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.

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.

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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.

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.

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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.

2 Like

March 11, 2021 at 12:04am

#1006171

With the Flow

So, okay, I'm still not completely coherent, but I'm not going to break my streak now.

Why a Traffic Flow Suddenly Turns Into a Traffic Jam
Those aggravating slowdowns aren’t one driver’s fault. They’re everybody’s fault.

I've always wondered about this, myself, and I've taken actual traffic engineering classes.

Few experiences on the road are more perplexing than phantom traffic jams.

I can think of a couple, like when cops pull you over because you have out of state tags and therefore they figure you're not going to come back to contest the ticket.

Because traffic quickly resumes its original speed, phantom traffic jams usually don’t cause major delays. But neither are they just minor nuisances. They are hot spots for accidents because they force unexpected braking. And the unsteady driving they cause is not good for your car, causing wear and tear and poor gas mileage.

Okay, that's a stretch.

In contrast, macroscopic models describe traffic as a fluid, in which cars are interpreted as fluid particles.

When I was in engineering school, I learned both fluid mechanics and traffic engineering. I noted then that the equations used to model traffic flow were the discrete versions of the continuous equations used to model fluid flow. Such an epiphany made me feel a whole lot smarter than I actually was.

This observation tells us that phantom jams are not the fault of individual drivers, but result instead from the collective behavior of all drivers on the road.

You know, there's a metaphor in there somewhere. Like... pick a societal problem. Any problem. Poverty, racism, global warming, whatever. It's no individual's fault, but the result of all of our collective behavior.

It's not a great metaphor, but there it is.

However, in reality, the flow is constantly exposed to small perturbations: imperfections on the asphalt, tiny hiccups of the engines, half-seconds of driver inattention, and so on. To predict the evolution of this traffic flow, the big question is to decide whether these small perturbations decay, or are amplified.

And this bit reminds me of the difference between laminar and turbulent flow. I can't be arsed to explain that right now. Google it.

Besides being an important mathematical case study, the phantom traffic jam is, perhaps, also an interesting and instructive social system.

Like I said. Metaphor. I mean, really, you have to read the article to get the full effect; I've had a few beers tonight and thus I'm probably not making nearly as much sense as I think I am. But this one's been hanging out in my queue for a while, and I'm pretty sure that I'm saying what I wanted to say about it.

Hopefully tomorrow I'll be more clear, but right now there's a turbulent traffic jam in my neurons.

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