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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Show Recent Comments

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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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

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

October 28, 2024 at 9:54am

#1079055

Waste Not

I don't know if I've used anything from Inverse before. This article caught my eye a while back.

The Rotten Secret Plaguing America’s Grocery Stores
Why, exactly, do grocery stores overstock food?

Just as the answer to any yes/no headline question is "no" by default, the answer to "why do [businesses] do [thing]" is almost always "money."

It's been a while since I saved this article, so let's see if I'm right.

We live on a planet where people still die of starvation, and yet we still waste so much food — it’s a problem, and not just for sustenance but the environment.

We can solve the problem, or we can learn to live with it. I did.

Part of it is just a category problem: Americans are used to seeing a wide and alluring variety of foods on shelves, and a lot of it, especially for produce and meats.

One of the first times I set foot in a Whole Fools, long before they merged with Amazon, I noticed the marketing gimmick of their produce section: an abundance of everything, all bins topped off and on the verge of overflowing. This was obviously a long time ago, but I remember thinking: "Some marketing expert decided that the store has to look like it'll never run out." I imagined stock clerks (or "associates" or whatever) being paid minimum wage to watch the produce section, and run in and replace every cantaloupe or courgette some customer put in their cart.

This is in contrast with other local grocery stores, who seemed to have no problem letting bins get nearly empty before replacing the contents. I imagine that's bad in a couple of ways. Some people, seeing that there are only three oranges left, will move on to buy something else, thinking, perhaps, "Those three must have been the ones no one else wanted," or, if they're more generous in their estimation of humans, "Let someone who really needs them have them." Others will grab all three, thinking something like "I'd better get that now," and then they don't eat them, thus simply moving the food waste problem from the producer to the consumer.

To be clear, though, I have no idea if there's any difference in food waste between WF and, say, Kroger.

So what do the stores do? They overpurchase, “knowing that some food waste must be built into their bottom line,” says Jennifer Molidor, senior food campaigner at the Center for Biological Diversity.

And what do they do with the unbought food? Do they donate it to homeless shelters? Not from what I've heard; they just toss it into a dumpster under a surveillance camera to make sure no one's getting free food.

Profit margins on perishable foods are so high that stores would rather overstock so as not to miss even one sale.

Yeah, I'm going to need a citation for that. But it does tie in to my "money" answer above.

On the high-tech side, retailers are starting to use artificial intelligence to better determine how much and when to order food items.

Using AI, huh? Well, that should calm consumer fears.

Another high-tech solution is “dynamic pricing,” or flexible price points that can shift depending on real-world market factors, in this case allowing stores to discount items that are getting close to the end of their shelf life.

That's actually pretty cool, in my opinion. Though stores have done a version of that for as long as I can remember, selling yesterday's bakery products, for example, at a discount. (The article does point that out.)

Still, when food does go unsold, it can go somewhere more productive than a landfill, and there, too, ReFED has seen progress. In the Pacific Coast Food Waste Commitment, signatories increased the percentage of unsold food that was composted by 28 percent and donated by 20 percent.

Okay, so maybe some of it does get donated. My inner cynic (which is like 90% of me) wants to know if the companies get tax breaks for doing so.

One challenge for dynamic pricing is how to reduce prices without a ton of extra labor. Some startups are experimenting with digital labels that could be easily changed.

If a price can be easily changed, it can be easily increased. When gas stations moved from physical signs to digital ones, their prices became more dynamic, too, probably because it takes less labor to change it.

There are also plenty of hurdles for finding useful places for food waste to go. For one thing, “landfilling in the United States is dirt cheap,” Sanders says, so doing just about anything else with unsold food costs more.

"Landfilling." "Dirt cheap." I see what they did there.

Also, again: money.

Donating food, meanwhile, not only comes with liability risk to keep the food well preserved, but it also requires new processes and labor for store employees to collect the food and new partnerships for how it gets picked up and where it goes.

Yeah, you have to wonder about liability in those situations. That's gotta affect their bottom line, too.

At some point, change will require mandatory measures, not just voluntary ones, to disrupt that dynamic. “We need laws and regulations from the government to hold industry accountable and make food waste prevention a requirement,” Molidor says.

Oh, more regulations? You just lost half your support.

Anyway, I don't have solutions. Like I said, I forced myself to get comfortable with food waste, because it's like when you were a kid and you wouldn't eat your vegetables and your mom was like "Think of all the starving children in Ethiopia!" (or whatever place was well-known for having famine when you were a kid). And you pushed the plate toward her and said "Send it to them!"

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