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 6 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 8 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.

August 19, 2020 at 12:10am

#991137

Mind Map

Today's link is a bit esoteric, and I wouldn't blame anyone for giving it a miss. But I find this stuff fascinating, so I waded through it, and of course I have comments -- else I wouldn't bother posting it.

Skip to the bottom for Mini-Contest results!

http://nautil.us/blog/new-evidence-for-the-geometry-of-thought

New Evidence for the Strange Geometry of Thought

In 2014, the Swedish philosopher and cognitive scientist Peter Gärdenfors went to Krakow, Poland, for a conference on the mind.

Philosopher and cognitive scientist? That's a dangerous combination.

In his talk, “The Geometry of Thinking,” he suggested that humans are able to do things that today’s powerful computers can’t do—like learn language quickly and generalize from particulars with ease (to see, in other words, without much training, that lions and tigers are four-legged felines)—because we, unlike our computers, represent information in geometrical space.

So, if I'm reading this right, the evidence suggests that we store concepts the same way we store, for example, the layout of our homes.

He argued that the brain represents concepts in the same way that it represents space and your location, by using the same neural circuitry for the brain’s “inner GPS.”

You know, we were talking about inventions yesterday, and as I re-read this article to talk about it, I came to the conclusion that there are, in general, two broad categories of inventions: those that can be compared to other things, and those to which we compare other things. To use yesterday's example, a wheel would be one of the latter. So, apparently, is GPS -- a truly world-changing invention, though one that's built on a multitude of earlier inventions.

The hippocampus’ place and grid cells, in other words, map not only physical space but conceptual space. It appears that our representation of objects and concepts is very tightly linked with our representation of space.

This makes all kinds of sense when you think about it (not that "making sense" means it's necessarily right). At the risk of delving into the kind of evolutionary just-so stories that I despise, our remote pre-hominid ancestors, lacking the concept of "concept," would have known nothing but their environments. It shouldn't be surprising, then, that our brains use the same sorts of brain configurations, adapting them for other purposes.

But again, I don't know; that's pure speculation on my part.

The article goes into some of the studies backing up the mapping idea.

Yet the mind is not just capable of conceptual abstraction but also flexibility—it can represent a wide range of concepts. To be able to do this, the regions of the brain involved need to be able to switch between concepts without any informational cross-contamination: It wouldn’t be ideal if our concept for bird, for example, were affected by our concept for car.

It will be when I finally get the flying car I've been promised.

Scientists still need to experimentally verify the link between the hippocampus and higher-order cognitive functions in humans. fMRI studies like the ones from the group in Oxford are, as yet, only suggestive.

I'm just including this here for anyone who's still with me. This isn't settled science; just an interesting idea with some evidence for it.

One reason the concept is intriguing is the potential impact on AI development:

Gärdenfors’ theory highlights a fruitful path, not only for cognitive scientists, but for neurologists and machine-learning researchers. It is a kind of incomplete, generic sketch on a canvas that invites refinement and elaboration. Cognitive spaces are, as Gärdenfors and Bellmund put it, a “domain-general format for human thinking,” an “overarching framework” that can help unravel the causes of neurodegenerative diseases, like Alzheimer’s, and “to inform novel architectures in artificial intelligence.”

Oh, yeah, and also to help us deal with brain dysfunction. Maybe we could even figure out how to get more people to appreciate science.

Mini-Contest results!

All great responses yesterday! Though I have to admit I'm just a little disappointed that no one said "sliced bread." Let's see, we have:

The Human Genome Project -- while I wouldn't consider that an invention, it's certainly an innovation, and a major one at that. As Sumojo

noted, it has improved human life to a vast extent.

Sterilization -- Lazy Writer est 4/24/2008

pointed out that the idea that implements and people should be, you know, clean, that's a big deal. Saved a lot of lives. I don't know why they didn't think of it sooner, to be honest; it shouldn't require knowledge of microorganisms to know that dirty things make people sick.

Writing, as per Zhen

-- I think I did a blog entry a while back on that subject. Or maybe a newsletter? If not, I'll have to do one. This one's foundational, the almost-hidden basis for a multitude of other inventions.

Printing Press -- Paul

noted that this extended the above to a wider audience.

Vaccinations (and beer!) -- NaNoNette

, I'm a big fan of both (though if I had to choose, it would be beer). Both have saved lives. No, I'm not joking -- as with the sterilization one above, the process of making beer required that water be boiled, which had the unintended (pre-germ theory) effect of destroying harmful germs. Beer, for much of human history, was often safer than water to drink.

And Pumpkin Spice Sox

mostly reiterated other comments, but added "autocorrect," which I'd argue is a mixed blessing, but it's certainly made it easier for people. And yes, beer.

Now, given that it's me here, I'm really tempted to go with beer (and I will, in fact, go have a beer when I'm done posting this). There are suggestions out there that civilization is the result of people doing what needed to be done to make beer. Of course, whether "civilization" is, on balance, better or worse is arguable. (I'm arguing "better" because I like not having to hunt my dinner.) But in terms of Most Important Invention or Innovation, I'mma have to agree with writing. After all, without it, what would I do when I'm not drinking beer?

Oh, right, binge-watching Star Trek. But wait -- that show had scripts. No writing, no scripts, no Star Trek, and no blog entry on this or any other day -- a bleak alternate universe indeed. Maybe even just as bleak as one without beer. So this time the Merit Badge goes to Zhen . I'll send you the MB before I start drinking, promise!

Everyone else, thanks! Great comments, and there will be another opportunity in a few days.

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