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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Show Recent Comments

10 Most Recent Blog Comments:

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').

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.

October 19, 2019 at 12:08am

#968099

Birthed in Beer

https://www.theguardian.com/technology/2016/jul/15/how-the-internet-was-invented...

How the internet was invented

In 40 years, the internet has morphed from a military communication network into a vast global cyberspace. And it all started in a California beer garden

You will note the prominent feature of the subhead is the phrase "in a California beer garden." Remember a while back I posted on the link between bars and invention?

In the kingdom of apps and unicorns, Rossotti’s is a rarity. This beer garden in the heart of Silicon Valley has been standing on the same spot since 1852. It isn’t disruptive; it doesn’t scale. But for more than 150 years, it has done one thing and done it well: it has given Californians a good place to get drunk.

And it hasn't achieved National Historic Monument status because...?

It doesn’t seem a likely spot for a major act of innovation. But 40 years ago this August, a small team of scientists set up a computer terminal at one of its picnic tables and conducted an extraordinary experiment. Over plastic cups of beer, they proved that a strange idea called the internet could work.

As far as proof of concept demonstrations go, I've seen worse.

If you had walked into Rossotti’s beer garden on 27 August 1976, you would have seen the following: seven men and one woman at a table, hovering around a computer terminal, the woman typing.

Typical 1970s scenario. It's always the woman typing. That's what they're for, don't you know?

The fact that we think of the internet as a world of its own, as a place we can be “in” or “on” – this too is the legacy of Don Nielson and his fellow scientists. By binding different networks together so seamlessly, they made the internet feel like a single space. Strictly speaking, this is an illusion. The internet is composed of many, many networks: when I go to Google’s website, my data must traverse 11 different routers before it arrives. But the internet is a master weaver: it conceals its stitches extremely well. We’re left with the sensation of a boundless, borderless digital universe – cyberspace, as we used to call it. Forty years ago, this universe first flickered into existence in the foothills outside of Palo Alto, and has been expanding ever since.

And so the internet, like civilization itself, owes its existence to beer.

Okay, so I had to do a search on Rossotti's. Turns out it was recently restyled.

https://www.paloaltoonline.com/news/2019/10/10/the-best-version-of-rossottis-tha...

Zott's may have new floors, clean bathrooms, free WiFi and artisan wood-fired pizza, but the spirit of the place largely lives on.

Heh. Free WiFi. And so we come full circle.

2 Like • Comment • View 2 comments

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