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.

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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

August 23, 2024 at 10:40am

#1075620

Unsolved, Unresolved

Today's link isn't a podcast. I don't do podcasts. Shouldn't they call them something else now that the iPod is no more? Anyway, from Vox, a collection of stuff from a bunch of podcasts:

17 astounding scientific mysteries that researchers can’t yet solve
What is the universe made out of? How should we define death? Where did dogs come from? And more!

The key word in the headline there is "yet."

“Whatever we know is provisional,” Priya Natarajan, a Yale physicist, told us about research on dark matter. But the sentiment also applies to science overall. “It is apt to change. What motivates people like me to continue doing science is the fact that it keeps opening up more and more questions. Nothing is ultimately resolved.”

This is, of course, a good thing. But "nothing is ultimately resolved" can be misleading.

Unexplainable isn’t about how scientists don’t know anything.

There's a tendency in humans to indulge in binary thinking. That is, believing that since science can't tell us everything, then it can't tell us anything. This is, of course, arrant nonsense.

We’re drawn to questions because they are optimistic. They invite us to dream of a better world in which they are answered, where the gaps between questions and our capabilities to answer them are smaller.

Ugh. Optimistic? Maybe I should stop asking questions. Besides, finding out that the universe will definitely end is hardly optimistic. I mean, what's the point of doing anything if it's just going to wind down and stop one day? I mean, sure, trillions of years from now, but that's not so long, is it?

Here are some of the questions that astounded us the most.

I won't list all of them. That would take too long. Just ones I have comments on or can make jokes about.

1) What is the universe made out of?

String.

String theory: the Universe is a big ball of string, and God is a cat.

(Okay, this section is actually about dark matter, unsurprisingly, but my string theory pun is funnier.)

2) How did life start on Earth?

Lots of people think it started elsewhere and migrated to Earth. I have another article about that in the queue. Which is fine and may be a testable hypothesis, but, either way, it doesn't get at the real question, which is how did life start, query, full stop.

3) How did dogs evolve from wolves?

That's not the best-phrased question. It's not like wolves haven't been evolving for the past 20,000 years or whatever. Though that's hardly long enough for a lot of natural selection to take place. Artificial selection, sure, which is why we have both chihuahuas and Great Danes and call them both dogs. It's like the idea that we evolved from chimps: no, we and chimps (and bonobos) share a common ancestor.

5) What will animals look like in the future?

I'm guessing the answer is "animals."

10) Is there anything alive in the human poop left on the moon?

You know, I get that this question is academically important. It is also unique on this list because while we don't know the answer, we know precisely how to find the answer, and all it'll take will be another trip to the moon, this time carrying pooper-scoopers.

11) Was there an advanced civilization on Earth before humans?

I know it's kinda vogue-y to ask that question right now, but the simple truth is, if there were, we'd know. We might not find their artifacts or other archaeological remnants, but any "advanced" civilization worth that adjective is going to need energy and produce waste. That waste would show up all over the globe, just like our waste is doing now.

Besides, if there had been, all the oil would have been gone before we started digging for it.

No. There wasn't.

12) What is the definition of “life”?

Yeah, that's not a science question so much as a philosophy question. As I've noted before, though, science can inform philosophy.

14) What did dinosaurs sound like?

I hope they figure this one out, and it turns out they meowed.

17) How will everything end?

With some questions as yet unresolved, I'm sure.

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