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 2 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 4 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.

August 20, 2021 at 12:28am

#1015876

Star Wrecked

I'm going to take a break from my usual format today to talk about drinks.

Specifically, Star Trek-themed drinks.

Now, drinking has been a part of Star Trek since the beginning. While the abominations based on synthehol are featured in most episodes of TNG, the show has otherwise been alcohol-positive. Consider, for example, this TOS episode where an invading alien is defeated when Scotty tricks him into a drinking contest.

"What is it?"

"Well, it's, um..." *sniff* "It's green."

They even, quite famously, made up their own multicultural drinks, the most famous probably being Saurian brandy (featured in that clip), Klingon bloodwine, and Romulan ale.

With the recent spate of new Trek shows -- Discovery, Picard, and the animated Lower Decks -- it has been my tradition to whip up a batch of Romulan ale to accompany my Trek viewing. Yes, I'm still slowly getting through my rewatch of all the earlier episodes, but that doesn't stop me from partaking in the new stuff. I know opinions are mixed about them, as per usual, but I like all of them.

But I digress. Romulan Ale, of course, doesn't exist, because the Romulan Empire doesn't exist. That doesn't mean we can't make our own version. The defining characteristic of Romulan ale is that it's bright blue, so obviously you need something like Blue Curaçao. So I found this recipe

in a web search.

In short (keeping in mind that this is not meant to be consumed in one sitting):

2 Cups Blue Curacao
1 Cup Clear Rum, no flavor or spice
1/2 Cup Grain Alcohol – (NOTE – You may substitute Vodka if you can’t get or do not want the Grain Alcohol – We use it in Geek Manor because we’re nuts).
1/2 Viso Will – (NOTE – If you can’t get Viso Will, you can use any Cranberry Lemonade, just know the color will be off, since Viso does not color their drinks).

Now, there are several problems with that recipe:

1. Way, way, way too much Blue Curaçao. It doesn't take much to make the drink blue, and I find its sweet orangey taste to be cloying.
2. Grain alcohol is, sadly, not available in Virginia. This might be my fault; it used to be available, but I used too much of it back in college and the authorities found out and banned it. (It is, oddly enough, readily available in Utah.)
3. What in the name of Kahless is Viso Will?
4. As it is "ale," there should be some carbonation involved. Doesn't have to be a lot, but it should be there.

So I made some modifications.

"But Waltz, didn't you just do a Comedy newsletter editorial saying you never make substitutions in a recipe?"

Yes, I did. But that's just for food. This is the important stuff: drinks.

So here it is, my recipe for the perfect Romulan Ale. Tastes vary, of course, and you should make it to your taste if you're so inclined.

1 Cup Blue Curaçao
1 Cup clear rum (this part I kept)
1 Cup vodka (and just to be utterly clear, this is unflavored vodka, not the teenager stuff)
1 1/2 Cups Sprite (or Sprite Zero if you're concerned about calories) (that's a joke)
1 1/2 Cups Club Soda (or plain seltzer)

Pour everything into one of the beer growlers you have sitting around. Close the growler and turn it gently a couple of times to mix (DO NOT SHAKE). Refrigerate overnight. Pour into beer glass.

The result:

Or the closest we can get to it without trading with the Romulans

Or the closest we can get to it without trading with the Romulans

Now, at that link they also refer to a Cloaking Device, which they describe as half Romulan Ale, half absinthe.

The problem is that absinthe is a) warp-capable and b) overpowering of any other flavors. So I make mine more like 2 parts RA and 1 part absinthe, and I pour it over ice instead of shooting it because it's better at making me undetectable.

It also looks pretty:

Romulan Ale + Absinthe = Ahead Warp Factor 6

Romulan Ale + Absinthe = Ahead Warp Factor 6

And fair warning, you will achieve warp speed while cloaked.

My friend also invented a drink we're calling Lower Drunks (in homage to the show Lower Decks), but that's not my recipe to share. Perhaps another time.

8 Like • Comment • View 9 comments

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