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:

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 15, 2024 at 7:42am

#1078334

Pushing the Button

This BBC article (whose headline resembles a Cracked headline) is a few years old now, but all that means is they might be missing a few recent ones.

The moments that could have accidentally ended humanity

In recent history, a few individuals have made decisions that could, in theory, have unleashed killer aliens or set Earth's atmosphere on fire. What can they tell us about attitudes to the existential risks we face today?

Already, I have a bad feeling about this. Sounds like an Agenda, which I don't expect from the BBC.

In the late 1960s, Nasa faced a decision that could have shaped the fate of our species. Following the Apollo 11 Moon landings, the three astronauts were waiting to be picked up inside their capsule floating in the Pacific Ocean – and they were hot and uncomfortable. Nasa officials decided to make things more pleasant for their three national heroes. The downside? There was a small possibility of unleashing deadly alien microbes on Earth.

No. No, there wasn't.

They thought there might be such a possibility, so I understand the caution, but in hindsight, there was never any danger from that.

A couple of decades beforehand, a group of scientists and military officials stood at a similar turning point. As they waited to watch the first atomic weapon test, they were aware of a potentially catastrophic outcome. There was a chance that their experiments might accidentally ignite the atmosphere and destroy all life on the planet.

Again: no, there wasn't. They thought there might be such a possibility, and they went on with the tests anyway, which says more about humans than I care to admit.

This sort of writing demonstrates some sort of fallacy that, well, I don't know what it's called, and it's hard to back-search these things.

The people in those cases were acting on imperfect knowledge. I mean, we all are, all the time, but not usually with imaginary stakes so high.

Let me maybe give you an example. You're walking through the woods one day, and you come across a pedestal with a single, big, red button on it. There's no writing, no pictograms, nothing to indicate what (if anything) pressing the button might do. From cartoons, you know that pressing the Big Red Button is generally a Bad Idea, but those are cartoons, not reality.

You're human, and you're faced with a mysterious button in a strange location to find a button. The urge to push it is real, palpable. How can you, curious ape, just... walk past it? You can't. Billions of years of evolution can't be overcome by the moral lessons in cartoons, not easily. Maybe it opens a secret cache with 10 million dollars in nonsequential, unmarked bills. Maybe a genie would come out to grant wishes, and you could wish for more than $10M. Okay, that's unlikely. Maybe nothing will happen. That seems most likely. Maybe a piano will drop onto your head when you press it, killing you. Or maybe it connects to the gigantic planet-exploding device hidden deep in the core. Wow, that's really unlikely.

What you're doing, whether you realize it or not, is risk assessment. Each possible event has a probability. Each possible event also has a desirability. Assigning probabilities is tough, and it's even tougher without Bayes' Theorem to help you assess the probability through the lens of prior knowledge. It's also very difficult if you have no way of knowing what every possible outcome is. But the point is, if you can do that, even roughly, and multiply the probability by the desirability, you can more easily rank your choices. Finding a stash of money? Low probability but desirable. Blowing up the planet? Really low probability, but undesirable.

In the end, billions of years of evolution win out, and you press the button.

Only through hindsight can you really know if you made the right choice or not.

Getting back to the examples from the article, yes, of course, I'd heard about them before. Take the astronauts thing. High probability: no deadly space microbes. Result: no disaster. Low probability: deadly microbes. Result: Astronauts die, Earth saved. Or, Result: Everyone dies. That last one would be Bad, so I don't blame them for hedging with the "deadly microbes" scenario.

It's kind of like the Trolley Problem, only your action shifts the trolley onto a random track where a random number of people are tied.

I don't know if I'm explaining all this very well, but my main point is, the article does a really bad job distinguishing "what we thought the probability was at the time" and "what the probability actually was, in hindsight."

The remainder of the article does a much better job explaining risk and our reaction to it, but the way it's described at the beginning stuck in my craw, so I could no more resist posting about it than you could resist pushing that Big Red Button.

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