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.

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').

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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.

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.

2 Like

October 3, 2020 at 12:03am

#994891

That's Not Kosher!

And now for something completely different... a recipe

(And an actual emoticon)

Merit Badge Mini-Contest below!

https://www.thekitchn.com/challah-french-toast-264194

Recipe: Best-Ever Challah French Toast

This came up from my list at random, but it's serendipitous because not only is it Shabbat right now (aka the Sabbath, which of course happens every week) but it's also Sukkot, a harvest festival that only comes 'round once a year, like most harvest festivals. It's kind of like the Jewish version of Oktoberfest, with less beer and more freezing your ass off outside at night. But with food.

Couple of things to start off.

Food as Cultural Appropriation: Get the fuck out of here with that shit. Eat what you like. And I can practically guarantee you that if you like bread, you'll like challah. Besides, I bet that unless you have dietary crap going on, you've already eaten bagels, pastrami sandwiches, and latkes. Anyway, this comes from my ancestral culture and I give you permission.

Pronunciation: Since the etymology of the name is Hebrew, it's pronounced with a sound we don't have in English. This is rendered as "ch." Now, "ch" can be pronounced like in chop, or like in Charlotte, or, and this is the important one here, like in loch or chutzpah. You make the sound by basically half-clearing your throat. If you really, really can't make the sound, then pronouncing it with a hard "k" is barely acceptable, though that will out you as a goy. But if I can learn how to pronounce a French "r," you can pronounce the Hebrew "ch." Might take some practice, though.

One final thing before we get on to this delicious and probably addictive recipe:

https://en.wikipedia.org/wiki/Challah

There. Now you know more than I do about this tasty bread.

Growing up, challah French toast was the only French toast I knew.

Same here, although unlike the author, my parents couldn't cook to save their lives. But with enough sugar, when you're a kid, who cares?

We would slather softened butter over thick slices of fresh challah on Friday night for Shabbat, and my dad would use the rest of the loaf to make mile-high stacks of French toast the next morning...

You know how I know the author doesn't come from an Orthodox family?

It wasn’t until a diner served me French toast made with a piece of limp wheat sandwich bread that I realized how spoiled I had been.

Seriously, it's possible to make decent French toast with other breads, but some of the crap that passes itself off as French toast is actually crap. (French toast, by the way, isn't all that French and actually predates what we now know as France -- but "Gaulish Bread" just doesn't have the ring to it. I hear that in actual France, it's called pain perdu, or "lost bread.")

Trust me, once you’ve had challah French toast, you’ll realize there’s no other way to eat it.

This is truth incarnate. So beware: you will be spoiled by this, just like I can no longer drink cheap-ass American adjunct light beer.

3 Steps for the Absolute Best Challah French Toast

And really, this stuff is dead easy to create. Even I can be arsed to whip up a batch from time to time, and I'm the laziest imaginable cook.

1. Cut thick slices of bread, then dry them out in the oven.

This is ideal, yes, but a lot of times the reason you're making French toast is you have bread that's going stale and needs to be eaten before it becomes duck chow or hockey pucks. And stale bread may not need the oven treatment. You probably know better than I do whether to do this step or not. I usually just use the stale bread.

2. Use full-fat dairy for the very best custard.

Seriously, if you're already committed to French toast, the paltry extra calories you get from using actual milk instead of watery skim milk won't make a thimbleful of difference.

3. Fry the French toast in butter, not oil.

I approve of this message.

The article ends with the actual recipe, which seems like a big block of text but is actually just padded for rank beginners so they don't make rookie mistakes like letting the butter (or the toast itself) burn.

For maximum sacrilege, I recommend serving with a heap of bacon strips. Unless you're actually Jewish, I promise this won't piss off God. The pig might not be very happy about it, though.

Besides, I once overheard some idiot complaining about cultural appropriation while chomping down on a bacon, egg and cheese bagel. Irony is utterly lost on some people.

Merit Badge Mini-Contest!

And hey, now that it's fully October and I'm feeling better than I was yesterday, tell me about your favorite breakfast food in a comment below. The one that makes me the hungriest gets its chef de cuisine a Merit Badge. As always, you have until the end of the day today, October 3, WDC time.

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