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 8 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

November 28, 2018 at 1:14am

#946444

Stout

I haven't always liked beer.

This is probably because of one of my first experiences with beer, which involved a bunch of other underage drinkers, a houseboat, and this swill.

I suppose I should be grateful for that, because if I actually enjoyed drinking when I was in college, I might not have graduated. As it is, I drank to be social, and I got used to the mass-produced crap that passes for beer in the US, which has the distinction of being cheap enough for fraternities to buy it by the keg.

Like many college students, I drank to get drunk.

And then, one day, around this time of year oh, about maybe 25 years ago, I went to a local brewery - one of the first in this area - with some friends. I don't remember exactly what or who convinced me to try it, but there was a Russian Imperial Stout brewed on premises.

Rarely does a person change their opinions when presented with logic or facts. Occasionally, an appeal to emotion will work, but even then, you're facing down the weight of that person's supposedly secure knowledge, which will, studies have shown, persist even in the face of overwhelming evidence to the contrary. Their ego gets so wrapped up in being right that their minds simply do not process conflicting data. It's dismissed: fake news.

I try not to be that kind of person. I'm sure I must have some deeply-held personal beliefs to which I stubbornly cling, because everyone does, but the nature of these things is that unless they're pointed out to me, I won't even know that I'm wrong. But once it is pointed out to me, I do make an honest effort to revise my thinking. Sometimes, though, it's not logic nor emotion, but experience that proves to be the trigger to a new way of thinking.

And this particular beer, this... nectar of the gods, it changed my perception of everything.

No more was beer to be a cheap way to alter consciousness. From the moment the dark, magical liquid touched my lips, I was a changed man. It was the beginning of my alcohol-positive lifestyle: not about getting drunk, or being social, or succumbing to peer pressure, but about enjoyment, about immersing oneself into the experience of the sublime. Not about drinking to excess, but about drinking to access - to access the joy that can be so elusive in sober life.

It is a state that I term beerenity, although it can be achieved through other libations also.

True happiness is rare and fragile, but I found mine, and it was all because I was willing to try something new. Naturally, sometimes when I try something new, it doesn't work out so well; but sometimes it does, and I never know until I try.

8 Like • Comment • View 4 comments

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