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:

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').

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.

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

1 Like

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

October 14, 2024 at 9:27am

#1078266

DeTECHtion

From Cracked, a countdown of the unseen:

5 Ways to Detect the Invisible Stuff Hiding From Us
Surround yourself with the right sensors, and they might just save us all

All around you are signals you have no way of perceiving on your own. For example, did you know that invisible waves travel through the air, transmitting words and images, decodable by the right device?

We only see a remarkably thin slice of the electromagnetic spectrum. There are good evolutionary reasons for this, but why interrupt a fact-comedy with pure fact?

Not everything imperceptible is on the EM spectrum, though.

5. Seismometers Spot Quakes and Save Trains

Not a single person has died in an accident aboard Japan’s bullet trains.

There was a movie a couple of years back called Bullet Train, set on a Japanese bullet train. Action/comedy, so the title is a pun. It's not much of a spoiler to say that bad things happen on the train and people die. But not in an accident caused by an earthquake. I had the opportunity to watch the movie again on a recent long flight, and it was even better than I thought it was the first time.

Even in 2011, when that 9.0 earthquake hit Japan, collapsing a bunch of bullet train stations and ripping up the track, seismometers on the ocean floor detected the first tremors and shut down the trains in time.

EM waves, including radio, travel faster than seismic waves, so these early detection systems aren't magic, but science.

4. The Mountain Pool of Ultrapure Water

Japan is also home to Super-Kamiokande, an observatory for detecting neutrinos. It’s a tank a hundred feet tall and hundred feet wide, located in a mine a thousand feet under a mountain.

Neutrinos are the second hardest things to detect, after dark matter. Well, maybe third, after dark matter and something so invisible we don't even know it exists, yet.

Also, I'm pretty sure the tank is measured in SI units, as is its depth.

Neutrinos travel at the speed of light and pass through the planet without affecting much of anything, but we still manage to detect them using Super-Kamiokande.

Far be it from me to quibble about something on a comedy site, but I'm going to do it anyway: Neutrinos have mass, and therefore don't quite travel at the speed of light. But, you know... close enough. (There was some sensationalist reporting a while back on neutrinos that appeared to exceed the speed of light, but that turned out to be a mistake.)

3. ShotSpotter, the Gunshot Detector

Sometimes, you hear a gunshot outside, and you shrug your shoulders, saying, “Eh, what can you do?”

Emigrate?

Starting in 1997, several cities in the U.S. set up a system of sensors that pick up the sounds of gunfire and relay the location of the source directly to police. In D.C., for example, the sensors picked up some 40,000 shots in a 20-square-mile area over the course of seven years.

That's the sound of freedom.

2. The Detectors that Detect Radar Detector Detectors

Naturally, such devices use waves of their own, which means they’re vulnerable to being spotted by radar detector detectors. Of course, that means you should equip your radar detector with a radar detector detector detector, which will shut your device down as soon as it realizes it’s being spotted.

And so the arms race continues.

Pretty sure my state is still the only one in the US where radar detectors are illegal, but I can't be arsed to find out for sure. Yes, I drive around a good bit. No, I don't have a radar detector.

1. The Parkinson’s Sniffer

This one may seem out of place, being a mutant superpower instead of a tech device. But if you read to the end, you see that they used the mutant's superpower to create the tech needed for a Parkinson's test.

No mention in the article of radiation and Geiger counters, but I suppose most people know about that one, anyway. In any case, all of these were known invisible things. It's the unknown invisible things you really have to worry about.

3 Like • Comment • View 1 comment

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