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.

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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

August 30, 2024 at 10:59am

#1075912

Floury Language

I expect to participate in "Blog Week Birthday Bastion 2024"

[13+] every day next week, and this will likely be the last link I comment on for a while. So savor it. Which should be easy enough, because it's about food.

Do You Really Need to Bake With a Scale?
I am legally obligated to tell you the truth, even if you don't want to hear it.

You're probably better off baking with an oven, I think.

So listen to me, your baking therapist: You need to stop the negative self-talk. It’s not you—it’s your measuring cup. And it’s letting you down!

Which is what I've been saying. But no, I get it, I'm not an expert. My advice is that taking my advice isn't always a good idea.

But I've railed on this numerous times, most recently here: "Waxy Scaley"

When I tried using measuring cups on five separate occasions. I got five totally different weights. This! Is! Not! Okay!

I think we get the point without all the bangs.

We all measure flour differently. Some of us dip-and-sweep, others fluff-and-spoon, a few of us live on the edge and simply tilt the bag over the measuring cup, hoping for the best, and there are those who shall not be named who use liquid measuring cups (please don’t).

No mention of sifting? Shame.

With a baking scale, however, 125 grams of flour (the generally accepted standard weight of 1 cup all-purpose flour) remains 125 grams no matter where and who in the world you are.

I could quibble about this. Unless you're using a balance scale, you're measuring weight, not mass. The same mass weighs a different amount depending on where you are on or off the planet.

But I won't. Accuracy is indeed important, as is precision, but it's possible to take it too far. The trick is knowing when you're taking it too far.

Why am I so focused on flour? Because every crumbly cookie, sunken cake, or leaden bread that made you believe you’re a bad baker can almost always be traced back to too much or too little flour.

Not mentioned in the article: the kind of flour you're using can also be important. You know why biscuits (American biscuits, not what the UK calls cookies) are better in the South? It's not because we're better bakers. It's that the proper flour for biscuits is generally only available in the American south. This scientific study

supports my assertion; also note their use of grams, not cups (though their mixing of SI and Imperial units made me twitch).

But the measuring thing holds true for other ingredients. Baking soda is an essential ingredient in many baked goods, and it works because of a chemical reaction, and chemical reactions are usually driven by mass, not volume.

Flour has a structural purpose in baking; given enough moisture and heat it forms a protein network that lends physical shape to your treats.

That protein network is mostly gluten. This is one reason why gluten-free baked things taste like mushy ass.

The benefits of a baking scale go far beyond flour, though. Ever come across measurements like “a scant ½ cup”? (I’m sorry, what?). It’s never an issue when a recipe is written with weights.

I don't think I've ever seen that (in my defense, I don't bake often). But what I have seen is stuff like "heaping tablespoon." Come on.

Or how about measuring sticky peanut butter or molasses or honey?

Which was the focus of the last blog entry I did on this, the one I linked up there.

Or what about when a recipe calls for chocolate or nuts? With so many variables—mini chips vs. regular chips vs. fèves, finely chopped vs. coarsely chopped nuts—a scale is the only way to ensure consistency.

While I agree with this, there are some ingredients, like chocolate chips for the eponymous cookies, where the actual quantity is more a matter of personal preference. The point remains, though: a standard cup of Hershey's Kisses (not great chocolate, but they're basically giant chocolate chips), if you can even measure something that bulky in cups, is not going to weigh the same as a cup of mini chocolate chips.

This has to do with solids vs. air voids, but that's just on a bigger scale (pun intended) than the flour problem.

Even if a favorite recipe doesn’t list weights, you can use this immensely handy conversion chart from King Arthur Flour to figure them out yourself.

Half my work in cooking seems to be finding these conversions online.

As an aside, it doesn't matter if it's grams or ounces. As long as you're on Earth, it's close enough. Just don't confuse weight ounces with fluid ounces.

Once you understand the life-changing magic of baking with a scale, you may never, ever use measuring cups again. Hold onto those spoons, though: For very small amounts, like teaspoons, I prefer to stick to volume measures. Most scales measure in 1-gram increments, so they just aren’t sensitive enough to pick up the nuances of something like ¼ teaspoon baking soda.

I used to have a 0.01-gram kitchen scale. Bought it from Amazon. Later, I realized that this probably put me on a drug-dealer watch list. No, assholes, I'm just using it to get the precise amount of cornstarch measured out.

By "used to have," I mean it fell off the counter once, and I replaced it with a scale that's less precise but has greater capacity. And is more durable.

The article ends with a link to a hamantaschen recipe, for which we are in the wrong time of year. But that's never stopped me before.

4 Like • Comment

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