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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September 6, 2022 at 12:01am

#1037379

The Heat is On

It has always perturbed me that they call the illness a "cold" when it sometimes results in an elevated body temperature.

No, 98.6 Degrees Isn't The Normal Human Body Temperature

But that's really not the point of today's discussion.

98.6 degrees. That's what we grow up being told the temperature of the human body is—if you use Fahrenheit, anyway; that same temperature is 37 in Celsius. That’s a weirdly specific value.

Not only is it weirdly specific, but different parts of the body necessarily have a different temperature. Balls, for example, are generally cooler. But as far as I know, no one has proposed taking a guy's temperature by sticking a needle into his balls, and if they did, 99% of us would run screaming in terror.

The other 1% would go, "How much do I have to pay?"

98.6 degrees actually represents the average temperature of a whole bunch of people. One scientist named Carl Wunderlich came up with that number in 1851 by taking people's temperatures repeatedly. He studied 25,000 people many times each, taking around a million measurements, and 98.6 was the mean of his many readings.

And here we have another example of the hazards of assuming that the mean is representative of the range. It's kind of like saying "The average debt of an American citizen is $40k" (it's not actually that number, but that's irrelevant), so you assume every American has $40k in debt, when the reality is some have lots more debt, and others actually have money.

Your temperature also fluctuates quite a bit even over the course of each day.

Especially if you venture (shudder) outside.

Though Wunderlich had a lot of data, it wasn't very good data. He took temperatures of the armpit, which isn't the best spot compared to the mouth or anywhere more internal.

To be fair, if he'd wanted to study 25,000 people by sticking a thermometer where the sun don't shine, 24,750 of them would run screaming in terror, and the other 250 would ask that he please use a bigger thermometer.

Other scientists have tried calculating the mean human body temperature since Wunderlich did. Invariably, they come up with an answer lower than 98.6. The mean human body temperature is below 98 degrees. In fact, it seems that every time scientists calculate a new mean, they come up with a slightly lower value than a few decades before.

Oh no, we're turning cold-blooded as a species! Oh, wait...

That's because when you examine thousands of people, a few of them are bound to be a little sick and have higher than normal temperatures. The healthier that people in general get, the fewer of these feverish people slip into the sample, and the lower the mean temperature falls.

And also probably some of the people studied are secretly our alien reptilian overlords in disguise.

Incidentally, the Fahrenheit scale originally used human body temperature as a benchmark, along with a particular mixture of water, salt, and ammonium chloride for 0. Why that particular mixture? Whatever; it's all arbitrary anyway.

Clearly, the Celsius scale makes more sense; it's much easier to calibrate a thermometer using ice water at 0 and boiling water at 100. But even in places that use Celsius (e.g. pretty much everywhere that's not the US), they sometimes quote temperatures in F just for the shock value. "It's 100 degrees in London today!"

Today, though, the temperature scale is based on—get this, now—the Boltzmann constant, the Planck constant, and the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom.

So next time you're running a fever, ask a quantum physicist about it.

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