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 3 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 5 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.

September 2, 2020 at 12:01am

#992094

Mission: Difficult

Sometimes I probably take things too literally.

PROMPT September 2nd

Be inspired by this quote: "Man often becomes what he believes himself to be. If I keep on saying to myself that I cannot do a certain thing, it is possible that I may end by really becoming incapable of doing it. On the contrary, if I have the belief that I can do it, I shall surely acquire the capacity to do it even if I may not have it at the beginning." - Mahatma Gandhi

I assert that I cannot fly without special equipment. Granted that I could fall, and possibly have some amount of control over the fall, but I could not take off, fly, and land where I desire like, say, Superman.

Neither can anyone else. Not even Gandhi. No matter how much you think you can, no matter how motivated you are, no matter what you convince yourself of, there's these persnickety things called the laws of physics that absolutely prevent it. I have seen exactly zero evidence to contradict me on this, and I'm certain that people have tried. I know I have.

I also assert that I cannot physically live forever, or tunnel through a mountain without tools, or move a thing with telekinesis. No amount of belief, no deep well of self-confidence, no pleading with the universe to make it so, can change this.

So on the face of it, Gandhi was, in this instance, absolutely wrong.

"Oh, but Waltz, that just means that no one has tried hard enough, or that we're so convinced of natural laws that we limit ourselves accordingly. We say to ourselves, these things are impossible, and so they are impossible."

Nope. They're just physically impossible.

Now, we can get around these things, depending on what your actual objective is. For a long time, powered flight was considered impossible, until it wasn't. If the purpose of tunneling through a mountain is to lay train tracks, we've done that -- with tools. If I need to move something, and it's light enough or I have the right equipment, I can move it - no need for telekinesis. As for living forever, well, I covered that in my last two newsletters; no need to belabor it. The point being that while no, I can't personally fly like Superman, I can get on a jet and go to Belgium. Well. I could if there weren't a fucking pandemic going on.

Having covered that we need to accept that there are limitations, though, I will concede that, if a thing is possible at all, belief in oneself is a good beginning to get it done. So many people are bad at math, for example. "I'm so bad at math! I could never understand physics." That's self-limitation. They have a mental block, convinced that there's something they can never understand, and so it is. But if you approach it going, "My teachers sucked. I know I can learn math. I'll get a book and watch videos," then maybe, just maybe, you can learn something new.

Neither of these observations -- that the possible is possible and the impossible is impossible -- is particularly new or meaningful. What interests me is not the possible made manifest by belief in oneself, or the impossible that can never happen, but the cognitive space in between the two.

One of the marvelous things about humanity is that we can conceive of the impossible, or even the improbable. Because I'm still rewatching old episodes of Star Trek, I'll use warp drive as an example. The speed of light is a known limitation on acceleration: nothing -- no matter, no energy, no information of any kind -- can accelerate in space past the speed of light.

Humans, who despise limitations of any sort (except, apparently, when they impose them on other humans), upon learning of this, immediately started imagining ways around it. One such imagined method is the aforementioned warp drive. Another is wormholes - folding space between two distant points like you're folding a piece of paper to bring the opposite edges together. One guy even managed to come up with a... possibly... plausible workaround: The Alcubierre Drive.

. All we'd apparently need is to be able to create negative mass and effectively limitless energy, which... well, we don't know if that's impossible or not. I'm certain we'll try, if we aren't already. Because that's what humans do.

All of this shows is that while we certainly have physical limitations, it is possible to imagine the impossible. And, as a result of imagining it, sometimes we can even do it. In fact, I'd argue that without being able to imagine things beyond the realm of possibility, we certainly do limit ourselves. As Commander Swanbeck (Anthony Hopkins) in Mission:Impossible II said to Ethan Hunt (Tom Cruise): "Mr. Hunt, this isn't mission difficult, it's mission impossible. "Difficult" should be a walk in the park for you."

We can't all be Tom Cruise. At least, not without that franchise's mask-creation technology. Let's get to work on that.

Now, look, I've been doing Merit Badge Mini-Contests about once a week. I can't usually do it more often than that because, well, 10,000 gps is 10,000 gps, and I'd eventually run out of funds. But this week, we get a free Merit Badge to give away every day for Birthday Week. I can't think of any better use for such a boon than to use it to bribereward people for commenting in my blog. So go ahead and comment below; I'll pick one to receive a badge.

I'm also going to give today's free one to one of yesterday's commenters... let's see... I liked LostGhost: Seeking & Learning

's comment, and I'm fairly sure she hasn't commented in here in a while, if at all, so she gets today's.

5 Like • Comment • View 15 comments

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