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

November 18, 2024 at 10:25am

#1080127

Losing Weight Is Easy

Big Think offers up another way for people to annoy pedants, and vice-versa.

It’s important that weight and mass are not the same
Here on Earth, we commonly use terms like weight (in pounds) and mass (in kilograms) as though they’re interchangeable. They’re not.

By "we," I suppose they mostly mean "Americans." But even people in countries that use SI units will use kg as if it's a unit of weight. This is fine, as far as I'm concerned, because damn near 100% of us live on the Earth's surface. And while there are gravity variations due to latitude, elevation, or anomaly (such as the one from "Anomalies"

a couple of weeks ago), they mostly don't make much difference unless you're a scientist who requires greater precision.

Still, as a pedant, I think it's important to know the difference. Also, as a pedant, I think it's important to keep it to oneself if one does not wish to be uninvited from social gatherings for being a pedant.

Since this is my blog, I'm making an exception here.

Conventionally, here on the surface of the Earth, we can convert between the two using only a minimal amount of effort: 1 kilogram is 2.205 pounds, and vice versa, 2.205 pounds converts to 1 kilogram. Going back-and-forth requires only multiplication or division, which seems easy enough.

"Easy?" Have you met people? Some of them break out in hives if you ask them to add 10% to something.

Incidentally, I don't usually bother with the 0.005. For most practical purposes, 2.2 is close enough and much easier to do head-math with.

A kilogram is an example of mass, not of weight, while a pound is an example of a weight, not of a mass. It’s only here on the surface of the Earth, where we’re at rest relative to the rotating Earth, that these two concepts can rightfully be used interchangeably.

I mean, as long as we're being pedantic, it's a really stupendously big universe and it wouldn't surprise me if there are other planets with the same acceleration due to gravity as that of the Earth.

But for nearly 100% of the volume of the universe, weight is irrelevant and only mass matters (pun absolutely intended).

If you release an object from rest and allow it to fall, it falls straight down, accelerating at a constant rate. It gains speed directly proportional to the amount of time that it’s been falling, and the distance it covers is proportional to the amount of time squared that the object has been falling.

Again, if you're going to get technical about it, you'd add "without air resistance."

This phenomenon, however, appears to not depend on mass or weight. A light object will fall just as quickly as a heavy object, especially if air resistance isn’t a factor.

There it is.

Someone did that on the Moon,

incidentally. This proved two things: 1) Physics is right; 2) at least that particular Moon expedition wasn't faked on an Earthbound sound stage.

The article proceeds to go into a lengthy (and weighty) discussion of the differences between scales and balances, and how the former measures weight while the latter measures mass. It's remarkably math-light.

Then:

We can sum up the difference succinctly: your mass is an inherent quality of the atoms that make up your body, but your weight is dependent on how those atoms accelerate under the influence of all the factors and forces acting on it.

Which I suppose is the main point of the article; everything else exists to support it.

There are plenty of physics textbooks (and physics teachers) that ignore this difference, and simply state that your weight, W, always obeys the equation W = mg. This is incorrect; it is only true when you are at rest on the surface of the Earth.

There's a popular trick question: "Which weighs more, a pound of feathers or a pound of lead?" It's a trick for several reasons.

We often talk about “watching our weight” or “trying to lose weight,” but if that was truly your goal, you could simply go to a higher elevation, move to a different planet, or even get into an elevator and wait for the door to close after you hit the “down” button.

Yes, "simply... move to a different planet."

Sounds like a good idea for other reasons, these days.

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