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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 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 10 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.

July 17, 2020 at 12:01am

#988362

Driver's Seat

It shouldn't surprise anyone to know that music is, indeed, very important to me.

PROMPT July 17th

How important is the role of music in your life and in your creative process? Do you write to music? If so, what kind and why do you think it helps?

There was a prompt here a while back that asked the age-old question: if you had to choose between losing sight or hearing, which would you choose? And it was a difficult choice because, while I dislike about 75% of all sounds, the other 25% is music and I'd hate to live without that. On the other hand, I couldn't play video games if I were blind.

Anyway, this prompt isn't that one, but I mention that to illustrate my mindset when it comes to sounds. I've known people who insist on having the TV on all the time for "background noise." This is anathema to me. If I want sound in the background, it's going to be music, not some jibber-jabber on the boob tube.

Another thing that surprised me when I found out about it is the number of people who don't listen to lyrics. I seriously can't wrap my head around that. While all aspects of music work together, I need to know what the lyrics are, and they'd better have some meaning or I won't like the song. This doesn't apply to instrumentals, obviously.

I've tried writing with music playing and without, and I can't say that it's made much of a difference. These days, mostly I just listen to music when I feel like it, which is not all the time. I'm perfectly content with silence, and I can usually tune out ambient noise except when the fucking neighbor mows their goddamn lawn three times a week.

There is one situation, though, when music is absolutely essential to me, and that is when I'm driving.

10 years ago this month, when I bought my current car, lots of things sold me on it, but the most important feature to me was its face-melting sound system. For the first time in my life, I wouldn't have to mortgage my house to pay someone to install an aftermarket stereo. One time, I bought a car at auction for $340, an old Toyota that was only firing on three cylinders and had exposed springs in the driver's seat, but it ran, and the very first thing I did was drive across the street to Crutchfield to buy a $900 stereo system with surround-sound speakers and a subwoofer that didn't really fit in the trunk but I made it work anyway.

Priorities, you know.

I know a lot of people prefer to listen to talk radio or audiobooks while driving, but to me those require too much concentration. Music fills the idle hole in my brain that's left over when driving only takes up a portion of my attention (albeit the main portion), while still letting my mind wander sometimes and get ideas.

Trying to imagine doing my cross-country road trips without music is like trying to imagine a movie without actors.

A long time ago, I had the idea to include a different song with every blog entry, but by the time I got back to blogging, other people had already run with the idea, so I only pursue it sometimes. But because it's appropriate, today I'll leave you with this, one of my all-time favorite songs:

So pick up your feet
got to move to the trick of the beat
there is no elite
just take your place in the driver's seat.

5 Like • Comment • View 8 comments

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