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 5 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 7 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 25, 2022 at 12:01am

#1035662

The Disconnected You

It is fitting that I'm just now getting around to sharing today's article, which is from 2014.

Why We Procrastinate
We think of our future selves as strangers.

Can I just say how refreshing it is to have the answer to the implied headline question right there in the subhead? Neat, short, concise even.

Of course, the article delves into much more detail.

The British philosopher Derek Parfit espoused a severely reductionist view of personal identity in his seminal book, Reasons and Persons: It does not exist, at least not in the way we usually consider it. We humans, Parfit argued, are not a consistent identity moving through time, but a chain of successive selves, each tangentially linked to, and yet distinct from, the previous and subsequent ones.

You know I find philosophy intriguing. But this? This sounds like a modern-day version of one of Zeno's Paradoxes.

Parfit’s view was controversial even among philosophers.

I'm sure it was. Not that it's definitely wrong, mind you; it's just that I can't think of a way to test it. Therefore, it stays in the realm of speculation—just another way of looking at things.

But psychologists are beginning to understand that it may accurately describe our attitudes towards our own decision-making: It turns out that we see our future selves as strangers.

Your future self is entirely theoretical. Your past self is not (except in the view of fringe philosophers). There's a big difference there.

That bright, shiny New Year’s resolution? If you feel perfectly justified in breaking it, it may be because it feels like it was a promise someone else made.

That's a non sequitur if I've ever seen one. In that case, you're breaking a promise your past self (aka you) made. Now, whether your past self has any business dictating stuff to your future self, well, that's another issue entirely.

Using fMRI, Hershfield and colleagues studied brain activity changes when people imagine their future and consider their present. They homed in on two areas of the brain called the medial prefrontal cortex and the rostral anterior cingulate cortex, which are more active when a subject thinks about himself than when he thinks of someone else. They found these same areas were more strongly activated when subjects thought of themselves today, than of themselves in the future. Their future self “felt” like somebody else. In fact, their neural activity when they described themselves in a decade was similar to that when they described Matt Damon or Natalie Portman. And subjects whose brain activity changed the most when they spoke about their future selves were the least likely to favor large long-term financial gains over small immediate ones.

I'd definitely keep a promise I made to Natalie Portman. Just saying.

The disconnect between our present and time-shifted selves has real implications for how we make decisions. We might choose to procrastinate, and let some other version of our self deal with problems or chores.

Well, absolutely. I don't want to deal with them. Let someone else do it. Sure, that someone else is future-me, but hey, I might get lucky and die before I have to deal with them.

Anne Wilson, a psychologist at Wilfrid Laurier University in Canada, has manipulated people’s perception of time by presenting participants with timelines scaled to make an upcoming event, such as a paper due date, seem either very close or far off. “Using a longer timeline makes people feel more connected to their future selves,” says Wilson. That, in turn, spurred students to finish their assignment earlier, saving their end-of-semester self the stress of banging it out at the last minute.

Wait, what? Some people don't get stuff done at the last minute?

Of course, the way we treat our future self is not necessarily negative: Since we think of our future self as someone else, our own decision making reflects how we treat other people.

"Other people?" You mean the game's NPCs?

So anyway, yeah, short article, interesting implications (if true). Eventually you might even get around to reading it. Not sure it has much to do directly with the Parfit philosophy from the first quote up there, but I suppose it fits that model.

Meanwhile, I'll just give myself a pat on the back for finishing writing this at 11:30 instead of 11:59.

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