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 19 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 21 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.

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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').

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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 17, 2020 at 12:05am

#975687

World Enough And Time

Since I'm finally getting around to learning a new language - however badly at this point - language articles get noticed more and thus stuck in my Blog Fodder List.

This is one of them.

https://theconversation.com/language-alters-our-experience-of-time-76761

Language alters our experience of time

Getting this bit out of the way first: no, time is not an illusion. But just like different people might estimate a mile differently (which is why we have measuring sticks), different people perceive the passage of time differently. Or the same person perceives it differently at different times.

And, to make another analogy, I've heard Europeans complain that Americans give distances in time. Like, the nearest city might be a half hour away, or a drive across the country takes three days. So at the very minimum, I'm prepared to accept that there are cultural differences in our perception of time.

My new study – which I worked on with linguist Emanuel Bylund – shows that bilinguals do indeed think about time differently, depending on the language context in which they are estimating the duration of events.

Again, no real surprise. Language gives us a particular way of looking at the world, and switching languages is like switching filters on a lens.

The logic in Aymara appears to be this: we can’t look into the future just like we can’t see behind us. The past is already known to us, we can see it just like anything else that appears in our field of vision, in front of us.

Which is the part I find really fascinating - the perception of time there is completely turned around from what we're used to. Like living in a mirror, where left is right and right is left.

A similar thing that I haven't seen too much written about is the direction of written language. Most Western scripts read from left to right, like we're doing right now. Hence we tend to order things in that direction: door number 1 is on the left, door number 2 in the middle, door number 3 on the right. To number them backwards seems at least weird, like you're about to launch a rocket via countdown.

But other languages, like Hebrew, read from right to left, so I imagine they'd order things the second way naturally.

There are interesting historical reasons for the differences in direction, and I suspect they both have to do with the predominance of right-handed people. If you're writing with ink on parchment, you want the wet ink to be in the clear, and not smudged by your hand as you go along - so you'd be inclined to pull your right hand to the right. Whereas if you're chiseling words on stone, and you're right-handed, you want to hold the hammer in your right hand and chisel in your left. Smudging isn't an issue, but angle might be. So, basically, left-handed people got the short end of the stick - sometimes literally.

None of which explains vertical scripts very well, and I don't have much knowledge about them. Anyway, to get back to what the article is saying, I think there are a lot of unconscious biases that we have, and our perception of time might very well be entangled with our perception of language.

It seems that by learning a new language, you suddenly become attuned to perceptual dimensions that you weren’t aware of before.

Again, if true (I don't know enough about any other language to have personal experience with this), that's really interesting.

But it also shows that bilinguals are more flexible thinkers and there is evidence to suggest that mentally going back and forth between different languages on a daily basis confers advantages on the ability to learn and multi-task, and even long term benefits for mental well-being.

Which relates back to one of my reasons for deciding to learn other languages at my advanced age. I mean, I almost never do anything for just one reason, and the primary reason has to do with booze (naturally), but why not make up for the brain cells lost to said booze by making my mind more flexible?

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