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:

Write_Mikey_Write!

about 5 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 7 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.

1 Like

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').

1 Like

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.

July 30, 2022 at 12:09am

#1035861

Grueling

On the heels of Turkey DrumStik

's last blog entry, "Foods and stoods"

, it's appropriate (and coincidental) that the random number generator directed me to this article about cooking.

Can’t cook, won’t cook? Here are the tips that saved me from a life of terrible meals
I used to survive on tinned lentils, microwaved eggs and the kindness of more gifted cooks. Then I learned to pull my weight in the kitchen – and if I can do it, so can anyone

Oh, look at Ms. Posh over here actually cooking lentils and eggs instead of ordering takeout like a proper lazy Brit.

(Guardian link so English English.)

(I'm going to refrain from making tired jokes about British food here. You can find them elsewhere, or make up your own.)

For most of my life, I have been a terrible cook. Some people say they are terrible cooks then whip up a perfectly palatable meal for four. I have given myself food poisoning, twice.

Only twice? And only yourself? You're doing better than a lot of people.

My regular diet used to include microwaved scrambled eggs, children’s lunchbox cheeses, tinned lentils mixed with tinned tomatoes, bowls of garden peas, and the “complete food” powder Huel.

You know, I've tried those microwave scrambled eggs. They're not completely terrible, unless you overcook them, at which point they're suitable for fixing tire (or tyre) leaks. Not nearly as good as actual eggs, of course.

Also, what the Huel is Huel?

For once I was actually arsed to look something up. here you go.

I haven't seen it marketed in the US, so I think it's primarily a British thing.

Pro: Their motto is "Don't be a dick," and it's in huge type on one of their walls.

Cons: Huel is a portmanteau of Human and fUEL, and their employees are called Hueligans.

According to a 2014 YouGov survey of 10,000 Britons, one of the largest ever conducted about food, 10% of us cannot cook a thing – equating to 5 million people. A smaller survey in 2018 found that 25% of respondents could only make three dishes (including boiled egg and soldiers, and porridge).

Translations:

Boiled egg and soldiers: Mostly a kid thing, soft-boiled eggs (the kind they make cups for) with strips of toast (the "soldiers").
Porridge: Oatmeal (usually). You knew that from stories.

She goes on to describe how to go from "I'm not a cook" to "I'm a cook":

See if you can identify the source of your belief that you are someone who can’t cook. You might uncover a false assumption – for example, that you don’t deserve to enjoy food, that any time not spent working is wasted, or that cooking is anti-intellectual or even women’s work.

It's work, I'll grant that. Sometimes too much work. In an effort to eat healthier, I've been buying more fresh fruits and vegetables, and, apart from bananas and carrots and maybe apples, they're a massive pain in the ass. Lettuce, for example. Gotta wash it, right? But then you have wet lettuce. So you try to dry it. Lettuce is wrinklier than an 80-year-old swimmer, with more folds in a head than an origami crane, so it's impossible to get all the water off. "So buy a salad spinner, Waltz." No. I don't have the room, and between me and my housemate there are already too many gadgets to clean.

But "women's work?" What is this, the 1920s? Hell, my dad grew up in that decade, and he only called it "women's work" when he wanted to get a rise out of my mom (it worked every time). Also, and I know I've said this before but I can't emphasize this enough, my mother was a lovely person but couldn't cook her way out of a shoe.

"Anti-intellectual?" I can't even.

Start by deciding that you can cook, “then prove it to yourself with small wins”, writes Clear.

Isn't that good advice for any task?

Practising my cooking felt a bit like practising my French with a native speaker who is also fluent in English: insisting on imposing my incompetence on others, at the expense of everyone’s enjoyment.

I've done that, too. But only with his permission, and not for very long.

After four months of living alone, I have learned that I cannot be without Greek yoghurt, kale, cannellini beans, peanut butter, sour cream, chilli flakes, spinach and frozen chapati breads.

