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 2 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 4 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.

December 21, 2021 at 12:02am

#1023471

Exit Night

With over 1800 entries in this blog, I'm pretty sure I've repeated a blog title before, though I haven't gone searching for duplicates, so I don't know when it might have happened.

This problem is only exacerbated on important dates, like my birthday or the winter solstice, both of which I tend to bang on about when they happen.

And today is the winter solstice, at 10:59 am in my time zone.

Hence the twisting of a Metallica lyric in my title today; I'm pretty sure I've never used that one.

It has been my personal tradition, for a while now, to stay awake between sunset and sunrise on the longest night of the year. I do this to remind myself that "even the darkest night will end, and the sun will rise."

As an aside, I can't be certain that tonight is the longest night. The information I found goes only to the minute, and the number of minutes is the same tonight and tomorrow night (though today will definitely be the shortest daylight of the year). I'm sure I'd know if they broke it down to the second. Either way, though, the difference, if there is one, is measured in seconds so it hardly matters. I'm only pointing this out because I like to get these technical matters straight.

This staying up through the longest night thing is something I try to do but don't always have the opportunity.

Sometimes, I'm traveling, and that's not conducive to staying up all night. Well, I'm not traveling this year.

I vaguely remember that one year, I had a cold, and so I didn't do it, because that would suck. One requires rest when one is ill. But I don't have so much as a sniffle right now (which is weird, because usually winter makes my sinuses act up even if I don't have a cold or you-know-what).

This year, my excuse is that I'm between eye surgeries -- as I noted, I had my left eye done on Thursday, and the right eye is scheduled to get poked at this coming Thursday. So no, I'm not going to attempt it tonight. My sleep is disrupted enough this week, as I'm pretty sure I mentioned a couple of days ago.

But I have to mark the occasion in my mind, anyway, and yesterday, I realized that I already had. The moment of sunrise on Solstice day is like a rebirth or renewal for me -- as I've said before, quoting Doctor Who, it's the moment I know that we're "halfway out of the dark." Well, I had a moment like that on Sunday, instead.

See, before I go see a movie, I sit at the theater bar and drink some beer. These past few months, I've had to take a picture of the menu on the wall and order from the image on my phone, because the writing on the wall was just too blurry for me to read -- but I could see fine up close. Being able to read beer menus on walls is an important life skill for me, and not being able to do it makes me feel like I'm too old and decrepit to be there. ("Don't they have printed menus?" "Yes, but they can't be arsed to keep them up to date.")

Well, before I went to see West Side Story, I sat down at the bar and... I could read everything on the menu. Even the small print, like ABV and prices. Though only out of my left eye.

So yeah. I'm halfway out of the dark.

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