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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Turkey DrumStik

about 5 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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').

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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

December 20, 2018 at 1:14am

#947845

December

Another turn of the wheel.

https://www.vox.com/science-and-health/2018/12/18/18144477/winter-solstice-2018

As I wrote last month, this is one of the things that, were we so inclined, we could all agree on. It affects the entire human race (at least for now; I'm still hoping for space travel), even if its effects depend on your current latitude.

I should add a couple of observations to the article I linked above.

First, while the vast majority of humans live in the Northern Hemisphere, that's no excuse for hemispherism. Most of us see it as the Winter Solstice but, as noted, it's the Summer Solstice in the south. To avoid being exclusionary, I've seen it referred to as the December Solstice. A better name - one that's not tied to our arbitrary calendar - might be the Capricorn Solstice. However, that would make the other solstice the Cancer Solstice, and I can see people objecting to that.

Second, unless I missed it (I tend to skip boring parts, which is one reason I don't like internet videos), it doesn't define what "solstice" really means. Yes, it says "Technically, the solstice occurs when the sun is directly over the Tropic of Capricorn, or 23.5° south latitude." But the word is from Latin: sol (sun) + stice (from a root that means 'stationary'). So literally it means like, "the time the sun stands still." Which is pretty cool. Of course, the sun doesn't actually stand still - figuratively or literally - it's a bit more complicated than that. From the point of view of any observer on Earth, each day when the sun reaches its highest point in the sky (known as local noon, which is usually not when the clock says noon), it appears slightly more to the south between late June and late December. As the calendar approaches the December solstice, the sun's apparent location at local noon drifts a bit further south each day. On the solstice itself, it's at its southernmost point. Technically, the solstice occurs at local noon at whatever longitude the sun happens to be crossing when it's at zenith at the Tropic of Capricorn.

Yes, I know. If I'd been reading this, I'd have skipped the last paragraph.

Third, did you see the gif in the linked article with the time-lapse of Earth from space? It's pretty cool. With a little digging, I found that the satellite that shot these pictures is a weather satellite in geostationary orbit. When I saw the gif, my first thought was "that satellite is directly over 0/0." 0/0 being the point where the Prime Meridian crosses the equator. It's in the Atlantic, off the coast of Nigeria. I'm not so jaded that I find photos taken from geostationary satellites to be a mundane thing. Really, I'm not sure whether it's over the actual origin or not, but it's pretty close.

Anyway, science in general is cool, astronomy in particular is fascinating, and culturally, we have a long history of marking the solstices and equinoxes. As I've mentioned before, I don't observe Christmas, so the day I care about is the solstice. It's a time to consider rebirth and renewal; the moment when, just as everything seems to be about to plunge into eternal darkness, there's the promise of more light to come. It's more interesting to me than the very arbitrary New Year's Day (the proximity of which to the solstice isn't accidental), and more meaningful.

In the past, I've marked it for myself by staying awake through the longest night, maintaining a vigil for the returning, invincible sun. I think I might revive that tradition for myself on Friday.

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