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 11 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 13 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.

March 16, 2022 at 12:23am

#1029004

It Was a Dark and Stormy Day

Yes, I'm running a little late today. Yesterday, because of *gestures vaguely at everything* I mixed spiced dark rum and ginger beer in copious quantities and drank it.

It was a Dark and Stormy day.

Consequently, I passed out completely, for several hours. I don't have a link, or a prompt. I suppose I could just wait until later, but where's the fun in that? I'm not sure if I'm hung over or still drunk. I do know that, thanks to the ginger beer I suppose, I'm feeling pretty good, all things considered.

Now, a proper Dark and Stormy also includes lime. But I didn't think ahead like that. Sometimes I'll buy limes, then not use them, and find them later, hard and shrunken, tucked way back in the fridge. Most of the time, though, I'll get it in my head to drink something that requires lime, such as a gin and tonic, and naturally, those times, there are no limes in the house.

I wish someone would invent stasis fields so that limes can keep fresh indefinitely. Forget warp drive, communicators, transporters, replicators, and world peace; the most important invention from Star Trek would be the stasis field.

Okay, it's actually from Niven, or possibly some SF writer before him, but my point remains.

Anyway, where was I? Oh, yeah, Dark and Stormy.

Technically, only Gosling's Black Seal rum can be used in a Dark 'n Stormy. Fortunately, I wasn't making a Dark 'n Stormy; I was making a Dark and Stormy. The rum I used was Kraken, which helpfully has a drawing of a multi-tentacled sea monster on the label, and is apparently made in Indiana; the ginger beer was from Stolichnaya, which, no, is not Russian; it's Polish, so don't give me shit about virtue signaling.

So even those ingredients were nonstandard; a proper Dark and Stormy should be made with Caribbean rum and Caribbean ginger beer.

At this moment in history, though, I don't give a shit.

Even in the depths of a binge, though, I'm still doing science: what, I wondered, is the best ratio of rum to ginger beer? I think most recipes give a proportion of 1:3 or 2:5. I tried 1:2, 1:3, and 1:1, among other mathematically in-between attempts. Admittedly, by the time I was done, I was finding it hard to do proportions. In the end, I suppose it's a matter of taste: do you want more spiced rum, or do you want more of the spicy, tingly taste of ginger beer?

1:3 seemed to do it for me, but like I said, I was lacking the cooling, tart effects of lime in the drink.

So that was my day. Very productive. How was yours?

4 Like • Comment • View 9 comments

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