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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Turkey DrumStik

about an hour ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

a day 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

a day 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').

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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

February 24, 2019 at 12:21am

#953061

Can You Dig It

I just have this to share today:

https://www.nytimes.com/2017/12/28/nyregion/new-york-subway-construction-costs.h...

Now, I know that not everyone is interested in infrastructure issues, or what's going on in New York. I'm a civil engineer with relatives there, so I try to keep up.

But the reason I'm sharing this story isn't the mismanagement, or the graft, or the corruption, or even the cost overruns. No, it's some of the excuses given for why subway construction in NYC costs 4-5 times as much as it does in other major cities.

“They’re claiming the age of the city is to blame?” asked Andy Mitchell, the former head of Crossrail, a project to build 13 miles of subway under the center of London, a city built 2,000 years ago. “Really?”

Well, that and a few of the labor union requirements the article details. Side note: I'm not taking a political stand for or against labor unions in general. But what that article describes is as pure New York City as it gets. You know. Empire State Building, Times Square, stock exchanges, Broadway, Brooklyn Bridge, union inefficiency.

But my absolute favorite part is toward the end when they compare it to a similar project in Paris.

Across the Atlantic Ocean, Paris is working on a project that brings the inefficiency of New York into stark relief.

The project, called the Line 14 extension, is similar to the Second Avenue subway. Both projects extend decades-old lines in the hopes of reducing systemwide overcrowding. Both involved digging through moderately hard soil just north of the city center to make a few miles of tunnel and a few stations about 80 feet underground. Both used tunnel-boring machines made by Herrenknecht. Both faced strict regulations, high density and demands from neighbors, which limited some construction to 12 hours per day.

But while the Second Avenue Subway cost $2.5 billion a mile, the Line 14 extension is on track to cost $450 million a mile.

There's an old joke that goes something like this: In Heaven, the French are the chefs, the British are the police, and the Germans are the engineers. In Hell, the British are the chefs, the Germans are the police, and the French are the engineers.

But it's looking more like: in Heaven the French are the engineers, and in Hell, the Americans are.

4 Like • Comment

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