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

about 10 hours 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.

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QPdoll

a day ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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.

1 Like

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

Sep 21, 2024 at 7:34am
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

Sep 21, 2024 at 6:46am
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

Sep 21, 2024 at 6:16am
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.

February 12, 2023 at 12:03am

#1044647

Revisited: "Crossing My Path"

Time for another trip to the olden days. This one's from way back in May of 2020: "Crossing My Path"

A 30DBC entry, this was written to a rather open-ended prompt: Start your entry today with the words: “I used to believe...”

I could have gone pretty much anywhere with this one. I chose what I consider to be a safe route; some of my former beliefs are profoundly embarrassing to me now. For instance, for a while there, when I was much younger (but old enough to vote), I thought that libertarians had the right ideas. It pains me to even admit that, but there it is.

As I noted to open the entry, though:

I used to believe that we stayed basically the same person all our lives. Then I read something recently, but didn't bother to save a link, about how some scientists did a study on young people, then studied them again when they were older, and what they found that Old You doesn't really have anything in common with Young You. I mean, we've known this from a physical standpoint - cells get replaced and all that - but it kind of calls into question the idea of continuity of consciousness, and identity itself.

I notice this even more with Drunk Me. He gets me the best presents from Amazon, though.

So I banged on, in that entry that feels so long ago but really isn't, about an article I'd found about superstition.

But what struck me today, revisiting this, was that I didn't really talk about "belief" itself.

It's one of those words that's a bit slippery. Sure, one can quote a dictionary definition ("an acceptance that a statement is true or that something exists" or "trust, faith, or confidence in someone or something"), but dictionaries don't capture all the nuances of a word. It can also mean a conviction, which has got to be confusing to those not used to English, because "conviction" also means that the legal system has determined that you've done a crime. You can have a conviction that you didn't do the crime, but you've still been convicted.

To me, there are three major kinds of belief:

1) a trust in someone (possibly yourself), as in "I believe in you!" This isn't the same thing as saying "I believe in aliens," because the existence of the "you" in that sentence isn't in question. Which leads us to my second kind of belief:

2) Certainty that something exists, despite a paucity of evidence, no evidence whatsoever, or evidence against. Belief in space aliens, God, Bigfoot, fairies, Russel's Teapot, etc. is in this category. And then there's:

3) Certainty that something exists, with solid evidence to back it up. Gravity, the (mostly)spherical shape of the Earth, evolution by natural selection, or cats, for example.

Wait, okay, there's another one: 4) to understand or remember something in a certain way, as in "I believe they said they would do that." I'm not really counting that one, though; it's got nothing to do with certainty.

So what we need is a different word for the third kind, in my opinion (or belief, if you will). Because saying "I believe in evolution" isn't equivalent to "I believe in leprechauns" in any way except linguistically. Assuming that it does leads to all sorts of confusion; for instance, one might say "I believe in climate change" and someone who doesn't accept the science might retort with "Sure, and I believe in Mothman."

The difference is substantial. It's why I bang on about science in here and try not to use the word "belief" in those contexts.

But I used to believe it wasn't that big a deal.

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