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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 19, 2022 at 12:02am

#1040820

Pronounced Differences

Today's article is about avid readers, which I'm sure all of my readers are. Else you wouldn't be reading this.

Words we think we know, but can't pronounce: the curse of the avid reader
Do you know how to say apropos? What about awry? We want to know which words you’ve mispronounced – and how you found out your mistake

This is from a couple of years ago, but that's probably irrelevant. I just found it last month. What is relevant is that it's an article about English pronunciation in the Guardian (British) written by an Australian woman, and it's well-known that the US, the UK, and Oz pronounce certain words different ways. "Privacy," for example. I think only the US pronounces that with a long i. So just keep that in mind.

When I mispronounced tinnitus (ti–nuh–tuhs is correct, ti-nai-tis is not) recently and was kindly corrected, my embarrassment was a fraction of when I said apropos (a–prow–pow instead of a-pruh-pow) to a large table of people in London when I was in my 20s. That day I was not kindly corrected, but only realised my mistake after howls of laughter and a whispered, “Maybe that’s how they say it in Australia?”

Now, see, I always thought it was ti-nai-tis. Some sources say both are correct. Officially, the first syllable should be emphasized. If enough people pronounce it the "wrong" way long enough, though, it becomes an alternative pronunciation.

As for "apropos," well, at least she didn't pronounce the s at the end, right?

Since then, I have learned that mispronunciation is often the downfall of people who read widely as children and form the incorrect pronunciation in their mind before actually hearing the word said aloud.

You know what? That shouldn't be embarrassing. It means you read. What should be embarrassing, but too often isn't, is the polar opposite: when you try to write something as it's pronounced, and you spell it wrong. I worked with a guy who kept writing stuff like "part in parcel" (should be "part and parcel"); "save and accept" (save and except), and "beckon call" for beck and call. Those errors aren't proof of illiteracy per se (he would write "per say"), but they do indicate a lack of interest in reading. And don't get me started on affect/effect.

What's even worse, of course, is mixing up things like it's and its; there, they're, and their; and your and you're.

Now, I'm not saying I always get everything right. Far from it. Only that I have more respect for people who mispronounce things because they read a lot than (not "then") I have for people who misspell things because they hardly read.

My ex-wife, for example, pronounced "picturesque" like "picture-skew." I thought it was adorable and never corrected her. Though in hindsight I should have used it against her in the divorce.

In short, I'd rather deal with someone who mispronounces "apropos" than with someone who writes it "apropoe."

Annals (not ay-nals), Hermione, misled (does not rhyme with thistled) and glower...

Look, it should be shining obvious that annals isn't pronounced the same way as anals. No one in the US knew how to pronounce Hermione until the first Harry Potter movie came out. I never thought misled rhymed with thistled. As for glower, well, honestly, I never was very sure about that one so I avoided saying it (turns out it's pronounced like flower).

A colleague pronounced facade with a k sound, another thought burial rhymed with Muriel and yet another was mortified to discover that segue was not pronounced seeg.

At least two of those are a result of not knowing the French influence on English.

English pronunciation can be tricky like that, anyway. We've borrowed so many words from other languages, words where you have to know a bit about the language to pronounce them correctly. Like, if you see the word "sake," you need to know if it's preceded by "oh for fuck's" or if you're talking about delicious Japanese rice wine.

French words very often leave English-speakers flummoxed. I’ve heard canapés pronounced in quite creative ways, and amuse-bouche, prix fixe and hors d’oeuvre have seen the odd food lover come a cropper.

Before I started learning French, I had a lot of fun deliberately mispronouncing French words. Canapés became can o' peas, for example, and hors d'œuvres became, to my vast personal amusement, horse doovers. And the surest way to annoy a French person is to say "Par-lezz-vouse fran-kais"

What word have you always mispronounced?

The article recommends commenting there with an answer to that. I wouldn't advise it as, again, this is over two years out of date. It also recommends tweeting same, which I definitely don't recommend right now.

But if you want to give me your examples below, feel free. Me? I don't know which words I'm mispronouncing. If I did, I wouldn't mispronounce them anymore. I know I used to think that rappelling (the practice of rock climbing with ropes) was like rapple-ing, but once I was laughed at and corrected I said ra-PELL-ing like you're supposed to. But that was back in high school.

I guess what we need is a verbal version of spell check. Something that makes a red squiggly line appear in your vision when you're about to mangle a word that you've only ever seen in print. Alas, we'll have to wait until cyborg technology is more advanced for that.

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