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:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

December 6, 2024 at 9:37am

#1080874

No Space for Time

The Big Think article that came up for me today involves math. Fair warning so you don't end up defenestrating your device.

Time: Yes, it’s a dimension, but no, it’s not like space
The fabric of spacetime is four-dimensional, with three for space and only one for time. But wow, time sure is different from space!

Time is different from space? I never would have guessed, what with them having different names and all.

When did you first realize that the shortest distance connecting any two points in space is going to be a straight line?

I'm not sure that's a fair question. It's something I'd consider intuitive. What's hard to grasp, sometimes, are the cases where the shortest distance between two points isn't a straight line, because that runs counter to our everyday experience.

In fact, that realization, as far as human knowledge is concerned, comes from a place we might not realize: the Pythagorean theorem.

"In fact," I think they've got this backwards. The Pythagorean Theorem may quantify the "shortest distance" intuition, but I'm pretty sure humans knew about the straight-line thing before they had numbers or geometry. (Also, the idea predated Pythagoras by hundreds or thousands of years; the Greeks didn't invent everything.)

Taking all three of these dimensions into account — so long as we assume that space is still flat and universal — how would we then figure out what the distance is between any two points in space? Perhaps surprisingly, we’d use the exact same method as we used in two dimensions, except with one extra dimension thrown in.

I feel like the only "surprising" thing here is that the math is basically the same.

Thinking about distances like we just did provides a great description of what we’ll wind up with if we consider flat, uncurved space on its own. What will happen when we fold time, as a dimension, into the equation as well? You might think, “Well, if time is just a dimension, too, then the distance between any two points in spacetime will work the same way.”

At which point, unsurprisingly, the math isn't basically the same.

There are two fundamental ways that time, as a dimension, is different from your typical spatial dimension. The first way is a small but straightforward difference: you can’t put space (which is a measurement of distance, with units like feet or meters) and time (which is a measurement of, well, time, with units like seconds or years) on the same footing as each other right from the start.

Feet? Footing? Get it? Haha.

Fortunately, one of the great revelations of Einstein’s theory of relativity was that there is an important, fundamental connection between distance and time: the speed of light.

Yes, and the invariance of that speed is still very, very hard to wrap your head around, because it, unlike the "straight line" thing, runs counter to everyday experience.

However, there’s also a second way that time is fundamentally different from space, as a dimension, and this second difference requires an enormous leap to understand. In fact, it’s a difference that eluded many of the greatest minds of the late 19th and early 20th centuries. The key idea is that all observers, objects, quanta, and other entities that populate our Universe all actually move through the fabric of the Universe — through both space and time — simultaneously.

Turns out that everything in the universe is moving at a constant speed... through spacetime. Once that was pointed out to me, a whole lot of other stuff started to make more sense.

It turns out that the faster (and the greater the amount) you move through space, the slower (and the lesser the amount) you move through time.

Like that, for instance.

There’s an even deeper insight to be gleaned from these thoughts, which initially eluded even Einstein himself. If you treat time as a dimension, multiply it by the speed of light, and — here’s the big leap — treat it as though it were an imaginary mathematical quantity, rather than a real one, then we can indeed define a “spacetime interval” in much the same fashion that we defined a distance interval earlier...

Great. Wonderful. Now we'll get "time is imaginary" on top of "time is an illusion" nonsense. I'd forestall this by pointing out that imaginary numbers aren't actually imaginary (or at least they're no more abstract than the "real" numbers), but that's not going to stop the airy pseudophilosophy.

There is, of course, quite a bit more at the link. I'm not sure if it's really that useful; I feel like there's either too much or not enough math to keep people interested enough to follow the arguments. But if all we can get out of it is "spacetime is weird, and time is different from space," maybe that's good enough.

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