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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.
October 12, 2022 at 12:02am October 12, 2022 at 12:02am
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It goes without saying that science requires scientists. Today's article features one I hadn't heard of before.
A middle-aged Scottish mathematician rises ahead of the sun to spend a couple of hours with Newton before the day punctuates her thinking with the constant interruptions of mothering four children and managing a bustling household.
Meanwhile, I have trouble finding the time to read any science texts, and I don't have any such distractions (cats don't count).
“A man can always command his time under the plea of business,” Mary Somerville (December 26, 1780–November 28, 1872) would later write in her memoir; “a woman is not allowed any such excuse.”
Well, at least that's improved since then.
When her parents realized that the household candle supply had thinned because Mary had been staying up at night to read Euclid, they promptly confiscated her candles.
But not, apparently, Euclid.
Mary was undeterred. Having already committed the first six books of Euclid to memory, she spent her nights adventuring in mathematics in the bright private chamber of her mind.
"But girls can't do math." Pfft.
When Somerville was forty-six, she published her first scientific paper — a study of the magnetic properties of violet rays — which earned her praise from the inventor of the kaleidoscope, Sir David Brewster, as “the most extraordinary woman in Europe — a mathematician of the very first rank with all the gentleness of a woman.”
Well, at least he included the important part, that she was feminine.
As for the paper about violet rays, that's also something I'd never heard of. Learning two new things in one day nearly broke my brain. If you're in the same situation I was, here. 
Lord Brougham, the influential founder of the newly established Society for the Diffusion of Useful Knowledge — with which Thoreau would take issue thirty-some years later by making a case for “the diffusion of useful ignorance,” comprising “knowledge useful in a higher sense” — was so impressed that he asked Somerville to translate a mathematical treatise by Pierre-Simon Laplace, “the Newton of France.”
Thoreau was a willfully ignorant ass.
As the months unspooled into years, Somerville supported herself as a mathematics tutor to the children of the wealthy. One of her students was a little girl named Ada, daughter of the mathematically inclined baroness Annabella Milbanke and the only legitimate child of the sybarite poet Lord Byron — a little girl would would grow to be, thanks to Somerville’s introduction to Charles Babbage, the world’s first computer programmer.
Now Lovelace, I'd heard of.
In The Mechanism of the Heavens, published in 1831 after years of work, Somerville hadn’t merely translated the math, but had expanded upon it and made it comprehensible to lay readers, popularizing Laplace’s esoteric ideas.
For a scientist, it is often enough to just do science. Being able to make it comprehensible to people almost as ignorant as Thoreau, though, that takes real skill.
Years later, Edgeworth would write admiringly of Somerville that “while her head is up among the stars, her feet are firm upon the earth.”
One of Springsteen's first lyrics went, "My feet they finally took root in the earth / but I got me a nice little place in the stars." I wonder now if that echo was intentional.
Don't laugh; he's much better-read than you think.
In 1834, Somerville published her next major treatise, On the Connexion of the Physical Sciences — an elegant and erudite weaving together of the previously fragmented fields of astronomy, mathematics, physics, geology, and chemistry. It quickly became one of the scientific best sellers of the century and earned Somerville pathbreaking admission into the Royal Astronomical Society the following year, alongside the astronomer Caroline Herschel — the first women admitted as members of the venerable institution.
Before you go looking it up, yes, Caroline Herschel was related to famous astronomer William Herschel; she was his sister. For some reason (I wonder why) you never hear about her discoveries.
To be fair, William discovered an actual planet. I won't name it here because you'll make a pun out of it. But she did make significant astronomical discoveries, mostly nebulae and the like (this was before astronomers figured out that galaxies were galaxies and not nebulae).
But I digress.
When Maria Mitchell — America’s first professional female astronomer and the first woman employed by the U.S. government for a professional task — traveled to Europe to meet the Old World’s greatest scientific luminaries, her Quaker shyness could barely contain the thrill of meeting her great hero. She spent three afternoons with Somerville in Scotland and left feeling that “no one can make the acquaintance of this remarkable woman without increased admiration for her.” In her journal, Mitchell described Somerville as “small, very,” with bright blue eyes and strong features, looking twenty years younger than her seventy-seven years, her diminished hearing the only giveaway of her age. “Mrs. Somerville talks with all the readiness and clearness of a man, but with no other masculine characteristic,” Mitchell wrote. “She is very gentle and womanly… chatty and sociable, without the least pretence, or the least coldness.”
Even other women couldn't help commenting on woman stuff.
Now, here's the bit about how the term "scientist" came to be:
Months after the publication of Somerville’s Connexion, the English polymath William Whewell — then master of Trinity College, where Newton had once been a fellow, and previously pivotal in making Somerville’s Laplace book a requirement of the university’s higher mathematics curriculum — wrote a laudatory review of her work, in which he coined the word scientist to refer to her. The commonly used term up to that point — “man of science” — clearly couldn’t apply to a woman, nor to what Whewell considered “the peculiar illumination” of the female mind: the ability to synthesize ideas and connect seemingly disparate disciplines into a clear lens on reality. Because he couldn’t call her a physicist, a geologist, or a chemist — she had written with deep knowledge of all these disciplines and more — Whewell unified them all into scientist.
Thus presaging the feminist nomenclature of the late 20th century, such as appending "-person" to occupations that previously ended in "-man."
It could have been worse. It could have been "woman of science." I think if that had happened, progress in equality might have been stunted.
That bit about "the peculiar illumination of the female mind" might give you pause, but "peculiar" didn't have the negative connotations then that it does today. We use it to mean strange or weird, but then it was more like "special." But "special" itself was ruined when people stopped using other words to describe the mentally deficient.
Whewell saw the full dimension of Somerville’s singular genius as a connector and cross-pollinator of ideas across disciplines. “Everything is naturally related and interconnected,” Ada Lovelace would write a decade later. Maria Mitchell celebrated Somerville’s book as a masterwork containing “vast collections of facts in all branches of Physical Science, connected together by the delicate web of Mrs. Somerville’s own thought, showing an amount and variety of learning to be compared only to that of Humboldt.”
Our view of science, as our view of many things, tends to be compartmentalized. Biology is distinct from chemistry. Cosmology is a different study from astronomy. That sort of thing. In reality, things aren't so clear-cut. Biology is the result of chemistry, which itself relies on quantum physics; cosmology and astronomy are likewise intertwined. Our neat little packages turn out to have frayed, fractal edges.
If it took a woman to see that, to tear down the boundaries between disciplines, well, that in itself would be an argument for inclusivity, regardless of ideas about equality and gender roles.
Above all, Somerville possessed the defining mark of the great scientist and the great human being — the ability to hold one’s opinions with firm but unfisted fingers, remaining receptive to novel theories and willing to change one’s mind in light of new evidence.
And you don't have to be a scientist to live your life with that philosophy. It's the only reason I hold out any hope that things will improve: that people might change their opinions when presented with new evidence. Some don't, preferring to get stuck in their ruts.
As Wilde noted, "We're all in the gutter, but some of us are looking at the stars." |
© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.
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