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 14 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 16 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.

April 9, 2020 at 12:01am

#980583

Learning the Language

I want to believe that this is true, so I'm going to accept it uncritically.

https://theconversation.com/youre-never-too-old-to-become-fluent-in-a-foreign-la...

You’re Never Too Old to Become Fluent in a Foreign Language

Oh, wait, no, accepting things uncritically is something I strive to avoid.

A range of headlines – from the BBC to the Daily Mail and The Guardian – all trumpeted the depressing message that it’s impossible to become fluent in a foreign language after around age ten.

If the Daily Mail said that there's a beer shortage, I would instantly doubt the existence of shortages. And of beer.

All of these reports dramatically misrepresented the findings from the study, and the message they sent is flat-out wrong.

On the other hand, I trust the BBC -- and yet, few outlets get science right.

So to sum up so far, there was a study on second language learning, and British news and tabloid headlines blared that one cannot become fluent in another language after age 10, but they all got the science wrong. News reporters getting science wrong? Quelle surprise!

For one thing, the words “fluency” or “fluent” never even appear in the original study, published in the journal Cognition.

That alone should throw the reporting into suspicion.

To be fluent in another language means that you can communicate with relative ease, that is, without it being a real strain on either the speaker or the listener.

There are dialects of English that I have trouble decoding, especially when spoken. One time, at a flea market in Florida, I just had to stop and gawk in awe at an exchange between two booth-tenders. I'm pretty sure they were both speaking English; at least, I made out a word or two here and there that was definitely English. It was just so rapid-fire, and with weird cadences, that I couldn't get the meaning. I don't know; maybe they were brothers with their own secret language, or something. More likely, they were speaking Floridian.

But to contradict the tabloids, this article goes on to say:

Learners of any age can achieve a brilliant, even nativelike, command of the vocabulary of another language, including such challenging structures as idioms or proverbs.

Whew. I hate to think my French studies are going to be entirely useless.

The puzzling thing about older learners – something the authors of the new study also found – is that they seem to have more problems mastering some, but not all, grammatical phenomena.

I certainly have problems deciphering emoji.

What is new about the Cognition study is that, by the usual standards of linguistic investigations, it uses a dataset of unprecedented size. Through an internet grammar quiz shared on social media, the authors collected almost 700,000 responses, two thirds of them from people for whom English was a second language.

Oh. Yeah, I'm not sure the large data set can overcome the biases here.

They found that the accuracy of the responses on the grammar quiz declined sharply for learners who began studying English after the age of 17, a long way off the age of ten, which is the age most of media reports focused on.

Crap. 17, 10, all that looks the same from my perspective. I thought they meant "older" as in "decrepit."

But the claim that its findings suggest that after age ten you are too old to learn a foreign language fluently is one of the worst misrepresentations of a scientific outcome that I have ever seen.

Which is why all science reporting needs to be scrutinized. Of course, most of us don't have the time to fact-check every article that comes our way. I certainly don't. But if something doesn't pass the bullshit sniff test, it's not okay to go repeating it like it's gospel.

You can become a perfectly fluent speaker of a foreign language at any age, and small imperfections of grammar or accent often just add to the charm.

I don't know about "perfectly" fluent. As for my own case, we'll see -- I haven't yet gotten to the point where I can think in French; I still have to translate back and forth to English. I'm not sure if I can ever get there. But I'm not going to let a "you can't do it at your age" stop me. In fact, I'm just contrary enough that it might be enough to make me do it out of spite.

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