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 10 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 12 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').

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.

May 15, 2022 at 12:23am

#1032420

Waters of Life

I might have overindulged earlier, which is why I'm running a little late today. Just as well that this one doesn't require a lot of brain power, because I'm pretty short on that right now.

No Longer Ignored, Mumbai’s Ornate Drinking Fountains Are Bubbling Back to Life

I simply found this story interesting. It's related to civil engineering, my profession, but even without that I just think it's cool.

“While studying these fountains early on, I realized that they were unique, but neglected and not really spoken of,” he says. “The city seemed to have forgotten them.” It pained him when he read about the demolition of pyaaus, or water fountains, and made it his mission to protect as many as possible.

While the person quoted here, Rahul Chemburkar, is an architect and not a civil engineer, the professions are related. Architects get to be more artistic, though.

Water supply (along with waste management) is one of the most important aspects of being able to live in cities.

“It is our moral duty to protect this part of the city’s heritage,” he says. “And if we could revive this idea [of public drinking fountains], it would be in contrast to the plastic bottles we drink from.”

Access to clean water is a basic human need. It's harder in cities. What I like about this article is that in Mumbai, it's more than just a public works project; it's part of the city's character.

This is where you'll have to click on the link to actually see pictures and drawings of the fountains in question. They are marvels of civic art.

The Keshavji Nayak pyaau, inaugurated in 1876, looks nothing like a public drinking fountain—it resembles a pavilion or shrine, with red sandstone pilasters, a cupola covered in carved peacocks, and statues of bulls at the entrance, a feature seen in many Hindu temples.

For instance, there's a pretty cool before-and-after pic of that one.

Beyond restoration, Chemburkar conducts frequent walks and lectures, via the Mumbai Pyaau Project, and sketches postcards of the fountains. I went on a walk last March, before the pandemic, and, along with architecture students, tourists, curious locals, learned about structures that I would have never noticed. The walk also nudged the group to contemplate today’s water-supply challenges.

And of course, there's a lot more to water access than just having fountains available. Just ask the Southwest US right now. But I can't be arsed to get into the politics of it in my current condition, so I'll just leave this here as something cool to share.

3 Like • Comment • View 1 comment

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