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:

Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

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.

2 Like

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.

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

August 25, 2022 at 12:14am

#1036902

Sodium Cool

No, I'm not trying to turn this into a cooking blog. Ugh. But sometimes a food article catches my eye, and also sometimes the random number generator spits them out back-to-back. In this case, and the last one, the hook isn't cook, but science.

How to Use Baking Soda Like a Scientist
This ingredient belongs in both the laboratory and the kitchen.

Next time someone spouts off about not eating stuff with chemicals in it, you can point out that sodium bicarbonate is, by definition, a chemical, and they almost certainly eat delicious pancakes, which are made with baking powder (baking powder is baking soda with other stuff added). If that doesn't work, mention sodium chloride. If they're still being stubborn about it ("But I only eat organic salt"), point out that every ingredient contains chemicals. It still won't stop the ignorance, but at least you've gained the high ground.

Whipping up a recipe can feel awfully similar to conducting a science experiment. Either one could involve adjusting burners or measuring out various powders and liquids, all while carefully watching to make sure your project doesn’t explode, burn, or turn a funny color.

Be fair, now. Sometimes exploding is the point.

Cookbook author and food writer Nik Sharma happens to be both a cook and a scientist.

Oh, that name is so close to being an aptronym. Nik Shawarma would be an awesome name.

And most scientists are also cooks. It's not like they get paid enough to hire a full-time chef. The only question is whether they apply one activity to the other. The important thing is that he's also a writer.

In his writing, especially 2020’s The Flavor Equation, Sharma is educating the food world on the science behind the most common cooking techniques and ingredients.

Yeah, this is kind of a book promo. But it seems like a useful book.

The article then switches to the author/scientist/cook's point of view.

My fascination for cooking and chemistry developed simultaneously. It all started in my high school chemistry lab, during a lesson on the relationship between acids and bases.

When he was bitten by a radioactive papier-maché volcano?

Sodium bicarbonate—or baking soda—was one of the first ingredients that made me realize that a kitchen is, in essence, a laboratory.

Depends on your definition of "laboratory." Most cooks don't approach it from a scientific perspective, instead using recipes or their ancestral knowledge. Nothing wrong with that, as the goal is to provide something appetizing and edible, but it's not science.

The Rise of Baking Soda

Oh, ho ho ho. I see what you did there.

I approve.

Bakers have used carbonates as chemical leaveners since the Middle Ages. These substances release carbon dioxide bubbles when dissolved in water, or mixed with an acid. In a batter, this has a lightening, lifting effect.

It's interesting to me that this postdated the use of yeast, which also has the effect of leavening baked goods (in addition to the magic it works in delicious fermented beverages). In either case, something's producing carbon dioxide. I would have expected it to be the other way around, as people didn't actually know what yeast was until, like, microscopes. Yes, yeast was used for thousands of years before someone said "holy shit, it's alive!"

Sodium bicarbonate, NaHCO₃, is of course not a living organism.

Fun Fact of the Day: NaHCO₃ can be found in the wild, in a mineral called nahcolite. Yes, its name is a pun on the periodic table symbols involved. I find this amusing.

Baking soda is naturally alkaline, raising the pH when added to liquids or foods. Often, to reduce the acid in coffee or a very sour soup, I’ll stir in a tiny pinch of baking soda to neutralize and counteract the acidity.

Someone once told me to add a bit of baking soda to ground beef before frying it, to make it brown better. I figured it couldn't hurt, so I tried it (science!). I was displeased with the results, but it did produce a faster browning action. Not sure of the chemical reason for that, but since I don't plan to do it again, it's not near the top of my curiosity list.

If I’m cooking dried beans, I’ll first soak them overnight in a brine made with baking soda and salt, or cook pre-soaked beans with a smaller quantity of both. If you’ve ever cooked dried beans, only to have them turn out unpleasantly hard, this is the trick for you.

No way. I get my beans from a can, as God intended.

But the reason I'm quoting this line is to point out that "salt" as a culinary ingredient is almost exclusively sodium chloride, but scientists call any substance with a certain ionized crystalline form a "salt." Here, he's using the culinary definition, as baking soda is itself, chemically speaking, a salt.

Baking soda can also act as a catalyst in two important food reactions. A tiny pinch of baking soda to vegetables or meats while roasting or sautéing accelerates the rate of sugar caramelization, and supercharges the Maillard reaction, the rate at which the amino acids in proteins react with sugar.

Well, that's that low-level curiosity satisfied. You didn't think I'd actually leave you hanging, did you?

Every time I look at the jar of baking soda sitting inside my pantry, I smile.

Well, I hope you change it out every so often. Unlike table salt, it has a relatively short shelf life. No, that's not just Church & Dwight (the makers of Arm & Hammer, which is probably the best-known brand of baking soda, at least in the US) trying to get you to buy more of their product. Using it as a fridge freshener, now, that's them trying to get you to buy more of their product. There is little actual evidence that this works, but damn, they're good at marketing.

By the way, it occurred to me the other day that I never really explained why I bang on about marketing in here sometimes. It's because this is, ultimately, a writer's blog; writers tend to want to publish, and publishing requires effective marketing. At the same time, I despise marketing excesses. There's a balance. Kind of like with adding sodium bicarbonate to food: too much and it leaves a bad taste.

The article ends with a "Tips and Tricks" section, about which I only have a couple of comments:

A pinch of baking soda mixed into a glass of water acts as an antacid to reduce heartburn.

The jury's still out on whether sodium actually affects heart health. But to be on the safe side, I try not to overuse sodium salts, including NaCl and bicarb. No, "heartburn" doesn't really involve the heart; it's just that, having had a heart attack, I'm wary of certain overindulgences (booze doesn't usually contain sodium, except for margaritas). I'd check with an actual doctor before using this "trick" if you've got heart issues.

Outside cooking, baking soda has many uses. It can also be used as a mild soap along with vinegar to clean kitchen counters, and stubborn grease marks.

This, I can vouch for. It's also fun to watch the baking soda/vinegar reaction (the chemical result is sodium acetate, water, and carbon dioxide, but you know it as the middle school science fair "volcano" I referred to above).

I'm also told that it can remove stains from nonstick pan surfaces. I haven't had much luck with that; I have a pan that badly needs a deep cleaning, but nothing has worked yet.

Anyway, the real point here is: chemicals are your friends.

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