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 8 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 10 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.

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

March 19, 2024 at 9:18am

#1066562

Stardust

The equinox will occur later today (or early tomorrow if you live in certain time zones), so it's appropriate that this cosmological link resulted from my random die roll today. Warning: It's from Salon, a source I don't normally use and that might be problematic.

Sure, we're all made of stardust. But what does that really mean?
We often hear that our bodies contain elements from the stars. But how do we know this for sure?

Personally, I think it's the most meaningful, profound insight in all of science. As far as I know, most cultures have an origin story: we were shaped from trees; we appeared out of literal nothing; we were formed from clay; whatever. I use passive voice there because, usually, gods were involved, powerful entities that predated humanity. Some origin stories also attempt to explain where those creator gods came from, but you quickly run into the turtle problem: if Earth is a flat disc resting on the back of a giant turtle, what's holding up the turtle? Another turtle? Well, what's holding up that one? It's turtles all the way down.

Now, you could say, "But science runs into the same problem. If all that exists was created in a Big Bang, how and where did the Big Bang... well... bang?"

This is a fair question, however it's stated. And science doesn't have a definitive answer. The difference between that and creation myths is that science doesn't pretend to have a definitive answer.

In any case, one reason I find it meaningful is that it doesn't just focus on one human tribe, or even on humanity in general, but the entire universe; that is to say, everything. How humans came to be from non-humans is the focus of evolutionary biology, not cosmology. And how different tribes formed is in the purview of, among other disciplines, anthropology. I'll grant that the creation stories our ancestors came up with have a certain poetry to them (and say an awful lot about the cultures involved), but to me, they all read like "Just-So" stories. Or, well, vice-versa. But my point stands.

With that, I won't be quoting too much from the article. That was really the thing I most wanted to say.

As the poets at NASA put it, “from the carbon in our DNA to the calcium in our bones, nearly all of the elements in our bodies were forged in the fiery hearts and death throes of stars.”

Poetry again. It depends what you mean by "nearly all."

By mass, it's mostly oxygen (the "ox" in "Hydrox" from yesterday's entry). By number of atoms, we're mostly hydrogen (the "hydr" in "Hydrox" from yesterday's entry). You might also recognize these elements as the atoms of water, and I'm sure you've heard that we're well over 50% water, so that all tracks, though both hydrogen and oxygen are part of other molecules in our bodies, notably carbon-based ones.

Point being that hydrogen is mostly primordial, not forged in stars. Though an argument could be made that, at the very least, protons are mostly primordial, no matter how many of them are in a given nucleus.

And they’ve been around far longer than we have. Light elements started forming an estimated 14 billion years ago, actually, in the first few minutes after the Big Bang, though others didn’t come around till a few hundred thousand years later when the universe cooled down enough for electrons to stay in orbit around atomic nuclei.

That... doesn't quite mesh with my understanding, though I admit my understanding may be off. An element is defined by the number of protons in its nucleus, irrespective of the number of electrons associated with it. A proton is a hydrogen atom (technically, it's a hydrogen ion, but whatever). So once you have free-roaming protons, you have hydrogen. Therefore, it doesn't matter if a proton (or a bound pair or triad of protons, together with the appropriate number of neutrons) has electrons or not; it's still hydrogen, helium, or lithium, respectively.

In fairness, the article calls out these elements in the next paragraph; I just have quibbles with how the information is phrased.

But you don’t just kill a star and get an entire cupboard of elements suitable for whipping up whatever material good — whether Uranus or, with apologies, your anus — you’re after.

Apology absolutely not accepted. Seventh-planet puns are never funny. Well, almost never.

Apart from that, well, the article delves into the science of it all, and I won't rehash it further. Suffice it to say, for my purposes, that we know, insofar as we can ever truly know anything, that we are indeed stardust. So is your dog. And poison ivy. And plastic and concrete. The good, the bad, and the ugly.

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