Complex Numbers (Book)

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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/

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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.

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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.

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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.

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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.

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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').

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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.

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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

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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

October 14, 2023 at 9:18am

#1057351

Quantum Mind Tricks

The article today, from The Conversation, is about a year old, but that shouldn't matter.

Four common misconceptions about quantum physics

But debates about quantum mechanics – be they on chat forums, in the media or in science fiction – can often get muddled thanks to a number of persistent myths and misconceptions.

Oh, it's way worse than that. I think there are still authors out there promoting books about harnessing the power of the quantum realm with your mind, or some such gobbledygook.

Remember: the bolded and italicized bits below, taken straight from the article, are the misconceptions. So if you skim this entry, please don't walk away thinking I'm endorsing misinformation.

1. A cat can be dead and alive

Erwin Schrödinger could probably never have predicted that his thought experiment, Schrödinger’s cat, would attain internet meme status in the 21st century.

I've met people who knew about Schrödinger's Cat, but weren't aware that it was a thought experiment. They believed Schrödinger had actually stuffed a cat in a box with a quantum choice contraption. I'm not ragging on them, but I think it's important to note that, to the best of my knowledge, no cats were harmed (or not harmed, or a superposition of the two) in the pursuit of knowledge about quantum physics.

Which is way more than other branches of science can say.

It suggests that an unlucky feline stuck in a box with a kill switch triggered by a random quantum event – radioactive decay, for example – could be alive and dead at the same time, as long as we don’t open the box to check.

The obvious issue with this thought experiment is that, if it requires consciousness to collapse a quantum state, a human doesn't need to open the box; a cat possesses consciousness and knows it's alive (or doesn't know anything, if it's dead).

Is it really both alive and dead as long as we don’t open the box? Obviously, a cat is nothing like an individual photon in a controlled lab environment, it is much bigger and more complex.

And that's the non-obvious issue.

In any event, Schrödinger came up with his Rube Goldberg cat-quantumizing machine idea to refute certain ideas about quantum physics, not to demonstrate its truth.

2. Simple analogies can explain entanglement

This one was way more immediately relevant last year, when the article came out, because 2022's Nobel Prize in Physics was all about quantum entanglement (this year's was about attosecond pulses of light, which, well, look it up; it's cool as hell).

There's a lot to absorb here, and I can't really do this section justice with cherry-picked quotes, but the upshot of it is this: There's no suitable macro-world analogy for quantum entanglement.

I'd also add that QE doesn't imply superluminal information transfer, as some people insist it means. It's plenty weird, but it doesn't defy the cosmic speed limit.

3. Nature is unreal and ‘non-local’

Another reminder that the above heading is false. But in this case, I'd add "probably."

Despite Bell’s theorem, nature may well be real and local, if you allowed for breaking some other things we consider common sense, such as time moving forward.

I've banged on in here about time on numerous occasions. Suffice it to say that, in the quantum realm, common sense needs to go right out the window.

I hate the concept, anyway.

Put another way, quantum equations, insofar as I understand them, don't have a time arrow. Time, then, is best viewed as an emergent property of macroscopic matter. Which is fine; lots of perfectly real things, such as temperature or life itself, are emergent phenomena.

However, most options on the table — for example, time flowing backwards, or the absence of free will — are at least as absurd as giving up on the concept of local reality.

This sentence, of course, is one of the main reasons I saved this article. The absence of free will isn't absurd at all; it is, as far as I'm concerned anyway, settled science.

4. Nobody understands quantum mechanics

But I'm not always right. For instance, I've crafted similar sentences to this #4 heading. This is my chance to qualify it; I do believe it's correct in at least one sense.

A classic quote (attributed to physicist Richard Feynman, but in this form also paraphrasing Niels Bohr) surmises: “If you think you understand quantum mechanics, you don’t understand it.”

I believe that quote is correct for people like you, me, and the author of Using Quantum Jedi Mind Tricks to Win the Lottery and Get Laid (or other books to that effect).

Quantum physics is supposedly impossible to understand, including by physicists. But from a 21st-century perspective, quantum physics is neither mathematically nor conceptually particularly difficult for scientists. We understand it extremely well, to a point where we can predict quantum phenomena with high precision, simulate highly complex quantum systems and even start to build quantum computers.

And while this is true—the calculations are, from what I've heard, far more accurate than in any other branch of science—that doesn't mean there aren't still arguments over what it all means. That is, questions of interpretation, like "many-worlds," are still open.

Where the true difficulty lies, perhaps, is in how to reconcile quantum physics with our intuitive reality.

Fair, because they are very different. I certainly don't claim to have it all figured out (unlike some writers), but as with anything else, that's not going to stop me from blogging about it.

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