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

September 5, 2022 at 12:01am

#1037330

Out Of Time

I expect the answer to every headline question with a potential binary result to be "no."

Can we time travel? A theoretical physicist provides some answers

Time travel makes regular appearances in popular culture, with innumerable time travel storylines in movies, television and literature.

Yes, and most of them do something self-inconsistent for the sake of plot.

But it is a surprisingly old idea: one can argue that the Greek tragedy Oedipus Rex, written by Sophocles over 2,500 years ago, is the first time travel story.

Well, we'll have to go back and ask Sophocles, wouldn't we?

But is time travel in fact possible? Given the popularity of the concept, this is a legitimate question. As a theoretical physicist, I find that there are several possible answers to this question, not all of which are contradictory.

Clearly, a theoretical physicist has more knowledge in this area than I do, but that doesn't stop me from making comments.

The simplest answer is that time travel cannot be possible because if it was, we would already be doing it.

Even the simplest answer runs into a complex problem; in this case, tenses. Does he mean that we would have invented it by now, or we will-would invent it in the future, sending schlubs back to our time along with, presumably, the technology?

Of course, future-us (assuming there is one, which is a big assumption right now) might be walking among us unseen, undetected. So might aliens. Or fairies. Maybe the fairies are the time travelers. Maybe the alien fairies are time travelers.

All of which is great fodder for fiction writing, but, without evidence, I can no more accept any of that than I can Russell's Teapot.

One can argue that it is forbidden by the laws of physics, like the second law of thermodynamics or relativity.

As we do not know all the laws of physics, and probably never will, there's always going to be some room for someone to go "but we don't know everything and we've found loopholes in 'laws' before."

There is also the matter of time-travel paradoxes; we can — hypothetically — resolve these if free will is an illusion, if many worlds exist or if the past can only be witnessed but not experienced.

Free will is probably an illusion, but we can't write time travel stories based on that, because most people need to believe that free will is real. Also they like protagonists with some agency, and this takes away their agency.

Even just witnessing the past presents potential paradoxes. To witness an event, one must intercept some of the light from that event, light that, presumably, would have affected other atoms. Sure, it might be a really, really tiny change, but, well, that's all it takes for chaos to shift. Butterfly effect and all that.

Unless, of course, the interception was always part of the past and the future-us only will-did cause an effect that had already happened.

I told you tenses were insufficient.

We can actually design time machines, but most of these (in principle) successful proposals require negative energy, or negative mass, which does not seem to exist in our universe.

FTL drives suffer from the same problem. But hey, we can't rule out creating exotic matter at some point.

Mathematical physicist Frank Tipler conceptualized a time machine that does not involve negative mass, but requires more energy than exists in the universe.

So we will have to learn to tap the energy of other universes.

Time travel also violates the second law of thermodynamics, which states that entropy or randomness must always increase. Time can only move in one direction — in other words, you cannot unscramble an egg.

Here's the thing, though: our brains are also constrained by the Second Law. Our perception "moves" in the same temporal "direction" as the egg-scrambling.

There is no doubt that if we could time travel freely, we run into the paradoxes. The best known is the “grandfather paradox”: one could hypothetically use a time machine to travel to the past and murder their grandfather before their father’s conception, thereby eliminating the possibility of their own birth. Logically, you cannot both exist and not exist.

Plenty of ways to get around paradoxes. The easiest is to accept that we don't have free will.

Kurt Vonnegut’s anti-war novel Slaughterhouse-Five, published in 1969, describes how to evade the grandfather paradox. If free will simply does not exist, it is not possible to kill one’s grandfather in the past, since he was not killed in the past. The novel’s protagonist, Billy Pilgrim, can only travel to other points on his world line (the timeline he exists in), but not to any other point in space-time, so he could not even contemplate killing his grandfather.

It's been a while since I read that. What I find amusing about Vonnegut is that he was absolutely a science fiction writer, but refused to describe himself as one, because so much of science fiction was space opera, and he was serious.

Could we allow for actual modifications of the past, so that we could go back and murder our grandfather — or Hitler?

The Hitler-killing thing isn't even a trope anymore; it's a cliché. But let's assume for the moment that we will invent time travel, and that time travel will-would allow future-us to alter the timeline. Logically, future-us would have a vested interest in keeping the timeline the way it is, if only to assure they will have come into being in the first place.

Or, just maybe, Hitler was the least of all possible evils. I mean, his regime almost developed an atomic bomb and a means of delivery over long distances. Perhaps future-us will have decided that losing 15 million people in conventional war and holocaust was better than losing a billion in nuclear holocaust. Trolley Problem, writ large.

If we will-would be able to go back and stop all of history's evil actors, then why were there evil actors?

In other words, what if we invent time travel and use it to go back and create the best of all possible worlds. And what if we're living in the result?

Time travel conjectures make one's head hurt. Especially because whenever I bring up that last argument, someone hits me upside the head. Sometimes that "someone" is me.

So is time travel possible? Probably not, but we don’t know for sure!

Yeah. I'm going to go with "no." But that's not going to stop me or anyone else from writing stories about it.

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