Greek yogurt: I eat this stuff from time to time because of its purported health benefits. It's basically liquid chalk.
Kale: I've ranted about kale before. I've eaten it. I'd rather not.
Peanut butter: I despise peanuts. I'm okay with real peanut butter, the kind that's not 60% sugar. I'd rather have almond butter, but I can afford it because I don't eat it that often.
Spinach: Not a pantry item. I'll buy a bag of spinach (pre-washed of course) for salads and omelets sometimes and it wilts within 48 hours, sometimes less. Frozen spinach is foul. Canned spinach isn't food unless you happen to be a one-eyed sailor. Also, spinach recipes crack me up: "5 cups raw spinach (1 tsp cooked)"

Self-help guru and entrepreneur Tim Ferriss surveyed more than 100,000 of his (mostly male) Facebook fans to discover what turned them off cooking and found an array of reasons: too many ingredients or tools, intimidating skills, different dishes finishing at different times, standing at the stove, food waste.

I've learned to be okay with food waste. Shitty of me, I know, but they just don't sell certain things in sizes I can use all of before they go off. My biggest problem with cooking? I almost always cook for myself, and it irks me that it'll take an hour to make something where the recipe specifically says 15 minutes, and then I spend all of 5 minutes eating it. Nothing should take longer to cook than to eat, in my opinion. I'm getting over that, but I still resent it sometimes.

One thing I'm really good at? Having everything finish at about the same time. I can't estimate quantities to save my life (I once measured what I thought was a cup of broccoli but was only a quarter cup, and what I thought was a teaspoon of oil but was actually half a cup), but I'm damn good at timing, so long as I ignore the fantasy times listed on the recipes.

For me, it’s my tendency to get distracted, especially if I am reading a recipe on my phone. I will flick to another browser tab and forget to stir, or put a pan on a high heat then wander off. The result is a dish that tastes like burnt pan bottom.

Yeah... that's definitely a you problem.

Then I got a smart speaker.

Aw HELL no.

Not every recipe is reliable. Cookbooks have at least gone through a process of recipe-testing and copy-editing; a top Google search result can just reflect good SEO. “Frankly, there are a lot of bad recipes out there,” Johansen says.

Yeah, my comment on the Stik's entry above reflects that: "Then there was the time I got something like "1 whole rotisserie chicken, 7 stalks of celery, 3 beefsteak tomatoes, (spices), wrap in a single 6" flour tortilla."

Either way, it is important to find a source that resonates with you.

You know what hacks me right off about recipes you find online? First they have to write a PhD dissertation on the dish. Scroll, scroll, scroll, scroll, come ON, scroll, scroll, oh THERE's the recipe. It involves five pans, a crockpot, six hours, an oven, a blender, a microwave, and spices that are only available in Thailand? Crap.

If the recipe calls for 45 minutes in the oven, but it looks done to you after 30 minutes, try it. If you really like garlic, double the suggested cloves and see.

Double? More like quadruple. Yeah, I like garlic.

For example: it’s important that aubergines are “very finely” sliced, but you can usually fudge onions – and always save the pasta water.

Aubergines (that's eggplant for us yanks) aren't food.

Incidentally, I've been wondering why Brits use the French words for eggplant and zucchini (aubergine, courgette) but not the French word for spinach. My best guess is that, being Brits, it would take approximately five microseconds for someone to shorten épinards to "nards," and that'll be all she wrote.

Fuck it. I'm calling them nards from now on.

As for onions, well, we pretty much agree with the French there: oignon.

Even the simplest meals can be elevated above a survival mindset: in Solo, Johansen dedicates a chapter to things on toast.

I suppose it's too much to hope for that the chapter be titled "Shite on a Shingle."

Part of scaling up in the kitchen, Johansen suggests, is finding “like-minded” people to cook for, who are less concerned about what’s on the menu than enjoying each other’s company.

Yeah, no. With all the different dietary preferences and rampant food allergies (and "allergies") out there, I'mma just keep cooking for myself, and sometimes my housemate.

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