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!

a day 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

a day 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 6, 2019 at 12:02am

#963785

Prism Sentence

Today, we're going to discuss something that isn't.

https://www.theatlantic.com/science/archive/2018/02/what-color-tennis-ball-green...

What Color Is a Tennis Ball?
An investigation into a surprisingly divisive question

We think of color as an intrinsic quality of a thing. It is not.

I was stunned. I’d gone from being so sure of myself to second-guessing my sanity in a matter of seconds. More than that, I could never have imagined the question of the color of a tennis ball—surely something we could all agree on, even in these times—would be so divisive.

Well, someone hasn't been paying attention to politics. Probably best for them.

An investigation was in order. It began with the most obvious first stop, Wikipedia. “Tennis balls are fluorescent yellow at major sporting events, but in recreational play can be virtually any color,” the page for tennis balls said. And it’s true. A quick spin through Amazon shows that tennis balls, at least according to the labels, come in yellow and green and purple and even pink with Hello Kitty’s face on them.

TOO FAR.

“I make this decision as much on the basis of what I think I know about tennis balls—that they are yellow—as I do on what color I recall that they looked when I last saw one,” he said. “In other words, like the color of a lot of objects, how we label [a tennis ball] is determined both by perceptual and cognitive factors: the actual physical light entering your eye and ... knowledge about what people have typically labeled the objects.”

When I was a kid, I remember having a deep, philosophical discussion with a friend of mine (we did that sort of thing), and I pointed out that I had no way of knowing whether the color I saw and labeled as "red" (or any other color) was the same as what he saw and labeled as "red." Like, if I could see through his eyes with his brain, maybe I'd see blue instead, or chartreuse, or black. This blew his mind. Much later, this argument was presented on the internet as something earth-shattering. I really should patent these ideas.

Conway pointed to bananas as an example of this phenomenon. Bananas can be a number of colors—green when they’re not yet ripe, brown when we’ve let them sit out for too long. But we label bananas yellow, and we do it because “this is their state when we care about them,” Conway said. “It is canonical among most people to call bananas yellow.”

Yeah, okay, but when you look at an individual banana, you can assign it a color. Green, yellow, or (in the case of the ones attracting fruit flies on my counter right now), brown.

In other words, humans are good at pointing at a yellow paint chip in a line of colorful chips and saying, that’s yellow. But if we’re shown a yellow paint chip alone and asked what color it is, we become less certain about calling it yellow.

Like many cities, we have a "Yellow Cab" taxi service here. Some of the cabs are yellow. Most are orange. At least one of them has a silver paint job. This is annoying, so I use Uber.

Conway took it a step further, suggesting that the way people see tennis balls could reveal something about their lifestyles. Night owls, for example, spend most of their time under artificial, warm light, which means they’d discount warm colors and see a tennis ball as green.

Well, that explains why my initial reaction to the headline was, "Green. Duh."

So now I'll go back to my original assertion: Color is not an intrinsic quality of a thing.

No, this doesn't mean it's an "illusion." I'm sick of people labeling things that are perceived differently by different people as an "illusion." Like, "Time is an illusion." No it fucking isn't. Yes, we each perceive it differently - mostly this is psychological and/or age-related; relativistic effects are really minor for us earth-bound clowns. But they're real. So is time. So, no, color isn't an illusion, but it does depend on certain quantum effects as well as one's perception.

I mean, we can talk about "the green grass," and I think most people associate green with grass. The only exceptions I know of are RG colorblind (duochromatic, technically), but even they have been inundated with social expectations like "the grass is green."

So imagine my horror, as a kid, when I heard about Kentucky bluegrass. No, not the music; the actual grass you can put on your lawn. I finally tracked some down at one point and scratched my head. Looked green to me.

I can't be arsed to get into the technicalities of it, but grass is green because that's the only color reflected from it when enough light of the right spectrum (e.g. the harsh glare of the accursed daystar) shines upon it. This has to do with electron orbits and quantized energy; like I said, not getting into it.

Point is, shine light of a different spectrum on grass, and it could appear to be another color. Or even if it's moonlight, which is highly attenuated sunlight reflecting off dark gray rock, and grass becomes gray or something darker.

And yes, the moon is mostly colored dark gray. That brings me to the other point about color, which is that it looks different depending on what other colors are around it. This is even less an intrinsic property of the thing, and more a matter of human nerve cells. Take a look at these, for example:

http://mentalfloss.com/article/54448/5-color-illusions-and-why-they-work

There are many other examples; a quick Google search will root them out.

So, yes, color is somewhat subjective. And arguments about it are idiotic. Unless something is colored Hello Kitty pink, in which case, AAAAAAH MY EYESSSSSS.

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