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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.

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October 15, 2019 at 12:12am
October 15, 2019 at 12:12am

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.

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