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:

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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

July 1, 2023 at 12:29am

#1051940

Moonshine

Going by Eastern Daylight Saving Time, there's a full moon coming up on July 3. The next one occurs on August 1, and the one after that, August 30.

There has already been media hype about the third one being a Blue Moon. As I've noted before, repeatedly, and with little effect, this is incorrect. The Blue Moon is not the second full moon in a calendar month. No, I will not stop pointing that out, and you're going to see it again from me as we inevitably get more articles about it in the near future (there's already another one in my queue).

But today's article is, in part, related to a different label, a more recent one: the "supermoon."

How Far Is the Moon From Earth? And Everything Else You Need To Know About Our Glorious Natural Satellite
What is it actually made of? How far is the moon from Earth? And is there really water up there?

I don't actually have a problem with the label "supermoon." Maybe it's a bit overhyped; all it means is that the full moon occurs near perigee, so the familiar cratered face seems a bit larger than usual.

Here’s everything you could possibly want to know about the moon.

Nonsense. There's always more to learn.

How Did the Moon Form?

Pick a culture, and they probably have a story about that.

The most commonly held theory for the moon’s creation is known as the “Giant Impact Hypothesis.”...A NASA study from 2019, for instance, cast doubt on the Giant Impact Theory. Specifically, scientists couldn’t find enough of a large group of Earth elements in the moon, even though the moon should contain them, per the Giant Impact Theory.

See? More to learn.

How Long Does It Take the Moon to Orbit Earth?

One month, duh. That's why we call it a month. But not the mostly arbitrary months of the Gregorian calendar.

It takes a little over 27 days—27.322, to be exact. Coincidentally, it also takes the moon 27 days to rotate on its own axis.

That's not "coincidence." Coincidence is that we're around at a time when the moon and sun are roughly the same apparent size in the sky. The moon is tidally locked; that's physics.

The moon’s orbit of Earth follows what scientists call an elliptical path, shaped more like an oval than a circle. So while we can’t see the moon spinning, we can see it change in size. It’s just a matter of perspective, but it reflects how the moon interacts with Earth. When the moon is farthest away from Earth, scientists refer to that as the “apogee,” and when it is closest, it’s at its “perigee.”

Ugh. Everyone calls it an elliptical path, not just scientists.

I feel obligated to point out that there is some variation in the face the moon presents to us; this is called "libration" (okay, that one's mostly scientists). So, sometimes, we can see a little more of one edge or the other.

How Far Away Is the Moon From Earth?

This is the bit that's related to the "supermoon" concept.

At its apogee, it’s 252,088 miles (405,696 kilometers) from Earth. At its perigee, it’s closer, at 225,623 miles (363,104 kilometers).

Perigee occurs once per orbit. It's only when the full moon happens close to perigee that we get the supermoon thing, but really, it's always slightly bigger at some point once a month.

That works out to an average of 238,855 miles (384,400 kilometers). That’s around 60 times the radius of Earth, or enough distance for 30 Earths in between.

I read somewhere (can't be arsed to look it up now) that it's also enough space for all the other planets in between. Which in reality would be catastrophic; this is just a size comparison.

How Does the Moon Affect the Tides?

...seriously? Okay, sure, the actual physics is a bit complicated (it took me a while to fully grok why there's a tidal bulge on the opposite side of the Earth, as well as the moon-side, and the article glosses right over that bit), but I thought it was common knowledge that the tides are caused by the moon.

What’s the Dark Side of the Moon?

A classic rock album by Pink Floyd.

A classic rock album by Pink Floyd.

I TOLD YOU SO.

But seriously, there’s no actual dark side of the moon, because the moon rotates just like Earth; as the moon rotates around Earth, it also rotates around the sun. This hidden region is better known as “the far side of the moon.”

Uggggghhhhh. I mean, okay, not completely wrong, but just wrong enough to annoy me. Just like half of Earth always faces away from the sun, giving it a "dark side" (we call it "night"), the same is true for the moon, which is how phases work. There is a bit of a complication, because when the dark side faces Earth (near the new moon), it's lit up by reflected sunlight from Earth. Earthshine. (I have to wonder if our future lunar colonists will make illicit hooch and call it earthshine.) There is a dark side, but it cycles through every month.

But yeah, don't confuse the far side (not the comic, but the hidden face of the moon) with the dark side... except during a full moon, when they're basically the same.

Or, as is noted on the aforementioned album, which I've quoted here multiple times: "There is no dark side of the moon really. Matter of fact it's all dark."

Which it is. From a certain point of view.

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