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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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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.

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