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.

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

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

July 6, 2023 at 8:43am

#1052229

Pedalers

Of course the bicycle moved us forward. I've never seen one with a reverse gear.

10 Ways the Bicycle Moved Us Forward

The article, from Mental Floss, is from 2016, but I doubt there have been massive changes in our knowledge of transportation history since then.

I'm entirely too lazy to retype section headers. They're in ALL CAPS.

1. IT REVOLUTIONIZED HOW PEOPLE HOOK UP, A CENTURY BEFORE TINDER.

I've been on the internet long enough to remember when it was all, "Don't hook up with anyone from the internet." Now it's "Only hook up with someone from the internet."

The effect on romance was profound: Long-distance courtships were possible.

This section has a distinctly British slant, so remember: In England, 100 miles is a long way; in the US, 100 years is a long time.

2. IT SHOWED THE WORLD THAT AFRICAN AMERICANS BELONGED.

And this section is slanted American, like people of African ancestry didn't live anywhere else that there were bicycles.

3. IT GAVE US THE NATION'S FIRST PAVED HIGHWAYS.

Having studied transportation engineering, I knew this. It's ironic now, as automobile drivers tend to think the road belongs to them and them alone, and many of them treat cyclists as annoying obstacles.

4. IT DEMONSTRATED THAT SPACE IS SAFE FOR ASTRONAUTS.

In 1973, the Skylab space station crew became the first people to pedal in space (on stationary bikes).

That's a stretch.

5. IT EMANCIPATED WOMEN FROM THE HOME (AND THEIR WARDROBE).

Not only did women love the freedom, they loved the dress, ditching heavy skirts for bloomers to work the pedals.

I've made jokes before about how if life made sense, it'd be men's bikes that didn't have the ball-racking upper tie bar. But the historical reason for the design of chick bikes has nothing to do with genitalia, but with fashion.

6. IT HELPED HUNDREDS OF JEWS ESCAPE THE HOLOCAUST.

Bartali helped Jews escape the country, carrying counterfeit identity papers in the frame and handlebars of his bicycle on “training rides.” If stopped by police for search, he’d ask them not to touch his “specially calibrated” bicycle.

These days, that would result in the cops disassembling the bike into its component parts, bending every single resulting piece, beating the rider for resisting arrest, and then charging the dude for their time. At least, that's how it would work here in the US.

7. IT BROUGHT LIFE-SAVING CINEMA TO PEOPLE IN REMOTE PLACES.

In 2013, a charity wanted to screen educational films in Malawi to spread info on HIV prevention, modern farming, and other issues. But most villages lacked electricity or gas for a generator. Enter Colin Tonks of Electric Pedals, who built a pedal-driven cinema that fits in two backpacks and weighs less than 40 pounds—perfect for toting to remote spots.

Seems to me there might be easier ways to get the information there, but what do I know?

8. IT CREATED THE PERFECT URBAN AMBULANCE.

In 1993, London ambulance driver Tom Lynch was stuck in traffic on his way to a call when he started thinking about how much faster he could get there on a bike.

More importantly, bikes make great urban pizza delivery vehicles.

9. IT PROVIDES CHEAP, CLEAN POWER FOR LOCAL ECONOMIES.

"Cheap" only if labor is "cheap."

10. IT HELPS ALLEVIATE SYMPTOMS OF PARKINSON'S DISEASE.

In 2003, biomedical engineer Jay Alberts was on a high-effort weeklong tandem ride with a friend who has Parkinson’s when the friend’s symptoms nearly disappeared. Alberts then studied a group of Parkinson’s patients who rode at an intense 80-90 rpm clip on a tandem bike, and had a 35 percent improvement in symptoms.

What a dilemma: do hard work, or continue to experience symptoms. I'm honestly not sure which would be worse.

5 Like • Comment • View 1 comment

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