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:

Write_Mikey_Write!

about 22 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 23 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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').

1 Like

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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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