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:

Elycia Lee ☮

about 9 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!

2 Like

Wordsmitty ✍️

about 22 hours 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.

1 Like

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.

1 Like

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.

2 Like

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.

2 Like

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.

2 Like

Brandiwyn🎶

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

Love it when stars align!

2 Like

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.

1 Like

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.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

January 3, 2020 at 12:54am

#972528

Beer for Breakfast

PROMPT January 3rd

Make a list of 20 random things (nouns). Try as hard as you can to make each item on your list as unrelated to the other items as possible. If your muse moves you, you’re welcome to add whatever commentary you like.

psychology
volume
county
appearance
permission
initiative
combination
estate
reception
drama
philosophy
length
procedure
attention
honey
debt
presentation
platform
song
population

I cheated. I literally selected the nouns at random

rather than coming up with them out of my head, which, let's be real here, wouldn't have been random.

It's just part of my natural psychology to be drawn to random numbers. More than anything else, the sheer volume of dice rolls is what attracted me to playing D&D, even from 'way back when I was living out in the back end of a rural county with few people around. When I'm not doing this blogging challenge, randomness makes an appearance through my selection from a list of articles I found interesting in some way.

Once I decide to do something at random, I rarely give myself permission to stray from the results. If I wanted to do something on my own initiative, I wouldn't have randomized the situation. For example, yesterday I somehow found myself at a brewery, staring at a tap list of 22 different beers from which I could select a combination of four for a tasting flight. Well... I could have bought more than one tasting flight, but doing so could negatively affect my estate. So the selection process can be dicey (pun intended, of course) - as much as I like beer, I give some styles a better reception than others.

Without going into too much drama, when it comes to beer, in general, I'm a contrarian - what I like, such as Belgian styles, few others do; what I don't like - sours for example - seem to be popular. I'm comfortable with this philosophy, though, and I'll go to any length to defend my own point of view. Thing is, I'll try almost anything, and when faced with a limit of four out of 22 possible beers, I have a certain procedure for these situations: roll the virtual dice, or in this case, use an app on my phone that is specifically for generating x numbers out of y possibilities, with or without attention paid to whether or not repeated numbers are valid (in this case, not).

I've been pleasantly surprised by this method; I've found beers tasting of lemon, of honey, of blackberry; but mostly they taste like beer - delicious, refreshing beer. I don't do it too often because I'd end up going into debt, but I do enjoy the presentation of a flight of different-colored beers all in a row or a ring on a platform specifically designed to highlight their variation and their luster. The right combination of beers - chosen carefully, or selected at random as I described - is as gorgeous as a song, even if the vast majority of the population would never agree with me.

Well, there. I did it. 20 truly random words, all shoehorned into a praise of beer. I did end up getting more than one flight, and then a couple of pints of different brews, and then a tasting of the style I specifically went to the brewery for. I should have mentioned that this is an exceptionally good brewery.

Then I Ubered home and passed out. Yes, I'm still drunk, but I did this entry anyway.

For anyone who doesn't know what I mean by a "flight" of beer, here's a representative picture from the internet.

8 Like • Comment • View 8 comments

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