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:

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 16, 2018 at 12:53am

#945662

Sure Thing

It's hard to be certain of almost anything.

I say "almost" because there is one thing we can all be absolutely certain of: "I exist." Not me, I, but every one of us I, each to their own mind.

Now, you can play games with that, saying that "I" is an illusion, but it's difficult to argue with Descartes: I'm pink; therefore, I'm spam. Er, I mean, I think; therefore, I am. It doesn't matter if we're meat suits, actors on a stage, avatars in a simulation, or disembodied brains trying out this thing called material existence; we simply are. Well, I am. I'm not so sure about the rest of you lot.

In life, we all like to make statements with certainty, and find things we feel we can be absolutely sure of. Apart from my own existence, though, I'd put a degree of certainty on everything. Like, I'm sitting here looking at a kitten. Does the kitten exist, or is she a figment of my imagination? The scratches on my arm and the fact that I see her tell me she probably exists. Like, 10^-50 uncertainty, and I'd call that as certain as I can get about anything. I'll call this the kitten level of certainty - basically, indisputable proof.

Now, I understand that you've never seen this kitten (you should; she's incredibly cute), so your level of certainty that this kitten exists is - and should be - much lower. You only have my word on it, and I could be deluded or lyingcreating a fictional story to support a point. Her name is Katushka, and she actually belongs to my housemate, if it can be said that a cat belongs to anyone at all.

Does Afghanistan exist? I once met a person who said he was from Afghanistan. I've met a few people who claim to have been there and returned. I've seen it outlined on a map. I've heard reporters talking about it. It's usually first on the list of countries on dropdown menus on the internet. But I've never been there, myself, so I'd only give it a 10^-10 probability of nonexistence. After all, the only thing a map proves (to kitten levels of certainty) is that the map itself exists.

On the other end of the spectrum, if you were to ask me if my mother were alive, I'd have to give that a near-zero probability - the polar opposite of kitten level certainty. I mean, I watched her die, there's a death certificate, I saw her body in the coffin, and I was there when they lowered it into the ground. But memory could be playing tricks on me.

So if anyone were to ask me if I'm certain of - I don't know - the existence of the sun, the best I'll be able to do is kitten level certainty - and that's during the day if I'm outside. At night, it becomes slightly less certain.

I've been burned too many times by saying I'm certain about some fact or other, only to find out later that I misremembered or was misled. So I've learned - much to the annoyance of non-pedants everywhere - never to be 100% certain of anything.

This might become a problem if I'm ever in court. "Mr. Waltz, are you absolutely certain you saw someone drive off in your car?" "Well, I'm 99.999999999999% sure, but I suppose it could have driven off on its own, or maybe someone slipped me some acid." "So, you're not 100% certain."

So when I see something like "Only 66% of 18-24 year olds in the US are certain the Earth is round," it doesn't surprise me much (assuming that's even an actual statistic and not something made up out of thin air). But it worries me. Not because it implies that 34% of 18-24 year olds aren't completely certain the world is round (that statement says nothing about the level of certainty of the other 34%; it could range from "I believe the Earth is flat" to "I'm almost completely sure the Earth is round") - but because 66% of them are so absolutely sure of themselves that they don't stop to think that absolute certainty is (almost) absolutely impossible. I've never personally seen the curvature of the Earth, but I have seen ships disappear over the horizon, and I've looked at plenty of evidence, and heard statements from scientists and astronauts - so I'm going to give this one Afghanistan levels of certainty. In addition, "round" has a squishy definition when it comes to a planet. Obviously the Earth isn't perfectly round; it has ocean trenches and mountains, and an equatorial bulge - plus, if you consider the atmosphere to be part of Earth, its boundary isn't clearly defined.

Consequently, if you asked me "Are you certain the Earth is round?" well... I might answer "no" just to be pedantic about it.

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