Complex Numbers (Book)

Blog Calendar

◄ November ► 2022
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

November 7, 2022 at 12:02am

#1040359

It Was Aliens

When faced with the unexplained, obviously, the explanation is aliens.

The Guy Who Discovered DNA Thought Aliens Sent It

How did life on Earth start? No one knows.

But we have some pretty compelling hypotheses.

We know all about evolution and DNA replication...

No, we do not.

...much more than our ancestors did, but we still have nothing but theories when it comes to explaining how nonliving matter ever started living.

Cracked here is using "theories" in its layperson sense, not scientifically.

One theory says this process, abiogenesis, never happened on Earth at all. Life came to Earth fully formed as simple microbes from some other planet, then it spread and evolved. Earth has never had the conditions for abiogenesis as far as we can tell (whatever those conditions might be), but an alien planet could have those conditions.

Except that we're pretty sure Earth did have such conditions at one point.

I'm sure I've mentioned this before, but going to "aliens did it" just kicks the can down the road. What we know beyond any reasonable amount of doubt is that life, somehow, started; as evidence for that, well, look in a mirror: life on Earth exists. I'm not saying it definitely, absolutely, got its start from non-life here on Earth, but I think we'd need to rule that out before jumping to any extraterrestrial origin hypotheses.

Which doesn't mean that I think life doesn't exist elsewhere also. I'm just saying it's within the realm of possibility that nonliving matter began to do its organizational thing right here in our own oceans, leading eventually to the development of DNA, cells, mitochondria, and, eventually, Instagram influenzas, without aliens being involved in the process. Though I'm still not sure about the influenzas not being aliens.

Francis Crick—Nobel prize winner and part of the team who first observed the structure of DNA—weighed in on the subject in 1973.

Crick, of course, didn't work alone (James Watson got co-credited for the discovery, which happened almost exactly 70 years ago), and there's good reason to believe that it was Rosalind Franklin who actually discovered DNA as such, though for some reason—I can't quite put my finger on it—he got a Nobel Prize but she did not. None of this, however, diminishes Crick's contribution to the field. Going on about aliens might, though.

The idea that spores from a different planet just happened to make their way to Earth ("panspermia") is too unlikely, said Crick. But you know what he said is a lot more likely, and which we have to consider? A theory dubbed "directed panspermia": Aliens seeded life on Earth on purpose.

Again, sure, maybe. We don't know. But we shouldn't jump to that conclusion. (Also, it's entirely within the realm of possibility that spores can survive interstellar travel.) And it still doesn't answer the actual question, which is: how does nonlife become life? That is, whether it happened here or elsewhere.

Crick also had some more specific arguments beyond this speculation (at one point in his paper, he says, "the psychology of extraterrestrial societies is no better understood than terrestrial society," realizing his reasoning is getting kinda "out there"). Life evolved to require such rare elements as molybdenum, which would make more sense if it started somewhere in which that element was more common. Also, Crick discovered that all life shares a universal genetic code, which is odd but would make sense if we all evolved from one kind of germ that infected our planet. That germ's home planet, however, probably had numerous separate very different genetic codes.

That's a whole wheelbarrow of speculation right there. I'm certainly not an expert in biology, but there are multiple ways that rarer elements can get concentrated here on Earth through natural, pre-biotic processes. If that weren't the case, we wouldn't find deposits of said elements. All it would take would be one well-placed deposit. One straightforward way is a meteor hitting Earth, which I suppose actually qualifies as an extraterrestrial origin, though not "directed panspermia." Around the time that life started here on Earth, there were a lot more meteors flying around.

As for the "life shares a universal genetic code" thing, that can be explained by DNA out-competing any other potential blueprint for life, analogous to how sapiens eventually out-competed neanderthals.

We're all comforted to learn that aliens might have sent life to Earth, like in Prometheus.

No, I'm not comforted by that at all; I'd still want to know how life began in the first place, here or elsewhere. Not that comfort has anything to do with science. Besides, I didn't like that movie.

At least this article correctly labels this as "speculation." Speculation is an important part of science. But just because someone's a Nobel Prize winner doesn't mean they're always going to be right.

Occam's Razor indicates that we shouldn't needlessly multiply entities, and throwing sentient aliens into the mix absolutely multiplies entities unnecessarily. It's equivalent to saying "we don't know, so God did it." Sure, it could turn out that he was right all along. I'll be sure to ask the aliens when they finally show up here.

6 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.