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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QPdoll

about 8 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 11 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 22 hours ago
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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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November 14, 2023 at 9:27am

#1059476

Cheesy Jokes

A couple of weeks ago, in "Hunter's Moon"

, I wrote: "I have another article in the queue that addresses the 'green cheese' cliché. Maybe it'll turn up on the next full moon (which will be the Beaver Moon). But probably not."

Indeed, yesterday was a New Moon, precisely as far as we can get from a Full Moon, so we may not even get to gaze upon the satellite this evening; it's still quite close to the sun, from our perspective, and will set before the end of astronomical twilight. If you're lucky, maybe you'll be able to make out a thin crescent just after sunset.

But today's article, from Mental Floss, takes a look at the moon anyway.

Why Do People Say the Moon is Made of Cheese?
The moon-cheese nexus may have started with a fable about a hungry wolf and a crafty fox.

Everybody knows that Earth’s moon does not, in fact, consist of dairy products.

Considering the astounding number of people who insist that the Earth is flat (that is, a nonzero number of people), I would never say "everybody" in this context.

So where did the myth that the moon is made of cheese come from?

I'm just as interested in folklore as I am in science, and regular readers know I'm a sucker for origin stories.

Though the idea that the moon is made of cheese has been around for millennia, it’s doubtful that anyone ever actually believed it, at least not academically.

Academics also knew the Earth wasn't flat. That doesn't mean your average farmer, or whatever, didn't believe it. To be fair to those farmers, it's not like it mattered in terms of doing agriculture.

The earliest record of this bizarre notion comes from a medieval Slavic fable in which a ravenous wolf chases a seemingly hapless fox, hoping to score an easy meal.

Never. Trust. The. Fox.

But the best-known early citation dates to 1546, and can be found in The Proverbs of John Heywood. The document is a compendium of some of the author’s most famous sayings, such as “the more, the merrier,” “a penny for your thoughts,” and “Rome was not built in a day.” At one point, he jokingly states “the moon is made of greene cheese” (“greene” refers to the food’s age rather than its color).

At last, a time frame for the "a penny for your thoughts" proverb! People nowadays still use that, but they generally offer two pennies (which I suspect is related to "my two cents' worth" because in the US, "cent" and "penny" are interchangeable) "because of inflation." But what would inflation actually do to the value of a 1546 penny?

According to this handy website,

one pence in 1550 (they only do 0-ending years) would be roughly equivalent to one pound fourteen in 2017, apparently the latest year they have British financial data for. Also from that website, one pence wouldn't have purchased a single horse, sheep, cow, measure of wheat, or a day's wages, but apparently it could buy a thought.

So, because I'm not British, how much was £1.14 equivalent to in USD 2017? There's a website

for that, too: the conversion factor is, on average, 0.808 pounds to the dollar (currency conversions fluctuate almost daily; I believe this is the average used by the US for the purposes of taxing an American's income from the UK). So x/£1.14 = $1/£0.808 yields x=$1.41.

Finally, as you might have heard, inflation has run unusually high these past few years, so we now must convert 2017 USD to 2023 USD. Is there a website for that? Of course there is.

From it, we find that number to be about $1.76, using the numbers from October of both years because the November 2023 numbers aren't out yet.

These, are, necessarily, very rough estimates, especially when it comes to the purchasing power of a British pound in a pre-industrial society. And I won't guarantee that I did the math right, because it's too early in the morning. But according to this, today, saying "$1.76 for your thoughts" would buy roughly the same amount of thought (If you need that in 2023 UK pounds, that's on you). But one wonders if thoughts are really that valuable nowadays; maybe I'll just keep saying "a penny."

Wow, do I digress. Back to the "greene cheese" article:

The scientific community has never supported the claim, yet every children’s program from Tom and Jerry to Wallace and Gromit has made its fair share of moon-cheese references.

Well, duh. It's still a fun piece of folklore, and thus great joke fodder.

Even NASA couldn’t resist getting in on the joke. On April Fool’s Day 2002, the agency claimed to have “proven” once and for all that the moon was made of cheese by releasing a Photoshopped image with an expiration date printed on one of the moon’s craters.

Casein

point.

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