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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Writer_Mike

18 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 16 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 19 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

July 28, 2023 at 10:24am

#1053216

Learn from the Best

I'm not sure how much of today's article / book ad from way back in 2017 I'm on board with, but it's interesting anyway.

How Einstein Learned Physics

Wanting to understand how Einstein learned physics may, at first, seem as pointless as trying to fly by watching birds and flapping your arms really hard. How do you emulate someone who is synonymous with genius?

That's a bit like asking why you should bother to learn how to play piano when Billy Joel already exists.

Whatever Einstein did to learn, he clearly did something right, so there’s merit in trying to figure out what that was.

I suspect that a big part of it is he never asked, "When will we ever need this in life?" or "Why should I bother learning that?" I mean, okay, I don't know, maybe he did (though he would have asked in German), but I doubt it, or he wouldn't have achieved what he did.

One of the most common stories about Einstein is that he failed grade school math.

The downside of being an iconic historical figure is that, inevitably, myths accrete around you. Newton's apple, Washington's father's cherry tree, that sort of thing. These myths (some of which might have a germ of truth inside) tell us more about what we want to believe than about what's factual. In Einstein's case, the "failed grade school math" myth is misleading, but tells me that, collectively, we need to remind ourselves that the guy whose picture is in the dictionary next to "genius" was actually human.

As the article points out, he didn't fail math, and he was also human:

At the end of college, Einstein had the dubious distinction of graduating as the second-to-worst student in the class.

Before rejoicing at this accomplishment, keep in mind that this was Germany around the turn of the last century, with an educational system not exactly known for accommodating individual differences in aptitude. I'm not singling out Germany, though; the US and UK weren't substantially different.

The difficulty Einstein had was undoubtedly due in part to his non-conformist streak and rebellious attitude, which didn’t sit well in an academic environment.

It is better known today that some of the most talented students get bored easily when attending classes geared toward those more in the middle of the bell curve. Not that our current educational system is much better at accommodating that; it's just different than it was 125 or so years ago.

Given Einstein’s enormous contributions to physics, I think it’s now worthwhile to ask how he learned it.

I mean, sure? There's no such thing as useless knowledge, but if, say, we figured out how Neil Peart learned how to play the drums, we still wouldn't achieve his level of greatness.

Einstein learned physics, not by dutifully attending classes, but by obsessively playing with the ideas and equations on his own. Doing, not listening, was the starting point for how he learned physics.

There was some emphasis on "learning styles" a while back, which seems to have died down because the concept is... well, I want to say bullshit, but bullshit at least has a use as fertilizer. I still think some ways of learning are more effective than others for each individual; for example, if you want me to remember something, make it a joke or a song (Schoolhouse Rock was like candy for Kid Me).

That said, it's not a revolutionary idea to assert that doing something is a very effective way to learn. So is teaching the subject.

How do you know when you really understand something? Einstein’s method was to try prove the proposition himself!

Also not revolutionary, but that can lead to problems, like the guy who tried to "prove" the Earth was flat by building his own rocket. There's a hell of a lot of knowledge out there, and if you have to prove everything yourself, you'll never get anything else done.

There is still value in figuring some things out for yourself.

Intuition matters more than equations

I'm not sure that's the case, but go ahead and read that section of the article if you want. I have no doubt intuition is important, but as noted in the text:

Einstein’s own thoughts were that “intuition is nothing but the outcome of earlier intellectual experience.”

This next section seems contradictory to me:

Thinking requires a quiet space and deep focus

and

Einstein was a master of deep work. He had an incredible ability to focus, his son reporting:

“Even the loudest baby-crying didn’t seem to disturb Father,” adding, “He could go on with his work completely impervious to noise.”

Sounds more to me like the trick is to ignore the noise around you. Also, some people report thinking better when there's music playing.

Einstein’s most famous method for learning and discovering physics has to be the thought experiment.

Sure, but that technique has its limitations, and is mostly just a mental exercise unless you can also turn it into actual science.

Overturn common sense… with more common sense

Sigh. Common sense is neither, and it's the opposite of science.

While solitude and focus were essential components of how Einstein learned and did physics, it was often conversations with other people that provided his breakthroughs.

I don't find this particularly revolutionary, either. It's pretty common to bounce ideas off other people.

Be rebellious

Rebellion for the sake of rebellion is just angst. I think it's important to know when to rebel, a trick I have never mastered, myself.

But, after all this, there's at least one section that I wholeheartedly agree with:

All knowledge starts with curiosity

And that's why I keep reading and posting this stuff. You don't have to be a genius to be curious.

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