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!

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').

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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.

October 15, 2019 at 12:12am

#967880

Chaos

https://www.wired.com/story/uncertainty-isnt-always-problemit-can-be-solution/

Uncertainty Isn't Always a Problem—It Can Be the Solution

Many areas of technology—from pacemakers to space missions—deliberately create controlled amounts of uncertainty to make devices and processes work better.

We D&D players have known about the joys of uncertainty since the mid-70s.

In 1978 my wife and I, with our two sons, then aged 2 and 4, were traveling home after a year in Connecticut...

Writing 102: Start your article with a personal anecdote. Just... make it an interesting one. I skipped most of the first paragraph.

Life is a lottery. Uncertainty often breeds doubt, and doubt makes us feel uncomfortable, so we want to reduce, or better still eliminate, uncertainty.

Nah, bring it on. It's nice to be able to predict, say, eclipses, but wouldn't it be cool if they happened at random intervals? Okay, maybe not "cool" because that would eliminate the entire foundations of orbital mechanics. But interesting.

Uncertainty isn’t just a sign of human ignorance; it’s what the world is made of.

I've been saying this for years. Nothing is 100% certain. Once you embrace that, everything becomes easier.

Random numbers—more precisely, the pseudorandom numbers that computers can be told to generate—can be really useful.

Especially if you've managed to fight your way to the bottom of the dungeon and are now facing the final boss.

Biologists make a big song and dance about evolution being inherently random.

I follow biology blogs. My housemate is a biologist. I have never heard a biologist claim that evolution is inherently random. Perhaps some do; I can't follow everything. There are random, or at least semi-random, inputs, sure, but evolution is a deterministic process in the sense of reactions to semi-random events.

So the organisms keep improving, step by tiny step. That way, evolution simultaneously constructs the peaks of the fitness landscape, finds out where they are, and populates them with organisms. Evolution is a stochastic hill-climbing algorithm, implemented in wetware.

This is what happens when you step out of your wheelhouse. Organisms don't "keep improving;" they adapt. Sometimes they don't adapt fast enough and a species dies out. That can be seen as an improvement from a distance, but to the species involved, it's kind of the opposite.

Randomness comes in many forms, and chaos theory tells us that a butterfly flap can radically change the weather.

The "butterfly effect" is famous, and it's a decent way to get thinking about chaos theory, but the general understanding of it is wrong. What I mean is, not every butterfly ends up causing a hurricane, and even the flaps of a butterfly's wings are deterministic - just not predictable.

Snark and criticism aside - and keeping in mind that I'm hardly infallible on these points, either - it's a good article with interesting points about how we're figuring stuff out. I'm just not motivated to buy the guy's book.

2 Like • Comment

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