, and your computer camera is operating on the sly, or your home interior surveillance system has been hacked.

2 Like

Jayne

Jan 24, 2025 at 10:55am
In response to "Watch Out"

Yes, loonies have a loon on them.

Toonies? Polar bear. Just one.

2 Like

January 25, 2025 at 7:30am

#1082771

Let It Lie

I'm used to stepping into puddles with indeterminate depths, but this one's going to test the integrity of my hip-waders. From The Conversation, and also from last month, hence the theme:

The case for lying to kids about Santa – from a philosopher

I have a vivid memory of the moment I realised Santa didn’t exist. I was around six years old, it was the height of summer, and I was sitting on the step outside our back door, thinking about God. The existence of God, back then, was something that annoyed me: it meant that every Sunday, we had to go to church.

Then I realised: there isn’t actually any evidence God exists. I only think God exists, because this is something people have told me.

I'll be honest here: that was way younger than I was when I had that particular epiphany. Well, it wasn't an epiphany for me, and not just because the word itself has religious connotations. No, it was more a gradual realization in my case.

I remember bounding up, excited, ready to share with my family this wonderful news. No longer would we be forced to endure the drudgery of weekly Sunday schools and sermons. But then I remember checking myself and thinking, oh no. If God doesn’t exist, by the same logic, Santa must be made up as well.

I want to doubt that a six-year-old could leap to such logical conclusions, but peoples' minds work differently. This author went on to be, as the headline notes, a philosopher. I did not. I have too much of a sense of humor to be a philosopher. You know what we call a philosopher with a sense of humor? A comedian. Pays better, though that isn't saying much.

Perhaps this was the moment I became a philosopher (though I should say that as an adult, I no longer think that the analogy between God and Santa really holds).

You know what the difference is? In most cases, I think, parents know that Santa is mythology, while remaining somewhat certain about the existence of God.

Here's a philosophical question: Telling someone something that isn't true is called lying (again, as noted in the article's headline). But isn't there a difference between telling them something that you know isn't true, and telling them something that you believe is true, but isn't?

Like, if I'm absolutely convinced that it's about to rain. Clouds are gathering on the horizon, and there's that smell in the air and maybe the wind is picking up. So I turn to my imaginary friend and say, "It's about to pour down rain here." But over the next hour or so, the clouds move off to the north (or whatever), leaving the area where I'm standing completely dry. Did I lie? Obviously, I was wrong (hey, it happens, however rarely). But was it a lie when all signs supported my conclusion?

But now the tables have turned. Now I am a parent of young children, and I am the one enforcing hegemonic myths about Santa.

Ah, yes, yet another difficult decision avoided by me not wanting or having offspring.

We all do it, of course.

No, "we" don't. Even if he meant "we parents," not all parents do it.

Our culture expects parents, basically, to lie to our children that their presents were left by a jolly fat man who flies in a sleigh pulled by reindeer through the sky.

Hell, even my parents did that, and they weren't Christians. Santa's not known for leaving Hanukkah presents. That's jolly old Judah Maccabee.

We all surely want our children to grow up to be honest people. Shouldn’t we set a good example, as far as possible, by telling them the truth?

Except that, sometimes, honesty doesn't serve us very well. For instance, when someone asks us, "Does this dress make my ass look fat?" we don't necessarily want to tell the bare, unvarnished truth.

Here's the thing about lying: it demonstrates empathy. To tell a deliberate lie requires enough knowledge about the other person's mental configuration to be able to tell them what they want to hear. As with all of our superpowers, this can be used for good or evil. But someone who "cannot tell a lie," like the mythological version of George Washington, might very well lack empathy.

Amusingly, and possibly even ironically, the "cannot tell a lie" cherry-tree incident is itself a made-up story. A myth. A lie.

To which I would say: well, no. We shouldn’t be honest about Santa – at least not at first. It is morally OK, to the point of being actively morally good, for parents to participate in the grand Santa lie.

And the rest of the article explains why this particular author believes this.

If I felt compelled to tell my children everything, I would pull no punches in relating the wretched state of the world, of existence, of my still-deepening resignation that nothing positive can be done about it. I would inflict the full brunt of my money worries, my health concerns, my (mostly irrational) worries about them.

Oh, joy! More existential conundrums that I neatly avoided by means of a simple medical procedure.

Now, here's the fun part:

Just two days before this piece was published, the same outlet posted an article by a different philosopher (also from the UK, though Scotland, not England): "Why you shouldn’t lie to your children about Father Christmas, according to philosophers"

I'll just note that in the UK, "Santa Claus" and "Father Christmas" are essentially the same figure. The history there is kinda cool;

basically, Santa was a US thing after we divorced from the UK, but then the name went back across the pond and, eventually, the stories merged.

Point is, these articles, together, present two sides of the argument. That might leave parents even more confused than they were before they read the articles, but it just leaves me with a certain sense of smugness for never having to make that decision.

Well, there was the one time I was at a friend's house and accidentally dropped the truth bomb about Santa Claus within earshot of their six-year-old (or thereabouts). I haven't been invited back since. That then-six-year-old would be in college now. I wonder if she's studying philosophy.

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