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.

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.

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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.

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').

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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 17, 2022 at 12:01am

#1037775

On Time

Hey look: a headline question to which the answer is "yes."

Does time really exist?
We take for granted that time is real. But what if it's only an illusion, and a relative illusion at that? Does time even exist?

Astute readers who follow the above link will note that, above said headline, is an important string of characters; to wit: "April 26, 2022." In other words, they're asking if time exists right after specifying the time coordinate (to within 24 hours) when the article was published.

Still, even though I take a very materialistic stance on the subject, it's worth thinking about, and the article makes some interesting points. It also ends up agreeing with my philosophical stance, so there's that.

In a philosophical sense, we’re taught to doubt and question everything.

No, "we" are not. For various values of "we."

Even the reality of ourselves and our own experiences are up for debate, as we have to make certain assumptions about how trustworthy our sensors — and our own senses, for that matter — actually are in order to arrive at any satisfactory conclusions.

Stub your toe on the foot of a furniture and then tell me it doesn't exist.

I used to work with a guy who was devoutly religious. He kept showing me optical illusions, which of course I appreciate. But it dawned on me one day that he was sneakily trying to tell me, "Look here, you can't trust your senses. You can only trust in God." Whereas the conclusion I draw from optical illusions is: "Things can be different from what they seem. Therefore, you need science to explain why your senses are deceiving you." Because every one of those illusions had a very good explanation.

Sure, certain things might appear real, but isn’t it possible that those appearances are deceiving, and that quantities or concepts that we take for granted might be nothing more than very convincing illusions?

Again: if it can be measured, quantified, and tested; or if it gets in the way of your really very pain-sensitive toe, it's "real." The chair I'm sitting in keeps me from falling to the floor. The floor itself keeps me from falling to the center of the Earth. All due to gravity, which can't be seen, but can damn sure be measured. To claim otherwise is to muddle the distinction between "reality" and "illusion."

Sure, an argument can be made—and I've made it myself—that what we sense isn't the whole picture. That's fine. Zoom in far enough and everything's energy, space, and quantum fields. What we sense is what was useful for our ancestors, going back to the first molecule of proto-life, to find energy and avoid becoming someone else's source of energy for as long as possible. It may all—consciousness, solidity, time, space—be an emergent property of some other process, but that doesn't make it any less real.

We’ve learned lots of surprising and counterintuitive lessons from our investigations of time. Time is relative, not absolute.

True. Doesn't mean it's an illusion. Sure, your "now" is (very, very slightly) different from mine, but we both sense the flow of time.

And as I've also banged on about in here, if anything's an illusion, it's the concept of "now." You don't actually live in the present; only the past and the future are real. By the time you realize it's "now," it's not "now" anymore.

Time always marches forward, not backward, but we still lack an explanation for the arrow of time.

Eh. Maybe.

Thermodynamically, the Universe has an arrow of time, which “flows” in the same direction as increasing entropy.

We, as the sensor, are also subject to the laws of thermodynamics—which are, indeed, emergent properties; they are the result, not of a single particle or field, but of a collection of wave functions. Heat, for example, isn't the property of an individual atom, but of billions of them all bouncing off each other. Is heat, therefore, an illusion? If you think it is, you've got some 'splainin' to do to some really sweaty folks.

And when we investigate the Universe on a fundamental level, it turns out that time may not be fundamental at all.

"Not fundamental" doesn't equate to "not real."

When it comes to the question of existence, physics is very simple and straightforward about what it considers to be a satisfactory answer.

Can you measure it?
Can you quantify it?
Can you define it in a mathematically self-consistent way?
Is it, itself, an observable quantity, and do other observables depend on it in an inextricable way?

If your answers to these questions are all in the affirmative, there’s no way out of it: you’ve got yourself a quantity that exists.

Which is, to an extent, what I've been saying. Only the article is far more rigorous.

Then the article applies these questions to time:

You might think, then, that perhaps time itself is pathological [their word for, basically, nonsensical and not understood by science]. Sure, we can measure it, quantify it, and even observe both its passing and the consequences of its passing. But shouldn’t it matter that your measurements of “how much time has passed” between the start and end of an event depends entirely on where you are and how you’re moving when you’re making those observations?

My answer: no, because we can quantify and measure and define the differences in the way time works in different frames of reference. If we couldn't, GPS devices would simply not work. For example.

And I'm not going to paste the long-winded example, taken pretty much directly from Einstein, but spoiler: I wasn't too far off the mark.

But could it be the case that, perhaps, we only perceive time to exist, and that it isn’t, in fact, actually real?

We can consider this from a particular perspective: looking at the notions of symmetries in physics. After all, the laws of physics, at least as we know them, are time-symmetric.

The laws of physics are, yes. Most of them, anyway. The laws of thermodynamics, however, are not.

But there are two ways to identify a physical difference between progressing forward in time and backward in time. The first is by looking at reactions that proceed via the weak nuclear force, such as radioactive decays.

The article gets a little technical here, but it should be clear to anyone with a little knowledge in the area that uranium decays into other stuff, which then doesn't turn around and undecay back into uranium. For example.

The second way, however, is even more familiar to most of us. Every time you:

scramble an egg,
drop a full glass of water onto the ground and watch it shatter,
or simply open the door between a hot room and a cold one,

you are creating a situation where there will be a thermodynamic arrow of time.

And that process is also going on in your brain, which is observing all that.

However, there are two important caveats to this discussion. While it’s true that time is real, it’s important to keep the following facts in mind.

Too much text to copy here; you'll have to look at the article. Basically, we don't know everything. Well, duh. We knew that we don't know everything. That's what makes science so damn interesting.

Despite the popular trend to question the nature of time, its physical “realness” is not in doubt. Time is an integral part of the Universe, and the boundary between events that have been observed or measured to have a definitive outcome and those whose outcome has not yet been decided is the best way we have to define, precisely, what we mean by the moment of “now”.

And, again, philosophically, it's important to think about these things. But as I've said before, if time is an illusion, then space is also an illusion, which makes everything we sense an illusion, which in turn calls into question the definition of "illusion."

So, as I said above, this is a rare case of the answer to a headline question being a "yes." It is, of course, a qualified "yes," as with most things in science. But the fact that there's a headline that can be answered with a "yes" calls into question the very foundations of the laws of the universe. That's what we should be worried about, not that time might be illusory.

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