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

December 14, 2024 at 9:32am

#1081148

Fruits of Our Labors

Appropriately enough, the first entry after the completion of my five-year daily blogging streak is from Cracked:

How the Tomato Became Torn Between the Lands of Fruits and Vegetables
A confusing, red, plant-based chimera

Right, because the most important characteristic of a tomato is which category we pigeonhole it into. But, okay, I'll play along.

I don’t know what it is about the fruit-versus-vegetable designation of a tomato that I find so particularly annoying, but it twists in my brain like a knife.

That sounds serious. Maybe, instead, take a break and think about Pluto for a while.

As it sits today, the tomato is indeed, botanically a fruit. At the same time, it is legally a vegetable...

Yes, and my mom was, to me, my mother, but to my dad, she was his wife. So?

First, let’s stick to the science, which decidedly declares a tomato a fruit according to botanical guidelines.

Well, botanically, it's a berry. And, according to botany, strawberries, raspberries, and blackberries are not berries. Why this would matter to anyone trying to fix dinner or dessert, though, is beyond my comprehension.

Where the other side of the argument comes from is the culinary world, the place where most people are interacting with tomatoes on a daily basis. It’s also the dominant layman’s classification, probably due to the fact that it’s based in common fucking sense.

Here's where I usually rant about how common sense is usually wrong and needs to be superseded by science. But the classification of a tomato isn't like studying what its nutritional characteristics are. Categories and classifications are imposed from the outside and are supposed to help us make sense of the universe, like what the definition of "planet" or "mammal" should be. Then something like a platypus comes along to remind us that the universe fundamentally doesn't make sense and we shouldn't expect it to. Point is, we could just as well say "any topping on a Big Mac is officially a vegetable," which might settle the tomato question once and for all, but move the discussion to whether cheese should be called a vegetable or not.

And yet, no less of an authority than the Supreme Court has ruled differently. Unsurprisingly, it’s money-related, specifically to do with tariffs. In the late 1800s in America, the taxation on fruits and vegetables was starkly different. Fruits could be imported with impunity, while bringing in foreign veggies would demand a steep 10-percent tariff. An importer named John Nix saw opportunity in the science, and refused to pay tariffs on a shipment of tomatoes, since they were technically fruits. The case climbed all the way to the Supreme Court, where it was heard in 1893.

It also should come as no shock that, in some cases, a thing can be categorized as one thing in one context, and another thing in other contexts. Like, astronomers consider any element that's not hydrogen or helium to be a "metal." That works for astronomy. It doesn't work for structural engineering.

As I read it, the Supreme Court agrees with the people, issuing the legal equivalent of “sure, technically, but come on, dude.”

Leaving aside for the moment that botanists and biologists are also (usually) people, all that means is that, in the US, tomatoes are vegetables by legal definition. I vaguely remember some nonsense a while back about whether ketchup, which doesn't have to be made from tomatoes but usually is, should also be considered a vegetable for the purpose of school lunch nutrition or something.

Left unsettled, then, is still the question of whether a hot dog (with or without ketchup) is a sandwich, and I maintain that no, it's a taco.

3 Like • Comment • View 1 comment

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