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 an hour 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 3 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.

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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