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 an hour 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 3 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.

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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').

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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.

November 8, 2023 at 9:40am

#1059144

Harvey Wallbanger Day

Prompted by "Invalid Item"

, another alcohol-positive observance... sort of.

National Harvey Wallbanger Day

Believe it or not, there are drinks I haven't tried. This is one of them.

According to the above link, a Harvey Wallbanger is made with "vodka, Galliano liqueur, and orange juice." Proportions aren't given, but I'm confident that a quick internet search, which I can't be arsed to do, would provide numerous recipes.

Now, don't get me wrong: I appreciate cocktails or whatever else you want to call mixed drinks. One of the great joys of life is creating new ones, such as the Star Trek themed libations I've collected here: "Ten Forward"

[18+] And I'd drink a wallbanger if it were available; it's just that as long as there are other options, some things just aren't on my radar, and this is one of them.

To understand why that is, consider the most popular liquors, all of which are often used as the basis for cocktails: vodka, tequila, rum, gin, and whisk(e)y.

Of those Big Five, vodka is, by far, the least interesting. Unless you get flavored vodka, it's generally neutral in character. The advantage of this is, clearly (pun intended), that it mixes well with other beverages. I don't hate it. I don't love it. It's just there to put alcohol into something that doesn't naturally contain it, such as, for example, orange juice.

Sure, there's some variation in quality between different brands of vodka, but in general, it doesn't have much flavor of its own. This is good if you don't like the other available flavors, or if you're just trying to get drunk.

I have a higher purpose in mind, which is to enjoy what I'm drinking while I'm drinking it.

So, as I'm sure we all know, if you mix vodka and orange juice, you get a drink called a screwdriver in the US (In the UK, I think it's less creatively called "vodka and orange.") A quick search turned up the fun fact that this very simple cocktail wasn't really invented until WWII, which surprised me until I remembered I did an entry long ago about the origins of commercially produced orange juice, which I seem to recall weren't much earlier than that. I even bothered searching for that entry, and here it is, from last year: "Orange You Special"

As with the subject of today's entry, the origin of the name is disputed.

A Harvey Wallbanger, then, is basically a screwdriver with the addition of, specifically, Galliano, an Italian liqueur. And quoting from the headline link again: "The drink did not gain in popularity until the 1970's, when George Bednar of McKesson Imports Company promoted it, as a way to sell Galliano, which his company handled."

Thus, it's just another marketing gimmick. So was orange juice, if you read my earlier entry.

That's a good enough reason for me not to order one at a bar, and I have little desire to clutter my liquor cabinet with yet another flavored liqueur.

Besides, why get one of those when there are so many cocktail options involving gin, rum, tequila, or whisk(e)y?

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