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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Write_Mikey_Write!

about 5 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.

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Adherennium Plotting something

about 7 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.

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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.

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.

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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.

April 4, 2024 at 11:15am

#1067553

It Ain't Nothin'

One of the more philosophically interesting implications of physics as we know it is that there is no such thing as "nothing."

We are not empty
The concept of the atomic void is one of the most repeated mistakes in popular science. Molecules are packed with stuff

Well, except maybe for the contents of your savings account.

The empty atom picture is likely the most repeated mistake in popular science. It is unclear who created this myth, but it is sure that Carl Sagan, in his classic TV series Cosmos (1980), was crucial in popularising it.

Part of the problem is that everyday language fails in its ability to describe sometimes esoteric scientific concepts. Take the word "nothing." It's obviously an ancient portmanteau of "no" and "thing," implying that when you describe a space as "nothing," there's "no thing" there. But the word "thing" isn't exactly precisely defined, especially when you get down to subatomic scales and discover that all particles are actually energy, and "thing" can be used to describe, well, anything, from stars to light to dark matter... maybe you see the semantic problem.

In other words, maybe the problem isn't the science, but the language used to communicate it. Sagan was brilliant, and a strong communicator with excellent fashion sense, but, being human, he'd have made mistakes, and he was working with a 1980s knowledge of science, which has advanced a bit since then.

All it takes to question the idea of the empty-space idea is to attempt to bring two fully solid objects together. Swords, say. If swords were mostly empty space, they wouldn't be very effective at parrying other swords. Or cutting, when you're not quick enough to parry. Swords swing just fine through the air, though, with only minimal resistance (unless you swing them broad-side-on), but air isn't "nothing," either.

Most problems surrounding the description of the submolecular world come from frustrated attempts to reconcile conflicting pictures of waves and particles, leaving us with inconsistent chimeras such as particle-like nuclei surrounded by wave-like electrons. This image doesn’t capture quantum theory’s predictions.

Saying that light, for example, behaves "sometimes like a wave and sometimes like a particle," betrays our macroscopic bias. If we'd learned the quantum stuff first, we might be confused by the "things" that only behave like waves (sound, e.g.) or only like particles (ping-pong balls, e.g.).

To compensate, our conceptual reconstruction of matter at the submolecular level should consistently describe how nuclei and electrons behave when not observed – like the proverbial sound of a tree falling in the forest without anyone around.

Two problems there:

1) How do we experimentally verify how these "things" behave when not observed? By definition, we'd have to observe them to find out.

2) A tree falling in the forest absolutely makes a sound. Sound is waves in air, caused by a transfer of kinetic energy; it exists whether an ear is there to pick up on it or not. The only way a tree falling in the forest would not make a sound would be if the forest were in a vacuum, in which case we have way more immediate problems than philosophical koans.

Here’s a primer on how to think of the fundamental components of matter: a molecule is a stable collection of nuclei and electrons. If the collection contains a single nucleus, it is called an atom. Electrons are elementary particles with no internal structure and a negative electric charge. On the other hand, each nucleus is a combined system composed of several protons and a roughly equal number of neutrons. Each proton and neutron is 1,836 times more massive than an electron. The proton has a positive charge of the same magnitude as an electron’s negative charge, while neutrons, as their name hints, have no electric charge. Usually, but not necessarily, the total number of protons in a molecule equals the number of electrons, making molecules electrically neutral.

I could quibble about some of the details there—for example, a neutron is slightly more massive than a proton—but that's a remarkably good summary, so long as it's understood that some details need further embellishment.

The interior of the protons and neutrons is likely the most complex place in the Universe.

Yes, even more complex than crowds at a Taylor Swift concert.

Particles with the same electric charge sign repel each other. So additional interactions are required to hold protons close-packed in the nucleus. These interactions arise from quark and antiquark pairs called pions that constantly spill out of each proton and neutron to be absorbed by another such particle nearby.

It occurred to me later in life that the grade-school simplicity of the seemingly contradictory statements "like charges repel; opposite charges attract" and "atomic nuclei are positively charged while the electrons are negatively charged" should be way, way more confusing to middle-school students than it is. Positively-charged protons in a nucleus stick together, and electrons don't just fall in and join with a proton to neutralize the charges. This article does a pretty good job explaining the basic physics behind why that's not a contradiction. I won't quote it.

Not gonna lie; it gets a bit technical. It needs to, though, to avoid the inevitable wobbliness of everyday word definitions, like "thing."

The association between this mass concentration and the idea that atoms are empty stems from a flawed view that mass is the property of matter that fills a space. However, this concept does not hold up to close inspection, not even in our human-scale world. When we pile objects on top of each other, what keeps them separated is not their masses but the electric repulsion between the outmost electrons at their touching molecules.

This bit kind of answers the implied question in the headline, so I'm including it.

My criticism of the empty atom picture isn’t meant to shame people’s previous attempts to describe atoms and molecules to the public. On the contrary, I applaud their effort in this challenging enterprise. Our common language, intuitions and even basic reasoning processes are not adapted to face quantum theory, this alien world of strangeness surrounded by quirky landscapes we mostly cannot make sense of.

Which is what I've been trying to say. Also, the Ant-Man movies didn't get it right, either. Fun movies, but don't use them as quantum physics lectures.

In the end, though, the idea of "something" and "nothing" will stick with us in our everyday language, and they are, I think, generally adequate to describe, say, the difference between a planet and the mostly-vacuum of space.

But even vacuum has something in it.

Not so sure about some humans' brains, though.

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