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 11 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 13 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.

June 29, 2020 at 12:02am

#986742

Words Enough, And Time

June's almost over -- it seems so short and yet interminable at the same time -- and next month I'll be in "30-Day Blogging Challenge ON HIATUS"

[13+] again.

So, today, I'll talk about one of the reasons most of us are here: words.

https://www.mentalfloss.com/article/51150/12-old-words-survived-getting-fossiliz...

12 Old Words That Survived by Getting Fossilized in Idioms

Not going to repeat all 12 words here, just some highlights. And then it'll be your turn, for a chance to win a Merit Badge.

English has changed a lot in the last several hundred years, and there are many words once used that we would no longer recognize today. For whatever reason, we started pronouncing them differently, or stopped using them entirely, and they became obsolete.

I'm pretty sure I've talked about things like that before. I know I have a couple more in the queue, though it'll probably be at least August before I get to them.

There are some old words, however, that are nearly obsolete, but we still recognize them because they were lucky enough to get stuck in set phrases that have lasted across the centuries.

We writers like to deride clichés, and usually for good reason. But we tend to forget that almost every trite phrase was once poetic and meaningful.

1. Wend

This, to me, is one of the more interesting ones because of the history:

The past tense of wend was went and the past tense of go was gaed. People used both until the 15th century, when go became the preferred verb, except in the past tense where went hung on, leaving us with an outrageously irregular verb.

In case you were ever wondering why English has such different words for present and past tense of "go."

7. Fro

The fro in "to and fro" is a fossilized remnant of a Northern English or Scottish way of pronouncing from.

This one is probably obvious, or at least it was to me when I first encountered it. But it never made sense. I mean, if the o sounds at least rhymed, I could see it, but no, we get an oo sound and then an oh sound. They're only related because they happen to be represented by the same letter. "to and fro" makes precisely as much sense as "to and from." But at least there's historical reason for it, as this list demonstrates.

12. Shrift

We might not know what a shrift is anymore, but we know we don't want to get a short one. Shrift was a word for a confession, something it seems we might want to keep short, or a penance imposed by a priest, something we would definitely want to keep short. But the phrase "short shrift" came from the practice of allowing a little time for the condemned to make a confession before being executed.

Unlike #7 there, this one made some sense to keep around. No, it doesn't rhyme, but it has that nice double alliteration: short shrift.

So, mini-contest time!

Take a look at the full article at the link (it's short). Do you have a favorite word or expression (other than the ones at the link) that involves a rare or otherwise obsolete word? I'm sure there have to be others besides the twelve listed.

And if coming up with one stresses your brain too much -- it does mine; I'm not coming up with any, myself -- then I'll accept any word (or phrase) with an interesting history. Just keep it real; I'll (hopefully) be able to sniff out faux or folk etymologies.

The one I like best will earn the commenter a Merit Badge. This could be because it's funny, or interesting, or both. As usual, deadline's midnight WDC time, and please leave the comment here and not in the Newsfeed.

3 Like • Comment • View 8 comments

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