Complex Numbers (Book)

Blog Calendar

◄ August ► 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	31
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

August 9, 2023 at 10:26am

#1053868

Planet Ix

The random number gods bring us back to the solar system today, but this time we're talking about a planet that only exists in educated speculation.

The Hunt for Planet Nine
What will it take to find the biggest missing object in our solar system?

Before Pluto achieved dwarf planet status, that would have been Planet Ten. I will admit, Planet X sounds better than Planet IX. But maybe I'll start calling it Planet Ix.

At 9,200 feet, there is 20 percent less oxygen than at sea level, enough to take all the air from my lungs after just three steps.

Is it bad that I understood immediately why one would open an astronomy article with this sentence? I had to walk very slowly when I was at the peak of Haleakala, where a bunch of telescopes lurk.

This wasn't Maui, though, but that other island. And the actual telescope is a couple thousand feet higher.

We were at base camp on the dormant volcano Mauna Kea, on the big Island of Hawaii. The pair were here to use one of the most powerful telescopes in the world, called Subaru.

Which has nothing to do with the car company; both take their names from what we call the Pleiades.

Tomorrow night, December 3, marked the start of their sixth observing run and their next attempt to find the biggest missing object in our solar system, called — for the moment — Planet Nine.

Yeah, I'm gonna call it Ix.

The article is from Longreads, so... well, it's right there in the name. I won't be quoting a lot more of it, but I wanted to highlight a few relevant sections.

As a theorist Batygin feels that he’s already mathematically proven its existence. But it’s generally accepted that for a planet to be considered discovered in the field of astronomy, the theory must also be accompanied by a photograph.

I know I've pointed this out somewhere before, but a while back, a planet inside Mercury's orbit was "mathematically proven." They figured one had to be there based on perturbations in Mercury's orbits. Some people even claimed to have images of it, taken during eclipses or whatever. So they called it Vulcan, after the forge god (not to be confused with, well, you know). You think it was bad when Pluto was designated a dwarf planet? Imagine renumbering all the planets. Earth becomes Planet 4, for example.

Anyway, short version, there was no Vulcan; the images were false and Mercury's orbit was fully explained by Einstein.

I'm not saying the data pointing to Planet Ix is nothing but gravitational anomalies, but there's still a lot we don't understand. Which is why we do science.

Part of their job was first to try to find a solution less extreme— like a passing star or a galactic anomaly — than a giant undiscovered planet far off in the depths of the solar system, because, a hidden planet? That was absurd.

See, now, to me, a passing star or a galactic anomaly would be more weird than a hitherto unobserved planet. But that's why they get paid the big bucks.

(I'm joking; astronomers don't get paid the big bucks.)

After the observing run was complete, I asked the pair if they ever felt that trying to find Planet Nine was ridiculous, that the whole notion of a giant missing planet and the efforts they have gone to to find it ever make them feel defeated. They both gave me roughly the same response: no. Their answer brought to mind the French philosopher and writer Albert Camus. He thought a lot about the myth of Sisyphus and plucked his unfortunate mythical backstory away from the root of his actions, the eternal task of pushing a boulder up a mountain only to watch it fall back down again.

Right? Because the telescope is located at the summit of a mountain, and the astronomers are like Sisyphus, climbing and descending the mountain over and over again.

Like I said, a long read, and a lot of it is, pun intended, atmosphere. But I found it fascinating. Life is barely possible at the summit of that volcano; it has, as the article notes, barely enough oxygen to survive, let alone move around and do stuff.

When I went up Haleakala—from which you can see Mauna Kea off in the distance—I'd brought a bottle of Coke with me. When I finished it, up at the summit, I twisted the top back on and left it in the car, because I didn't want to add to any trash at the peak. After enjoying the cold, thin air for a couple of hours, we headed back down the mountain. Back near sea level, in Kehei, I went to toss the bottle into a bin, and saw that, despite the cap being tightly screwed on, it had crumpled, as if squeezed by a fist; this was physical proof of the vast difference in air pressure. And Haleakala is "only" at about 10,000 feet.

My point here is that outside of a thin membrane of habitability, proportionally thinner than an eggshell, the Universe is actively trying to kill us. And yet we continue to push that boundary to learn everything we can about this vast, uncaring expanse.

Whether we find proof of Planet Ix or not, there's something to be said for the lengths we'll go to in order to try.

6 Like • Comment • View 3 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.