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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Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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.

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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.

3 Like

October 18, 2018 at 12:38am

#943662

Alone

https://www.livescience.com/63843-look-harder-for-aliens.html

I like Star Trek and Doctor Who as much as anybody, and more than most, but the more I learn, the less I think we'll find spacefaring aliens anytime soon.

What it comes down to for me is only one factor in the Drake equation.

If you don't know, the Drake equation isn't anything like a real mathematical equation. It doesn't describe anything in physics or astronomy or any other science. It is merely a starting point for thinking about the possibilities for extraterrestrial life.

Let's take a look at it (from Wikipedia

):

The Drake equation is:

N = R_∗ ⋅ f_p ⋅ n_e ⋅ f_l ⋅ f_i ⋅ f_c ⋅ L

where:

N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);

and

R_∗ = the average rate of star formation in our galaxy
f_p = the fraction of those stars that have planets
n_e = the average number of planets that can potentially support life per star that has planets
f_l = the fraction of planets that could support life that actually develop life at some point
f_i = the fraction of planets with life that actually go on to develop intelligent life (civilizations)
f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space

Before I go any further, I'm going to call "intelligent life" anyone who can build spaceships. Like us. So we can stow the inevitable jokes about not finding intelligent life on Earth for a bit. And we're not talking about dolphins or cats or other superior species; just the ones who build spaceships.

My point of contention is f_i. We have exactly one data point for this: us. And it took 4-5 billion years from Earth's formation for us to appear, and Earth came into being about 10 billion years after the universe began. That's a long damn time, especially compared to the 60 or so years that we've been building spaceships.

And here's the thing that bugs me: some humans like to think of their species as the pinnacle of evolution, but the evidence for that is lacking. It's a matter of perspective, you see - certainly we've been successful from a species survival standpoint, but so have roaches. Intelligence - technological prowess - is obviously not a requirement for species success, and given the existence of nuclear weapons, I'd argue that it's no guarantee of it, either. The ecosystem would do just fine without us (some would argue it would do better, but that's not my point right now).

Nor is there anything that mandates the evolution of technological ability. Life went on long before that particular trait came into being.

So we don't know. It could happen a lot. Maybe it only happens rarely. With only one data point, we have no way of knowing. Perhaps it's a little like winning the lottery - you only have a one in (some astronomical number) chance of winning it, but once you've won it, your chance of having won it is unity. And we're looking at the cosmic lottery from the perspective of having won it.

Therefore, it hardly matters what the other terms are, if the chance of intelligent life developing is only one in 10¹⁰⁰. That's as close to zero as doesn't matter, even in a universe as inconceivably large as ours. And the Drake equation is focused on our galaxy alone. Even if every other fraction in the equation is effectively 1, and even if there's a trillion stars in the galaxy (estimates tend to be lower than this), if that one factor is 1 in 10¹⁰⁰, we could easily justify the number of technological civilizations in our galaxy to be 1.

Do I think there's other "intelligent" life out there, somewhere? Yes. I've said so before. Big universe and all that. But if it's not in our galaxy - nay, even if it is in our galaxy and, say, on the other side of the core - it might as well not exist for all that we'll ever be able to contact it or vice-versa. Absent, of course, revisions to the known laws of physics.

To be clear, even finding a goddamn microbe of non-terrestrial origin would be of earth-shattering importance (hopefully not literally). I think it likely that this will happen, given how short a time it was from the formation of Earth to the first stirrings of life thereon. But it hasn't happened yet, so we are still in the realm of speculation.

I hope I'm wrong. I really do. And I think it's important to keep looking, if only for the other things we can learn from the search. But maybe we also need to work on becoming the superior beings we wish we could believe in.

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