Complex Numbers (Book)

Blog Calendar

◄ November ►
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:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

Access Restricted

This is a private area. Access has been restricted and you may not view it under its current settings. If you feel you should have access, please contact the area's owner and request access.

Note: To contact a member via email, click the small envelope icon to the right of their handle. You must be logged into your account to send an email.

2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 21, 2024 at 10:23am

#1080285

Experimental Metaphysics

My random number generator likes to have a laugh at my expense sometimes; today, it came up with another Quanta piece. This one, however, doesn't involve scary numbers.

‘Metaphysical Experiments’ Probe Our Hidden Assumptions About Reality
Experiments that test physics and philosophy as “a single whole” may be our only route to surefire knowledge about the universe.

It does, however, discuss science and philosophy, as the headline warns.

Metaphysics is the branch of philosophy that deals in the deep scaffolding of the world: the nature of space, time, causation and existence, the foundations of reality itself.

Once something can be experimentally verified, though, it ceases to be metaphysics and becomes... physics. Or fact. Plenty of things we (for varying definitions of "we") know (for varying definitions of "know") now (for varying definitions of "now") were once in the realm of metaphysics.

It’s generally considered untestable, since metaphysical assumptions underlie all our efforts to conduct tests and interpret results.

And if it's untestable, it's not science. At least, not yet; the untestable can become testable. I've said before that philosophy guides science, while science informs philosophy. I stand by that assertion.

What the article (and it's a fairly long one) focuses on is what to do when we don't even know that we're using philosophy to guide science.

Intuitions we have about the way the world works rarely conflict with our everyday experience.

Well, yeah, because intuition is largely based on everyday experience.

But at the uncharted edges of experience — at high speeds and tiny scales — those intuitions cease to serve us, making it impossible for us to do science without confronting our philosophical assumptions head-on. Suddenly we find ourselves in a place where science and philosophy can no longer be neatly distinguished. A place, according to the physicist Eric Cavalcanti, called “experimental metaphysics.”

My first reaction is to say that the phrase is misleading. However, sufficient explanation can suffice to un-mislead it. If this article isn't sufficient explanation, I don't know what would be. A book, maybe? For once, I don't see a book ad hidden in the article.

You should definitely go to the link just to see the pictures of Cavalcanti, though. I think he and Orlando Bloom were separated at birth.

In experimental metaphysics, the tools of science can be used to test our philosophical worldviews, which in turn can be used to better understand science.

Okay, I don't have a problem with that.

THE DIVIDING LINE between science and philosophy has never been clear. Often, it’s drawn along testability.

Yeah, that's where I, an absolute amateur, generally draw it, at least in my head. I will note, though, that the system we call "science" developed out of what used to be called "natural philosophy."

However, astronomy developed out of astrology, and chemistry out of alchemy, so I'm not sure that's a win for natural philosophy.

As it turns out, though, the testability distinction doesn’t hold. Philosophers have long known that it’s impossible to prove a hypothesis.

Which is why scientists test for whether something can be falsified, not proven.

In 1906, though, the French physicist Pierre Duhem showed that falsifying a single hypothesis is impossible. Every piece of science is bound up in a tangled mesh of assumptions, he argued.

Dang ol' French, messing things up for everyone.

Take, for instance, the geometry of space-time. Immanuel Kant, the 18th-century philosopher, declared that the properties of space and time are not empirical questions. He thought not only that the geometry of space was necessarily Euclidean, meaning that a triangle’s interior angles add up to 180 degrees, but that this fact had to be “the basis of any future metaphysics.”

And this is why we don't take the words of philosophers as absolute truth. Especially those of Kant. (I will resist the obvious pun.)

The unit of empirical significance is a combination of science and philosophy. The thinker who saw this most clearly was the 20th-century Swiss mathematician Ferdinand Gonseth.

You will note that he wasn't French. He probably spoke the language, though.

For Gonseth, science and metaphysics are always in conversation with one another, with metaphysics providing the foundations on which science operates, science providing evidence that forces metaphysics to revise those foundations, and the two together adapting and changing like a living, breathing organism. As he said in a symposium he attended in Einstein’s honor, “Science and philosophy form a single whole.”

I don't think I'd ever heard of this dude before, but that sounds awfully familiar. Like it's very close to my own ideas, reached independently, as noted above. Which means I have to be careful, lest my own confirmation bias kick in.

The article goes on, like I said, for a while, with examples of the intersection of philosophy and science.

Then, what to me is a pretty important example:

Michele Besso, Einstein’s best friend and sounding board, was the only person Einstein credited with helping him come up with the theory of relativity. But Besso helped less with the physics than with the philosophy. Einstein had always been a realist, believing in a reality behind the scenes, independent of our observations, but Besso introduced him to the philosophical writings of Ernst Mach, who argued that a theory should only refer to measurable quantities. Mach, by way of Besso, encouraged Einstein to give up his metaphysical notions of absolute space, time and motion. The result was the special theory of relativity.

Also not French. But the real point here is: in order to come up with his famous, and now well-supported, theory of relativity, Einstein had to shed some basic assumptions that he, maybe, didn't even realize he had. Once he did that, he made history and changed both science and philosophy completely.

I won't quote more from the article, but I read the whole thing. I'll just say this: Einstein was famous for, among other things, popularizing the concept of the thought experiment, and the article leans heavily on that particular technique for figuring things out. And thought experiments, when based on known science, basically philosophy.

I rest my case.

3 Like • Comment

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