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 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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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2 Like

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

August 23, 2024 at 10:40am

#1075620

Unsolved, Unresolved

Today's link isn't a podcast. I don't do podcasts. Shouldn't they call them something else now that the iPod is no more? Anyway, from Vox, a collection of stuff from a bunch of podcasts:

17 astounding scientific mysteries that researchers can’t yet solve
What is the universe made out of? How should we define death? Where did dogs come from? And more!

The key word in the headline there is "yet."

“Whatever we know is provisional,” Priya Natarajan, a Yale physicist, told us about research on dark matter. But the sentiment also applies to science overall. “It is apt to change. What motivates people like me to continue doing science is the fact that it keeps opening up more and more questions. Nothing is ultimately resolved.”

This is, of course, a good thing. But "nothing is ultimately resolved" can be misleading.

Unexplainable isn’t about how scientists don’t know anything.

There's a tendency in humans to indulge in binary thinking. That is, believing that since science can't tell us everything, then it can't tell us anything. This is, of course, arrant nonsense.

We’re drawn to questions because they are optimistic. They invite us to dream of a better world in which they are answered, where the gaps between questions and our capabilities to answer them are smaller.

Ugh. Optimistic? Maybe I should stop asking questions. Besides, finding out that the universe will definitely end is hardly optimistic. I mean, what's the point of doing anything if it's just going to wind down and stop one day? I mean, sure, trillions of years from now, but that's not so long, is it?

Here are some of the questions that astounded us the most.

I won't list all of them. That would take too long. Just ones I have comments on or can make jokes about.

1) What is the universe made out of?

String.

String theory: the Universe is a big ball of string, and God is a cat.

(Okay, this section is actually about dark matter, unsurprisingly, but my string theory pun is funnier.)

2) How did life start on Earth?

Lots of people think it started elsewhere and migrated to Earth. I have another article about that in the queue. Which is fine and may be a testable hypothesis, but, either way, it doesn't get at the real question, which is how did life start, query, full stop.

3) How did dogs evolve from wolves?

That's not the best-phrased question. It's not like wolves haven't been evolving for the past 20,000 years or whatever. Though that's hardly long enough for a lot of natural selection to take place. Artificial selection, sure, which is why we have both chihuahuas and Great Danes and call them both dogs. It's like the idea that we evolved from chimps: no, we and chimps (and bonobos) share a common ancestor.

5) What will animals look like in the future?

I'm guessing the answer is "animals."

10) Is there anything alive in the human poop left on the moon?

You know, I get that this question is academically important. It is also unique on this list because while we don't know the answer, we know precisely how to find the answer, and all it'll take will be another trip to the moon, this time carrying pooper-scoopers.

11) Was there an advanced civilization on Earth before humans?

I know it's kinda vogue-y to ask that question right now, but the simple truth is, if there were, we'd know. We might not find their artifacts or other archaeological remnants, but any "advanced" civilization worth that adjective is going to need energy and produce waste. That waste would show up all over the globe, just like our waste is doing now.

Besides, if there had been, all the oil would have been gone before we started digging for it.

No. There wasn't.

12) What is the definition of “life”?

Yeah, that's not a science question so much as a philosophy question. As I've noted before, though, science can inform philosophy.

14) What did dinosaurs sound like?

I hope they figure this one out, and it turns out they meowed.

17) How will everything end?

With some questions as yet unresolved, I'm sure.

3 Like • Comment • View 2 comments

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