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:

tj-Merry Mischief Maker

about 9 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

September 8, 2024 at 9:02am

#1076471

Revisited: "Orange You Glad This Wasn't You?"

My first random selection of a past blog entry landed on one I'd already revisited. I knew that could happen, obviously, but I was still mulling over whether to do another revisit, or just skip it and roll again.

I've decided, instead, to point to the previous Revisited, thus: "Revisited: "Best Waste of Time EVER""

But that's all I'm going to say about it.

And then I rolled again, and got this discussion of an article, posted about four years ago: "Orange You Glad This Wasn't You?"

That's the one we'll look at today.

Something a little different today.

Yeah, back then, maybe it was "a little different," but I think now it looks like just another article I could riff on.

The link, to a 2020 bit about Tropicana's orange juice marketing disaster back in 2009, is still there... but when I looked, it said "Member-only story."

That's their right, of course, but nothing from Medium has ever captured my attention enough to become a "member," and to me, there are very few articles on the internet worth reading that aren't also worth sharing. So I don't bother. I pay WaPo for a subscription, because that's the newspaper I grew up with (if indeed I grew up at all), but that's about it.

Therefore, all I can do is talk about my own reactions. Even the specific "rebrand" they're talking about is hidden and forgotten.

So why am I linking something about orange juice? Because it's a product, and some marketing techniques apply to a broad range of products... like, for instance, your writing.

Not that I'll ever apply these techniques. But, being entirely altruistic and thinking only of my readers, I perform the public service of informing them.

Okay, no, I just find this stuff fascinating for whatever reason, and I'd rather learn from others' failures than from my own.

But maybe this will help someone else. Or, hey, maybe you just like orange juice.

The reality is a bit of both.

Coke's "New Coke" fiasco personally affected me. I don't drink orange juice, so I never noticed a rebrand.

Yes, almost 40 years later, and I still carry trauma from that horrible year.

The article helpfully includes "before" and "after" pics.

Not any more, if you're not a member.

Function is more important than form. Well... for me. Engineer, remember? But I can admit that when it comes to trying to sell shit, part of the function is to catch a viewer's eye (or ear in the case of radio). That requires form.

Much as I rag on myself for having no artistic talent (I don't) or not understanding marketing (I don't), I do have some background in graphic design and artistic composition. Part of being a photographer. I maintain that I wasn't very good at it, but I understand some of the theory, just like I understand some music theory even though I'm a terrible musician.

Then I quoted the following from the original article:

“Historically, we always show the outside of the orange. What was fascinating was that we had never shown the product called the juice.” Really? I mean, it’s juice. Give me a clear symbol of it, and I’m good to go. And what could be clearer than the actual fruit the juice is from?

On which I commented:

Somewhere in there, I think, is probably a metaphor for promoting one's writing. I'm not quite sure what it is, but I'm pretty sure it's in there.

Having had four years to think about it, I'm still unable to articulate what that metaphor is, but I'm still sure there is one, taunting me. I think it's encapsulated in the line, "Give me a clear symbol of it, and I’m good to go." People respond to symbols; that's kind of our thing as humans, to let one thing stand for another, or for an abstract concept.

But I'm reminded of what might have been my first lesson in marketing failures, which, to be fair, could have been an urban legend. The story went something like this: In some less-developed country, it was standard practice to show an image of the contents of the can on the label. If it's beans, for example, the can depicts beans. For carrots, there's a picture of carrots. That way, even those who can't read (a large portion of the population) would know what's supposed to be in the can. Anyway, the fail I heard about was that Gerber tried to enter that market, but couldn't... because their baby food cans had a picture of a baby.

Almost every barbecue joint I've ever seen uses a pig as a mascot. This is not because they're marketing to literal pigs. Metaphorical ones, maybe, but their customers generally don't root for truffles and oink.

In any case, I think I've discussed other marketing mistakes in here since then. That entry's comment section is also good for examples, because I turned it into a Merit Badge contest, which is something I haven't done for a while. Except for last week's free badge giveaway, of course.

But I can't leave this without noting that the header for this blog doesn't exactly depict what's in the can. So I guess I really do fail at marketing.

1 Like • Comment • View 1 comment

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