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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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.

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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.

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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.

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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.

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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

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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.

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October 14, 2021 at 12:02am

#1019301

In a Jam

Very short article today, which is good because I don't really have much energy to comment on it.

Mathematicians have solved traffic jams, and they’re begging cities to listen
Mathematicians are unimpressed by engineers’ solutions.

I know the headline talks about engineering and mathematics, but it's about something most of us have to deal with on a personal level.

Most traffic jams are unnecessary, and this deeply irks mathematicians who specialize in traffic flow.

Not as much as it irks drivers.

Krylatov would like to solve urban traffic jams forever, so much so that he has coauthored a book of new math approaches to traffic and ways to implement them.

I've said before that I don't mind people using articles to promote their book. This might be an exception. Few people who read this article are in a position to give a damn what the book says.

1. All drivers need to be on the same navigation system.

Yeah, that's going to happen. Maybe when people get over their fear of autonomous vehicles, which will be shortly after pigs fly and right before hell freezes over.

I had a rant prepared in my mind about that, but now it's going to have to wait.

2. Parking bans. Many urban roads are too narrow and cannot be physically widened.

What's the damn point of driving anywhere and suffering through traffic jams if there won't be a place to park?

3. Green lanes. For cities that want to increase electric car use, special lanes should be created for electric cars, providing an incentive for their use.

Or, you know, you could put the parking back in.

4. Digital twins. Traffic demands and available infrastructure can only be balanced with digital modeling that creates an entire “twin” of existing roadways.

I know I haven't messed with traffic engineering for many years, but this bit made no sense to me.

And just in case you were wondering, “The mathematical approach in this case is superior to the engineering and economic one.”

Look. Yes, it's been a while for me. But one thing hasn't changed since I went to engineering school: Engineers do not ignore mathematics. It's, like, an integral part of what they do. If a mathematician comes up with better traffic modeling ideas, I guarantee you it's not the engineers who aren't listening.

It's the politicians.

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