Complex Numbers (Book)

Blog Calendar

◄ February ► 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
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

February 4, 2023 at 12:01am

#1044216

Turning Over a New Leaf

Here's a confession:

I'm not entirely confident when it comes to picture prompts.

But I do feel like it's important to try new things, to, as the kids say, step outside of my comfort zone and do something I'm not sure of, like this month's round of "Journalistic Intentions"

[18+].

One thing I continue to do: pick prompts at random. Today, we have this lovely photo of a leaf under a blanket of water: Orange Submersion

I've seen reports that at least one person is having trouble with xlinks, so if that hyperlink doesn't work for you, I can provide the raw URL on request.

Ever wonder why water is (mostly) clear? I have. I mean, apart from when it's murky from suspended particles, or sometimes when mixed with delicious booze (which is itself, in its pure form, clear). And it's clearly (pun intended) not the same as air; you can almost always tell where one ends and the other begins. The surface of pure water is easy to identify, but, for me at least, damn near impossible to render in a drawing. But then, I've never been very good at drawing... but I digress.

The best answer to why water is clear that I've been able to come up with is that this is not the right question to ask.

Life crawled up onto land "only" about half a billion years ago. In comparison to the four billion or so years since life began on Earth, that's a significant fraction of time, but it means that life was changing and evolving for 7/8 of its history underwater. And some of that life, at least the animal portion, found a competitive edge in being able to directly sense prey or predators: in short, vision is a very useful sense to possess.

It's my understanding that eyes evolved several different times. That is, there's not one proto-organism that gradually turned light-sensitive cells (which many organisms have, not just animals) into an eyeball, which then split off into different species. No, the proto-organism might have been mostly blind, and some of its descendants developed vision in different ways: compound eyes like ours, or the simple eyes of arthropods and such, or whatever.

But most of those eyes originally evolved underwater. Thus, they developed in such a way that vision would be an evolutionary advantage, which means being sensitive to a range of the electromagnetic spectrum to which water is mostly transparent.

Water isn't clear because of some innate property of it; it's clear, to us, because our distant ancestors evolved a sense that allowed them to see in it. Water blocks some other wavelengths.

But the other thing it does is change the direction of light at its boundary. Stick your arm into a fish tank, and it'll appear to bend. At some angles, the light doesn't escape the water at all, but reflects off of it like a perfect mirror (this is a function of index of refraction, and it's also how fiber optic cables transmit data). The surface is also partially reflective when viewed from above, which is how you get the artistic dappled effect from ripples, like in today's picture prompt.

Leaf (pun absolutely intended) it to me to get all sciencey about a pretty picture. But I'm of the considered opinion that the knowledge gleaned from science can improve an aesthetic experience. I know the math and physics behind rainbows (or, well, I used to, and can easily find the information again), but that doesn't decrease one's visual impact.

And if today's discussion hurt your brain, just be glad I didn't go into the biology of what makes leaves turn color and fall off as winter approaches.

Another time, perhaps.

5 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.