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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Annette in France

about 3 hours ago
In response to "It's Hot in The South, Once Again!"

The quote is too small and low contrast to be read.

Elycia Lee ☮

about 13 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

a day ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

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Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

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Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

Sep 25, 2024 at 2:49pm
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

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November 24, 2019 at 12:18am

#970361

Really? You Want Me To Talk About Numbers? REALLY?

PROMPT November 24th

What numbers hold special meaning for you? Consider dates, times, ages, years, or anything else you can count.

Interesting thing about numbers. Well, there are lots of interesting things about numbers. But specifically, for the purpose of this prompt, there are no uninteresting numbers.

Bold claim, considering there's an infinity of numbers, yes?

For the record, I didn't come up with this. I heard it somewhere. I do this a lot - hear something and then forget the context. But I don't want people thinking this is original to me. Lots of things are, of course; just not this.

The argument for "there are no uninteresting numbers" goes something like this: Consider the set of positive integers (which are what most people think of when they think of "numbers," even though when I saw the prompt the first thing I thought of was π).

You can think of something interesting about a lot of ordinary integers. 1, for example, is, well 1. It's the multiplicative identity, among other things. 2 is interesting because it's the only even prime. 3 is interesting because it's the first odd prime (1 doesn't count as a prime number). 4 is interesting because it is the sum of its factors (2+2=2*2). And so on. Some numbers might be interesting for nonmathematical reasons, like 11 is interesting because "these go to 11" and 42 is interesting because it's the Answer to the Ultimate Question of Life, the Universe, and Everything (according to Douglas Adams).

But like I said, there exist a lot of numbers. An infinite number of numbers. Surely once you get past the numbers we can expect to encounter in our lives - past the hundreds, thousands, millions, billions, maybe? What about 8,345,232,098,712,450? A real mathematician might be able to rattle off something interesting about that particular string of digits, but I'm stumped.

So it stands to reason, then, that as you count upwards from 1, you'll eventually come to a number that's not interesting, right? You'd think so, but wait - that number is The First Uninteresting Number! That makes it interesting! Keep counting, and you reach the actual First Uninteresting Number, but hold on - that makes it interesting. And so on. And so on, out to infinity.

The upshot of this seeming paradox is that there are no uninteresting numbers.

So, what numbers have special meaning for me? As a fan of number theory, I should say "all of them," but that raises another paradox: if everything in a set is special, is anything in that set special?

I already mentioned a few above. 11, for example. That's from this scene:

And for some reason, more than just about any other scene from any other movie (with the possible exception of Star Wars), that bit of comedy genius has embedded itself forever into pop culture. It's become a meme in the sense that the word "meme" was supposed to have when Richard Dawkins coined the word, coincidentally (maybe) right around the same time that movie came out. For example, ever watch a video on the BBC website? Next time you do, take a good look at the video player. The volume goes to 11.

When I first noticed that, I couldn't stop laughing all day.

As far as dates go, well, my birthday, obviously. That's the most important day of the year.

With ages, I noticed something in my own life: I tend to go through phases in a 7 year cycle. Every 7 years, something changes the way I live or look at life. Of course, once I noticed this, it might have become self-fulfilling, but that's okay; I can think of worse ways to arrange one's life. Fortunately, I still have a couple of years to go before the next cycle.

Still, like I said way up there

, the first thing I thought of was π. It seems to be fundamental to everything in the universe, from the smallest scales up to the largest, and yet its definition is simple: the ratio of a circle's circumference to its diameter. There's something immensely powerful about such a number, and yet it's utterly impossible to know it to the last decimal place, because it has infinity decimal places.

And that's interesting.

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