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 21 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 23 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.

January 9, 2020 at 12:01am

#973008

Eat It

PROMPT January 9th

What are some of your favorite foods? Comfort food, ethnic food, meal, beverage, fruit, protein, unique food combination, etc - make us hungry!

Beer.

Well, okay, not really. But I have a brand to maintain, here; work with me.

For a little more than a year now, as I've noted perhaps more than readers would like, I've been on a mission to lose weight - a mission that has been mostly successful, although I seem to have plateaued since late November. The plateau shouldn't be surprising, considering that late November through the end of December is a feast time for nearly everyone. So I've been trying not to let it get in the way of my mission, just keeping up with the work. And hey, at least it's a plateau and not a backslide. Well, not much of a backslide. Well, not more than I had expected. Still, I'm right where I was before Thanksgiving, so I'm calling it a plateau. So there.

But it's probably notable that the reason I had to lose weight at all was because I'd gained it in the first place, and one of the reasons I gained it is that I have favorite foods, and I ate them.

I wrote a story here long ago that proposed, somewhat tongue-in-cheek (mmm, tongue), that the trick to successful dieting is to only eat foods that you don't like. While I was joking, I fear there was probably a kernel of truth in this, at least for me.

There are some lucky few individuals, I think, whose taste in food corresponds with what we know of as eating "healthy." These fortunate few actually enjoy, say, broccoli or quinoa, rather than knowing, intellectually, as I do, that such foods are to be sought out for their low calorie density, despite their repulsive taste. Well, okay, they're not repulsive to me, really; they're just far from my first choice. These people avoid fast food not out of principle, but because they simply dislike it. Rather than ordering a pizza, they have no problem whipping up a veggie stir-fry, for example

Obviously, I'm not one of those people, so I have to work at it. So I can't say I haven't had any of my favorite foods over the past year or so, but I've certainly reduced the intake thereof.

I identified two main things that made me eat what I shouldn't. First, laziness: I hate going grocery shopping, and cooking takes time and effort and forces one to clean up afterward - for me, this is a recipe for deciding to order out most of the time. And second, hedonism: I seek immediate pleasure over possible long-term benefits.

And when it comes to immediate pleasure, the foods I actually enjoy (as opposed to eating because I'm lazy) hurt me in two ways - they taste good, so I eat more of them, and they're high-calorie. These include, but are not limited to: pizza (specifically New York style pepperoni pizza), cheeseburgers, French fries, spaghetti, baguettes, curries, lasagna, Philly cheese steaks, tuna salad sandwiches, pastrami reubens, bagels with lox, and nachos.

I don't know if that's going to "make us hungry," but just writing that certainly made me hungry, so thanks a lot for that.

You'll note that absent from that list is anything resembling a vegetable or fruit (potatoes don't really count; they're mostly carbs). It's not that there aren't veggies or fruits that I like; it's just that, absent careful consideration on my part, eating a salad just gets in the way of my enjoyment of the true foods I listed above. Especially bread. Bread is food; everything else is optional.

Which is why a low- or no-carb diet would never work for me. Tell me I can never eat at Taco Bell again, and I'll probably be okay with that; their food objectively sucks, and it only appeals to my laziness. But tell me I can never eat bread again, and I won't see the point in living.

So yeah, nowadays I try to practice more moderation. I haven't ordered delivery food for a year. I've eaten at fast food places once or twice, because I was on the road and in a hurry, not out of sheer laziness. I mostly limit my restaurant meals to no more than one a week. I only bought (and subsequently devoured whole) a gourmet baguette once. No, for the most part I've been sticking to less delicious food in hopes that, maybe, just once in my life, working and sacrificing for something will actually pay off. It never has, so far, but I keep hearing good things about delayed gratification, so I figured it was worth a shot.

I'm still not sure it's worth it. As I've said before, what's the point of living if you have to give up that which makes life worth living?

I might have said something like this before in here, but I can't be arsed to look it up: Consider, for a moment, whatever it is in life that brings you the most joy. Your kids? Your dog? Travel? Money? Spouse? Movies? Music? It doesn't matter what it is; just consider it. Now, what if you are given hard evidence that if you give up that one thing, never see it or hear it or touch it ever again, then maybe - barring accidents - you could live a couple more years?

Could you do it? Seriously, think about it. Never interact with the thing you love most in the world, or at best once or twice a year, in exchange for the possibility, not the guarantee, but the possibility, of two or three more years of life? Well, that's me with food and booze.

You can tell me, "Well, just find joy in something else." And indeed I do. But nothing exceeds the level of happiness I get when I'm eating a good meal and with it, enjoying a great beer or a fine glass of wine. Well, nothing except also hearing decent music at the same time. I can maybe live with doing such a thing less often. Maybe. But never again? Not worth it.

Have a big dinner.
Have a light snack
If you don't like it, you can't send it back
Just eat it! Eat it!
Get yourself an egg and beat it!

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