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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 20 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 22 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.

February 4, 2020 at 12:09am

#974835

Life, The Universe, and Everything

It's an Internet truism that the answer to any headline that asks a yes/no question is, at least 95% of the time, "no."

http://nautil.us/issue/42/fakes/is-physical-law-an-alien-intelligence

Is Physical Law an Alien Intelligence?
Alien life could be so advanced it becomes indistinguishable from physics.

Terrestrial life is indistinguishable from physics. That is to say, drill down deep enough and life is a complex chemical reaction that's been going on for about 4 billion years, give or take, and the underpinning of chemistry is physics.

But, okay, I know that's not what the author meant.

After all, if the cosmos holds other life, and if some of that life has evolved beyond our own waypoints of complexity and technology, we should be considering some very extreme possibilities.

Considering, sure. Taking our speculations at face value? Not so much. Just because we can think something, that doesn't make it true.

...it would neatly explain why we have yet to see advanced intelligence in the cosmos around us, despite the sheer number of planets it could have arisen on—the so-called Fermi Paradox.

So would my ruminations about the Fermi "Paradox" in here, what, two entries ago? I'm under no illusion that I'm any more right than this speculation is, but at least my speculation passes the parsimony test.

Indeed, perhaps our universe is one of the new forms into which some other civilization transcribed its world.

Ah, it's the old "maybe the universe is a simulation" canard again. No, I can't prove that it is or isn't. But that's one problem with this hypothesis: it's not falsifiable. The other problem with it can be summed up as: So what? Assume that the universe is a sophisticated simulation. What changes? We're still obligated to act ethically. We're still curious about the universe. We've still got science to search for answers and understanding. Unless you think The Matrix was a documentary (spoiler alert: it was not), all it does is kick the Consciousness Question can down the road.

Let me explain further. Assume, again, that this universe in which we live is, indeed, someone's sophisticated simulation. Where, then, does this "someone" live and what are the laws governing their existence? It seems likely that, as technology advances, we will at some point be able to create our own simulations, of ever-increasing complexity. But there's a hard limit - not one born of technology, but woven into the very fabric of reality, and that is that no simulation can approach the complexity of the universe in which it was born. So we create a simulation, and the simula-tees evolve and make their own simulations of lower complexity, and so on until it becomes painfully obvious that a certain iteration is a simulation. Now, which explanation is more parsimonious: that we just happen to be at an intermediate stage of simulation complexity, one that seems incredibly complex indeed, rather than a "daughter" simulation that is, necessarily, less complex? Or that we're in an actual, real, physical universe?

Given all the available evidence, my money's on the actual, real, physical universe. Nothing else makes statistical sense. Sure, you can say "but maybe that hard limit was programmed in to just our simulation." This smacks of special pleading, and you end up bleeding out from a cut from Occam's Razor.

These possibilities might seem wholly untestable, because part of the conceit is that sufficiently advanced life will not just be unrecognizable as such, but will blend completely into the fabric of what we’ve thought of as nature. But viewed through the warped bottom of a beer glass, we can pick out a few cosmic phenomena that—at crazy as it sounds—might fit the requirements.

"warped bottom of a beer glass?" More like "warped haze of weed smoke."

In that case, dark matter could contain real complexity, and perhaps it is where all technologically advanced life ends up or where most life has always been.

Sure, and Halle Berry could show up at my door tomorrow, wearing a Catwoman suit and carrying a couple of growlers of craft beer. That is, technically, a possibility. Do you want to bet on it?

Or to take this a step further, perhaps the behavior of normal cosmic matter that we attribute to dark matter is brought on by something else altogether: a living state that manipulates luminous matter for its own purposes.

Considering that the evidence we have for dark matter, whatever it ends up being, extends to clusters of galaxies that existed billions of years ago, what this is implying is that intelligence sprang, like Athena from the head of Zeus, fully formed at or near the birth of the universe. Might as well try to turn God into a scientific hypothesis.

Consider that at present we have neither identified the dark-matter particles nor come up with a compelling alternative to our laws of physics that would account for the behavior of galaxies and clusters of galaxies. Would an explanation in terms of life be any less plausible than a failure of established laws?

Um. Yes. Unequivocally less plausible. "Established laws" are refined all the time. And the author, an astrophysicist, should already be aware that science hasn't called its theories "laws" since, like, Newton's time.

Once we start proposing that life could be part of the solution to cosmic mysteries, there’s no end to the fun possibilities.

Sure, and I'm a big fan of science fiction. That doesn't mean I think it reflects reality. Okay, so you've made a hypothesis. How do we test it? What solid predictions can it make?

Again, I'm not saying it's definitely wrong. Just that it's probably the least parsimonious alternative.

Okay, I'm not going to get into more of the article here. Read it for yourself; draw your own conclusions. If you want to argue from authority, well, that author is an authority, and I'm just another putz with a blog and a diminishing supply of neurons. Could be I'm wrong. But I maintain that just because we don't understand something, that's no reason to go multiplying entities endlessly.

On the other hand, speculation like this is a good thing, because at the very least, it provides hypotheses that could potentially be ruled out.

2 Like • Comment

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