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 9 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 11 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.

February 14, 2024 at 10:18am

#1064168

Other Stuff We Don't Know

On this, the darkest and gloomiest day of the darkest and gloomiest month (yes, I know that, technically, in this hemisphere, December is darker and gloomier, and the solstice is the darkest and gloomiest, but metaphorically, it's today), I have an article from Cracked that's about words, a light to pierce the dark gloom.

5 Language Mysteries No One’s Been Able to Solve

There exist, of course, far more than five. But at least these are five I hadn't heard of before.

Imagine if words didn’t mean what you think they do. You read a news report about some victim being murdered by a succubus, but what if a succubus isn’t really a demon? What if it’s some special type of Golden Retriever? With some old words, especially those that have been clumsily translated, we’re just taking shots in the dark.

Words often don't mean what people think they mean. You hear or read a word and, usually, you're too lazy to look it up in a dictionary (even now, when a dictionary involves simply typing or pasting the unknown word into a search engine). So you go from context. And sometimes, you get it wrong, and get laughed at. Other times, you use a word the correct way, and you get laughed at because everyone else thinks it means something different. "Decimate," e.g.

Anyway, the list:

5. What Does ‘Our Daily Bread’ Mean Anyway?

We used to have a bakery here called Le Pain Quotidien. Apparently, it was part of a chain (which I didn't know), and the chain was Belgian, not French. Whatever, even before I started learning French, I wondered if they were going for the Biblical reference, why not "Notre Pain Quotidien" instead? Maybe because it would confuse English speakers. That's never stopped the French, but it might give Belgians pause.

I don't miss it; there's a much better bakery in town, and one of my great thrills in life is to go in there and pronounce their offerings "pain de campagne" and "croissant" in the French way. (I'm easily amused.)

One story from the Bible tells of the time Jesus taught his followers how to pray. When you pray, said Jesus, you shouldn’t just go on repeating some set words...

His followers wrote down the words he said and repeated them verbatim for the next 2,000 years. They taught the prayer to their children, who would learn to recite it long before they had any idea what such phrases like “thy will be done” means. This is the opposite of what Jesus told them to do, but that’s organized religion for you.

And don't get me started on indoctrinating schoolchildren with the Pledge of Allegiance, starting when they're too young to know what "pledge" or "allegiance" (not to mention "republic" and "indivisible") mean.

That prayer is called the “Our Father” or “The Lord’s Prayer.” Though, people today don’t use the exact words that were said millennia ago, because not too many people speak Aramaic anymore.

Fun fact: some Jewish prayers are in Aramaic.

Other fun fact: Catholics, at least, were okay with using Latin for that prayer right up until the 20th century, and I'm pretty sure there are some orders who ignore Vatican II.

But I digress. This section points out that the word which is commonly translated as "our daily bread" might mean something else entirely.

Or maybe it said, “Give us this day our garlic bread.” That’s a prayer to unite people of all faiths.

4. One Mystery Verse of Dante’s ‘Inferno’

Education means different things in different eras. Today, to be a properly educated scholar of culture, you need to be able to explain where every Disney remake went wrong, even though you have never watched any of them and rightly never intend to. In the past, in certain circles, an educated person was someone who’d studied Greek and Latin.

Also, you need to know the convoluted overarching plot of all the Star Wars properties, as well as the various timelines of Star Trek. Not to mention Monty Python movies, The Hitchhiker's Guide to the Galaxy, and several others.

Perhaps you would have to translate the ancient works of Virgil, which sounds hard.

It is hard. Third and fourth year Latin in high school, as far as I can recall, was all about studying Virgil's Aenid.

Much easier, surely, would be translating a poem written 1,300 years after Virgil, featuring Virgil as a character.

Confession time: I've never read the Divine Comedy. Not even a translation.

But then you’d come upon a verse that goes like this: Raphèl mai amècche zabì almi. What does that mean?

We don’t know. It’s not Italian. Nor is it any other language we know of. The character who says it is Nimrod, a hunter from the Bible guarding the ninth circle of hell, and it seems like this line of his may be gibberish.

Even the name of Nimrod has changed meaning, thanks to Bugs Bunny. Pretty sure I've touched on that subject in here before.

In any case, the article speculates, but their guess is as good as yours.

3. The Vowel Vow of the Royal Habsburgs

The Hapsburgs, starting with King Frederick III in the 15th century, used the following motto: “A.E.I.O.U.”

This is what passed, in the 15th century, for wordplay.

Could mean a lot of things, but as the section notes:

Let’s be sure to keep an eye on any man born in Austria. Some of them seem to harbor sinister ambitions.

2. Who the Devil Is Betsy?

“Heavens to Betsy!” says the old exclamation. It’s a way of expressing shock. Descriptions of this idiom liken it to “for heaven’s sake,” a phrase that began as a euphemism...

I don't think this requires that much analysis. Betsy is, or was, a common nickname for Elizabeth, one of the most common girly names, along with about a thousand other nicknames like "Beth" or "Liz" or "Betty," not to mention variants in myriad other languages. The name Elizabeth is now synonymous with royalty, but like many other names of European origin, it derives from Hebrew via the Bible. And in Hebrew, the name (which was Aaron's wife's and John the Baptist's mama's name, though those were probably two different women given the centuries separating them) means something like "God is my oath." Combine that with the way "Heavens to Betsy" just rolls off the tongue, and you have a near-perfect minced oath.

Other speculations abound at the link.

1. Is Copacetic a Word at All?

Of course it's a word. Sure, someone made it up. This is the case with all words. Some were simply made up longer ago than others.

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