Complex Numbers (Book)

Blog Calendar

◄ August ► 2020
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:

Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 8 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

August 21, 2020 at 12:05am

#991258

Quantum Speculation

I looked over some earlier entries and discovered a trend to rag on Scientific American articles.

Today's, though, I have fewer problems with.

https://blogs.scientificamerican.com/observations/what-does-quantum-theory-actua...

What Does Quantum Theory Actually Tell Us about Reality?
Nearly a century after its founding, physicists and philosophers still don’t know—but they’re working on it

In essence, this seems to be a good introduction to quantum theory, at least at its very basic.

For a demonstration that overturned the great Isaac Newton’s ideas about the nature of light, it was staggeringly simple. It “may be repeated with great ease, wherever the sun shines,” the English physicist Thomas Young told the members of the Royal Society in London in November 1803, describing what is now known as a double-slit experiment, and Young wasn’t being overly melodramatic. He had come up with an elegant and decidedly homespun experiment to show light’s wavelike nature, and in doing so refuted Newton’s theory that light is made of corpuscles, or particles.

It does, however, toss words around without much precision. For starters, Newton developed several "theories," but the particle nature of light was more of an assumption. And I'd argue that it's more of a refinement than a refutation. Photons act like both, depending on context. Lots of other stuff that Newton came up with turned out to need refinement, but the basic math is still sound, especially for everyday experience.

But the birth of quantum physics in the early 1900s made it clear that light is made of tiny, indivisible units, or quanta, of energy, which we call photons. Young’s experiment, when done with single photons or even single particles of matter, such as electrons and neutrons, is a conundrum to behold, raising fundamental questions about the very nature of reality.

The old argument "is light a particle or a wave?" is usually phrased thus. Personally, I think they're using the wrong terms, as usual. As with Newton, "particles" and "waves" are functions of our everyday experience. The question isn't why light has properties of both; the question is why macroscopic effects are one or the other.

Some have even used it to argue that the quantum world is influenced by human consciousness, giving our minds an agency and a place in the ontology of the universe.

All these years and we're still stuck on that nonsense.

The article goes on to describe the double-slit experiment. I remember doing a basic one of those in physics class when I was in college. It's one thing to hear it described; it's another to actually see it (and do the damned math).

The photon is not real in the sense that a plane flying from San Francisco to New York is real.

That's... misleading. Both are "real." One behaves differently than the other, is all.

Werner Heisenberg, among others, interpreted the mathematics to mean that reality doesn’t exist until observed.

Also misleading, and probably not the case. No, reality doesn't require us hairless apes to observe it, or believe in it. Well, to be fair, there is a nonzero chance that the universe sprang into existence, fully formed, complete with a coherent history. If this event also happened to create humans, then there might be something to this. But while that chance is nonzero, it's represented by an unimaginably small number. Otherwise, the evidence suggests that reality existed before we did, and will continue to exist under most foreseeable conditions after we're all gone.

But quantum theory is entirely unclear about what constitutes a “measurement.” It simply postulates that the measuring device must be classical, without defining where such a boundary between the classical and quantum lies, thus leaving the door open for those who think that human consciousness needs to be invoked for collapse. Last May, Henry Stapp and colleagues argued, in this forum, that the double-slit experiment and its modern variants provide evidence that “a conscious observer may be indispensable” to make sense of the quantum realm and that a transpersonal mind underlies the material world.

Sigh. Yeah, I know. I wasn't going to talk about panpsychism again. But if everything is conscious, than anything can be the "observer," not just humans.

On a related note, you've probably heard of the Schrodinger's Cat thought experiment. Look it up if you haven't; I can't be arsed to explain the whole thing. The upshot of it is that Schrodinger postulated that without an observer, a cat in a box subject to the randomness of quantum phenomena is neither alive nor dead, but in a state of superposition, until an ape opens the box and collapses the cat's wave function.

What's always bothered me about this (and as far as I know, no one has actually performed the experiment, much to the relief of half the world's cat population) is that in his scenario, there is one conscious entity who "knows" whether the cat is alive or dead; to wit, the feline in the box.

Also, there are other ways of interpreting the double-slit experiment. Take the de Broglie-Bohm theory, which says that reality is both wave and particle.

There you go throwing words around again. There's an upper limit to where wave/particle duality holds sway. As this article notes, that limit is as yet undetermined. But it's definitely smaller than a golf ball. A golf ball is unquestionably (except to the most drug-addled philosophers, which would seem to be a lot of them) a reality-based object, and it doesn't act like a photon in the double-slit experiment.

Crucially, the theory does not need observers or measurements or a non-material consciousness.

So when drug-addled philosophers start talking about how nothing exists unless we observe it, give them different drugs. Well, who knows. Maybe they're right. But I doubt it.

If nothing else, these experiments are showing that we cannot yet make any claims about the nature of reality, even if the claims are well-motivated mathematically or philosophically.

Decent conclusion, anyway. "We don't know yet" is a great answer in science.

And given that neuroscientists and philosophers of mind don’t agree on the nature of consciousness, claims that it collapses wave functions are premature at best and misleading and wrong at worst.

I'd be a lot more concerned if neuroscientists and philosophers agreed on anything.

4 Like • Comment • View 1 comment

© 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.