Complex Numbers (Book)

Blog Calendar

◄ September ► 2023
S	M	T	W	T	F	S
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
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.

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

September 6, 2023 at 8:29am

#1055247

The Big One

The RNG gives us another planetary article today. This one's about the big guy.

Jupiter revealed
The Juno spacecraft has been circling Jupiter since 2016. Here are four things we’ve learned so far about the biggest planet in the solar system.

In the four years since NASA’s Juno spacecraft went into orbit around Jupiter, it has slowly been coaxing the king of all the planets to reveal its deepest secrets — an astonishing catalog that includes daisy chains of continent-sized cyclones circling both of Jupiter’s poles; vast hailstorms of ammonia-laden “mushballs”; a bloated, fuzzy core at the planet’s center; and a convoluted magnetic field like nothing else in the solar system.

The article is from 2020, so add another three years to that, as well as possibly other new discoveries that I can't be arsed to search for right now. I did manage to check whether the ceiling fan (I mean, look at the probe

) is still active, which, apparently, it is.

Named after the goddess Juno, who was Jupiter’s wife in Roman mythology, the spacecraft was launched in August 2011 with the goal of understanding the giant planet’s origin and evolution.

"Jupiter's wife" glosses over a whole slew of godly shenanigans, including a bunch of extramarital affairs (not all of them consensual), and that Juno was also his sister.

Meanwhile, here are four of Juno’s greatest hits to date.

None of which are the hits she took out on Jupiter's illegitimate children. Or, well, they would be illegitimate, except that gods play by different rules.

Around the planet’s south pole, Juno spied five cyclones, each wider than the United States, parked around a central cyclone of the same size. Not to be outdone, the north pole revealed eight similar cyclones encircling their own polar vortex.

For context, those are pretty small for Jupiter. Compare the best-known storm on Jupiter's cloudy surface, the Great Red Spot (not to be confused with a teenager's acne), which at its peak was something like 3 times the diameter of our entire planet. (Later in the article, the GRS is described as being "a bit wider than Earth;" as with many of us, it's been subject to shrinkage over the last century or so.)

Because Jupiter is what astronomers call a gas giant planet, there is no point asking what conditions are like on its surface: It doesn’t have one. Instead, the hydrogen and helium gas that make up the bulk of Jupiter’s atmosphere simply get denser and denser the farther down you go, until the hydrogen becomes a liquid metal.

As with my last planetary entry, the whole "unclear boundary between phases" thing is mind-boggling. This adds another level of boggle with the metallic hydrogen thing, which has got to make astronomers irate, because to them, hydrogen and helium are nonmetals, while everything else is a metal. This is distinct from the chemistry definition of a metal, which has something to do with free, shared electrons. Hippie socialist electrons.

Metallic hydrogen, as I understand it, only occurs at unimaginable pressures, such as those found near the center of enormous planets.

As a digression, there's a weird thing about planets that I don't fully understand. Well, there are lots of those, but in particular, the center of any planet is subject to high pressures. That part makes sense, like how water pressure increases with depth. But because the core is surrounded by planet, the net force of gravity there is near zero (the mass pulls equally from all directions). I haven't reconciled this in my mind yet.

Although Jupiter doesn’t have a surface, researchers had a running argument before Juno’s arrival as to whether the planet had a core — a solid ball of heavier elements gathered at the planet’s center.

Need to settle a bar bet? Build a billion-dollar ceiling fan and send it to Jupiter.

I joke, but the probe really did shed some light on that subject, though there are, of course, still mysteries.

As for the planet's Olympic-class magnetic field:

It’s as if someone took a bar magnet, bent it almost in half, frayed one end, split the other end, and then stuck the whole thing in the planet at a cockeyed angle. In the north is the frayed end: Rather than emerging around one central spot, the magnetic field sprouts like weeds along a long high-latitude band. In the south is the split end: Some of the field plunges back into the planet around the south pole while some is concentrated in a spot just south of the equator.

Jupiter is, believe it or not, quite different from Earth, so it shouldn't be that surprising that its magnetic field is weird. Still, the article goes a bit into just how weird, and, well, its weird.

Anyway, apparently, some science got done and we need to do more science.

As always.

Just don't tell Juno we're sending another probe; she might get jealous.

7 Like • Comment • View 2 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.