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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November 29, 2022 at 12:01am

#1041125

Freezing Hot

Science is hard.

Controversy Continues Over Whether Hot Water Freezes Faster Than Cold
Decades after a Tanzanian teenager initiated study of the “Mpemba effect,” the effort to confirm or refute it is leading physicists toward new theories about how substances relax to equilibrium.

Not all counterintuitive results involve quantum effects.

It sounds like one of the easiest experiments possible: Take two cups of water, one hot, one cold. Place both in a freezer and note which one freezes first. Common sense suggests that the colder water will.

And this is one reason I don't trust common sense.

But luminaries including Aristotle, Rene Descartes and Sir Francis Bacon have all observed that hot water may actually cool more quickly.

To be fair, Aristotle was wrong about a lot of things; those other dudes were right about a lot of things, but they were still subject to human biases and observational inconsistencies. This is why we do science: to counteract the effects of human bias, error, and "common sense."

The modern term for hot water freezing faster than cold water is the Mpemba effect, named after Erasto Mpemba, a Tanzanian teenager who, along with the physicist Denis Osborne, conducted the first systematic, scientific studies of it in the 1960s.

Probably you've seen those videos of idiots living in the snow who throw boiling water from its pot and it freezes solid before it hits the ground. This is not the same thing; that happens, they're pretty sure, at least partly because hot water evaporates quickly, and because throwing it increases the surface area, which means it can evaporate even more quickly.

They're not idiots for doing this; they're idiots for living where the world is trying to turn them into corpsicles.

No, this is talking about hunks of water under conditions identical except for the temperature. Even then, as it turns out, it's not always observed.

Over the past few years, as the controversy continues about whether the Mpemba effect occurs in water, the phenomenon has been spotted in other substances — crystalline polymers, icelike solids called clathrate hydrates, and manganite minerals cooling in a magnetic field.

All of which may have profound technological implications, but we want to know about water. You know, that stuff we literally can't live without.

“A glass of water stuck in a freezer seems simple,” said John Bechhoefer, a physicist at Simon Fraser University in Canada whose recent experiments are the most solid observations of the Mpemba effect to date. “But it’s actually not so simple once you start thinking about it.”

One cool thing (pun absolutely intended) about this article is how it highlights the international, multicultural process of science.

“My name is Erasto B. Mpemba, and I am going to tell you about my discovery, which was due to misusing a refrigerator.” Thus begins a 1969 paper in the journal Physics Education in which Mpemba described an incident at Magamba Secondary School in Tanzania when he and his classmates were making ice cream.

Okay, see, right there I'm already running into a contradiction. Ice cream has a high water content, sure, but it's not water. Most water isn't water, either; it contains various minerals. Even distilled water is rarely 100% pure, and even if it were, doing experiments with it doesn't necessarily translate to real-world applications.

Basically, I suspect that at least part of the problem with replicating the results here is that different trace elements in a sample of water can cause it to behave differently in certain circumstances, like when you're doing freezing experiments.

I don't know that for sure, though. I'm sure the scientists involved have already thought of that, but if so, the article doesn't say much about it.

Space was limited in the students’ refrigerator, and in the rush to nab the last available ice tray, Mpemba opted to skip waiting for his boiled-milk-and-sugar concoction to cool to room temperature like the other students had done. An hour and a half later, his mixture had frozen into ice cream, whereas those of his more patient classmates remained a thick liquid slurry. When Mpemba asked his physics teacher why this occurred, he was told, “You were confused. That cannot happen.”

Some teachers don't have open minds. Many kids do, at least until teachers close them.

Over the decades, scientists have offered a wide variety of theoretical explanations to explain the Mpemba effect. Water is a strange substance, less dense when solid than liquid, and with solid and liquid phases that can coexist at the same temperature.

And that's another problem: even if it happened with consistency, you have to come up with a reason why it happens, and then test that.

Still, Burridge and Linden’s findings highlight a key reason why the Mpemba effect, real or not, might be so hard to pin down: Temperature varies throughout a cup of rapidly cooling water because the water is out of equilibrium, and physicists understand very little about out-of-equilibrium systems.

Stacking on yet another problem.

Statistical physicist Marija Vucelja of the University of Virginia started wondering how common the phenomenon might be. “Is this like is a needle in a haystack, or could it be useful for optimal heating or cooling protocols?” she asked.

Just leaving this bit here because I like it when my alma mater is involved.

If nothing else, the theoretical and experimental work on the Mpemba effect has started giving physicists a handhold into nonequilibrium systems that they otherwise lack.

Another fun thing about science: sometimes a result, or even a failure, can have secondary benefits.

Fortunately for lazy-ass me, the article ends with a brief description of whatever happened to Mpemba himself:

After igniting a decades-long controversy with his teenage interrogations, Mpemba himself went on to study wildlife management, becoming a principal game officer in Tanzania’s Ministry of Natural Resources and Tourism before retiring.

So he didn't end up working on pure science, and the effect named after him might never had any impact on his life. Remember that next time some kid complains that they'll "never use" whatever they're studying at the moment; it's still important.

Osborne, discussing the results of their investigations together, took a lesson from the initial skepticism and dismissal that the schoolboy’s counterintuitive claim had faced: “It points to the danger of an authoritarian physics.”

And also to the danger of a stubborn "common sense."

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