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.

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.

August 14, 2022 at 12:01am

#1036469

Punday

We're going to have a bit of pun today. Difficulty: other language.

Why the Japanese Calendar Is Full of Unofficial Food Holidays
A linguistic quirk created both Banana Day and Strawberry Day.

Over the last few years, an increasing number of unofficial food holidays have popped up all over the Japanese calendar. Banana Day, Strawberry Day, Curry Day; the list goes on.

The only things better than food holidays are drink holidays. I like to think it's always Beer Day somewhere.

Each is celebrated with a flurry of online attention and clever marketing tie-ins from restaurants, trade groups, and food companies.

Because of course it is.

But how does a random date become Strawberry Day, for instance? Part of a larger trend known as dajare no hi, or (bad) pun days, most of these celebrations depend on wordplay, to which the Japanese language is particularly well-suited.

You know, if I'd known how much learning other languages would up my pun game, I'd have done it long ago. I only know a few Japanese words, mostly ones related to food, martial arts and cars, but even there, there are opportunities for puns. Nissan, the car company, for example: I've been calling it "23" because it sounds like the Japanese for 2 and 3.

That's not where the name came from, though. To make matters worse, 1 through 5 in Japanese is "ich ni san shi go," which I can never forget now because of its resemblance to "Each Nissan, she go."

I'm probably butchering Japanese here. If so, sumimasen. But I do know a bit more French, and I am still amused about the French word for seal (the animal, not the seventwo things holding back the apocalypse). I've probably mentioned this in here before, but just in case: The word is phoque, but it's pronounced just like a certain versatile Germanic curse. To make matters worse, if you want to say, "I'm going to feed the seals" in French, you'd say "Je vais donner à manger aux phoques." Since the final s isn't pronounced, and the "aux" is pronounced almost exactly like "oh," well, don't say that in front of your mom.

But I digress.

Take Banana no hi (Banana Day) for example. Broken down, the Japanese pronunciation—ba-na-na—corresponds to the Japanese for eight (ba) and seven (nana). The seventh day of the eighth month, August 7, thus becomes “Banana Day.”

Don't you dare scoff. We have Star Wars Day here. May the Fourth be with you.

The only surprising thing there is that it goes month/day like in the US and not day/month like in civilized countries.

Similarly, the Japanese for strawberry—ichigo—corresponds to the number one (ichi) and five (go). Depending on one’s location, Strawberry Day, or ichigo no hi, is marked on either January 15, or even the 15th day of each month.

Depending on how many strawberries there are for sale, I suppose.

In a 1999 paper for the Japanese Society for Language and Humor Studies, the linguist Heiyo Nagashima explained how puns have existed in Japanese since the time of the Manyoshu, a collection of poetry published in the later Nara Period (710-794).

If you have to study humor, it's not funny anymore. Regardless, this long history of puns is probably one reason I've always been interested in Japanese culture.

Japanese is also a heavily inferential language, with emphasis placed both on the ability of a writer to convey information as succinctly as possible, and that of a reader to infer the correct meaning. Without knowing that many food-day puns depend on dates, they can be something of a riddle.

And that's why I never started any kind of formal study of the language.

Pocky no hi: Pocky Day. Observed on November 11, since 11/11 resembles the long skinny shape of the popular sweet snack. Also known as Pocky and Pretz Day, which includes the savory version.

Which is probably for the best, not focusing on the other association of that date. And I'm glad this one came up out of my rotation today and not tomorrow, which is the anniversary of Japan's surrender at the end of WWII.

Niku no hi: Meat Day. Corresponding to two (ni) and nine (ku), Meat Day is commonly marked by stores and restaurants on the 29th of each month, although February 9 (2/9) is also often observed.

The use of the Japanese particle "no" is interesting, by the way. From what I understand, in these contexts, it indicates a kind of possessive. Like we use apostrophes.

These puns are not just confined to food. February 22, for example, is Cat Day. In Japanese, cats say nyan, rather than “meow,” which may also be read as the number two. 2022 was particularly notable for its 2/22/22 date, which was naturally dubbed “Super Cat Day.”

Any country with a Cat Day is okay by me.

How can both "nyan" and "ni" be "two?" you ask. Well, it's not unheard of for there to be several names for one concept, and also, these are transliterations.

While bad puns in Japanese produce the same eye-rolls as their English counterparts, there’s no denying that clever wordplay and special foods can make ordinary days into something worth celebrating.

A pun is only funny to the perpetrator. At least until the victims come after them with pitchforks and/or katana.

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