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:

Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

October 25, 2018 at 12:19am

#944140

Perchance to Dream

https://www.theguardian.com/lifeandstyle/2018/oct/22/cant-sleep-perhaps-youre-ov...

What are the signs that you may be overtired? If you get into bed at night and find your mind is still racing with what has been going on through the day, overtiredness could be to blame. During the day, you may find it difficult to concentrate or to see the wood for the trees in your professional and your personal lives. You perhaps catch a lot of colds, especially when you’re winding down for a holiday. You’re irritable and you find yourself reaching for sugary snacks to keep yourself going during the day.

I wrote a few weeks ago about sleep paralysis. It was tempting, when I was experiencing that more often, to avoid sleep as much as I could in an attempt to avoid that condition. But I noticed that, for me at least, sleep paralysis was more likely when I ran a sleep deficit, not getting enough sleep and staying awake too long. So I forced my rational mind to overcome the fear, not something that's easy to do.

For years, I never got enough sleep. I used to wish for time enough to catch up. Between working late and over weekends, having a social life, and wanting to play video games, the only thing that gave way in my life was sleep. I never really kept track of how long I'd sleep on an average night, but it couldn't have been more than 6 hours and was more likely closer to 5. And inevitably, around 3 to 5 pm, I'd crash hard at the office, sometimes even nodding off at my desk (never got caught doing that, at least as far as I know). My lack of sleep was only because there were so many things I wanted to do in a day that I simply couldn't fit them all in to 24 hours.

What I never got, though - what I never understood, even a little bit - was the idea that sleep is some sort of weakness. "I'll sleep when I'm dead" is something I've actually heard a lot of times, as if sleep was something to be ashamed of. I suppose it's just another expression of uniquely American puritanism - the idea that we have to be productive at all times, and sleep is not productive.

I never really wanted to be productive, just busy.

My sleep schedule is still fucked up, but instead of getting too little sleep, I'm getting - not too much, I suppose, about the standard 8 hours a day - but it's broken up. Sleep in late afternoon, around the time my system used to crash when I was working and not getting enough sleep, then wake up, do stuff, and finish the sleep cycle in early morning. That's why I keep doing these blog entries after midnight - I'm awake, my mind is as sharp as it gets, and I'm still usually hours away from another sleep time.

At least the sleep paralysis hasn't hit in a long time, but getting other people to understand and accept that this is my cycle is a bit of a chore.

I have the advantage of working for myself and generally being able to set my own schedule, but the vast majority of people don't. I wonder how many psychological problems could be reduced if people could just find their own cycle and abide by it.

3 Like • Comment • View 6 comments

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