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 14 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 16 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.

1 Like

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').

1 Like

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.

April 24, 2020 at 12:17am

#981887

The Big Not So Empty

Today, I'm going to talk about nothing.

http://nautil.us/issue/49/the-absurd/what-is-space

What Is Space?
It’s not what you think.

It's the Final Frontier. Duh.

Ask a group of physicists and philosophers to define “space” and you will likely be stuck in a long discussion that involves deep-sounding but meaningless word combinations such as “the very fabric of space-time itself is a physical manifestation of quantum entropy concepts woven together by the universal nature of location.”

I saw graffiti on a bathroom wall once, in college, that went something like:

And Jesus said, "Who am I?"

And his disciples said, "You are the eschatological manifestation of being, the very korygma of our existence."

And Jesus said, "What?"

...yes, this was in the philosophy department. The drama department had a penchant for adding two more arms to the final L of what was printed on the bumwad dispensers: "PRESS DOWN FOR NEW ROLL."

On second thought, maybe you should avoid starting deep conversations between philosophers and physicists.

This is essential life advice.

It turns out that the nature of space itself is one of the biggest and strangest mysteries in the universe. So get ready, because things are about to get ... spacey.

Pick up your favorite mind-altering substance and read the rest of the article.

Space is definitely not an empty void and it is definitely not just a relationship between matter. We know this because we have seen space do things that fit neither of those ideas. We have observed space bend and ripple and expand.

This is the part where your brain goes, “Whaaaaat ... ?”

The expansion of space is, itself, mind-blowing. An astronomer once described to me two variations of what could eventually happen to the universe. In the first variation, gravity pulls everything back together. In the second, in theory, the expansion of space (which is, after all, not just "out there," but everywhere, including within us) will eventually get to the point where not just gravity, but even the forces binding atoms together will be overcome, tearing everything apart.

"I don't know which one will happen," he said. "I'm torn."

I groaned appropriately.

Anyway, the article's worth reading, even though it gets pretty deep. No, because it gets pretty deep. That's why I recommended mind-altering substances above.

If your brain is not yet hurting from all these gooey space-bending concepts, here is another mystery about space: Is space flat or curved (and if it’s curved, which way does it curve)?

People get confused all the time about the "curvature of space." I think of it as analogous to the curvature of the Earth: under normal circumstances, we don't see it, and our brains think it's flat (and a bunch of people with small brains still think it's flat). But it curves in three dimensions. Space, then, curves in higher dimensions.

What would it mean for space to have a curvature? One way to visualize it is to pretend for a second that we live in a two-dimensional world, like being trapped in a sheet of paper. That means we can only move in two directions. Now, if that sheet we live in lies perfectly straight, we say that our space is flat.

But if for some reason that sheet of paper is bent, then we say that the space is curved.

Yeah, like that.

In this case, it turns out that we do have an answer, which is that space does appear to be “pretty flat,” as in space is within 0.4 percent of being flat. Scientists, through two very different methods, have calculated that the curvature of space (at least the space we can see) is very nearly zero.

"Very nearly zero" isn't "zero."

I love Star Trek, but when they get the physics wrong, it annoys me. One example is when the Next Generation crew found a Dyson sphere. A Dyson sphere is a giant shell an advanced civilization could build around a star, harnessing all of its energy and providing a great deal of lebensraum. It's named after Freeman Dyson, who first conceptualized it. Now, we're used to Earth, which, again, is roughly spherical, but our senses perceive it as flat. A Dyson sphere would be many orders of magnitude bigger than the Earth, so being on, or really anywhere near, its surface, it should appear even flatter. But no, this particular episode showed it curved.

That's far from the only time Trek sacrificed realism for the sake of appearances, but it stuck out in my mind.

Because as far as we know, the fact that we live in a flat universe is a gigantic cosmic-level coincidence.

Some things really are coincidences, like how we're living in a time when the moon appears to be about the same size as the sun in the sky, producing spectacular total eclipses every so often. A few million years earlier, and the moon would have appeared much bigger; eventually, there will no longer be total eclipses as the moon continues its slow divorce from Earth. As far as we can tell, this truly is a coincidence. The flat space thing? I don't know. Maybe. Maybe a flat universe (or mostly flat) is necessary to create the conditions under which our kind of life can evolve. Anthropic principle: we wouldn't be around to notice it if it were any different. But hey, I don't know.

Finally, you can ask whether space is actually made up of tiny discrete bits of space, like the pixels on a TV screen, or infinitely smooth, such that there are an infinite number of places you can be between two points in space?

Well, I guess that depends on whether or not we're living in a simulation.

That's a joke. We're not living in a simulation.

If space is quantized, that means that when we move across space we are actually jumping from small little locations to other small little locations. In this view, space is a network of connected nodes, like the stations in a subway system. Each node represents a location, and the connections between nodes represent the relationships between these locations (i.e., which one is next to which other one).

That would put an end to discussion of Zeno's Paradox. Maybe.

Some even suspect that the relationships between nodes of space are formed by the quantum entanglement of particles, but this is mathematical speculation by a bunch of overcaffeinated theorists.

Sometimes caffeine isn't the only drug involved.

Anyway, like I said, the article is worth reading (open it in a private window if you're hitting a paywall). Else I wouldn't have bothered to post it here.

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