Now I Know my ABCs

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sasha
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Now I Know my ABCs

Post by sasha » December 24th, 2021, 8:51 am

I think there's a broad misconception that mathematics is about numbers - adding them, subtracting them, multiplying and dividing... Isn't it just about numbers begetting other numbers?

Nope. That's just arithmetic. Arithmetic is often found at the tail end of mathematics, as an epilogue or addendum, or as a bit player in some riddle involving geometry or trig. It's what spelling is to writing. That's because numbers are too specific - useful for balancing your checkbook or calculating your taxes (the latter of dubious benefit at best). Mathematics is more about generality - discovering deeper rules governing the behavior of those prosaic numbers, and of finding non-obvious relationships between them. Mathematicians will apply general rules of logic using tokens standing in for actual numbers to see what happens. For example, everyone knows that 1+1=2, and that 2+2=4. To a mathematician, these are just specific instances of the general rule that "a + a = 2a" - that ANY number added to itself equals twice the number. (Here the multiplication by 2 is implied, though it’s sometimes explicitly stated with an “x”, a dot, or an asterisk.)

As with any other tribal group, certain cultural traditions and customs have evolved among the various clans within mathematics. While any letter can become a stand-in for any number, some have come to be used more in certain contexts than in others. Geometers seem especially fond of representing the side lengths of polygons with a, b, c, d, etc. "Consider a triangle with sides a, b, c..." Or, if their dissertation also involves the angles within the points of the triangle, they might instead use upper case A, B, and C for the sides, and lower case for the angles. Things get a little tricky when dealing with shapes of more than four sides, because "e" is the retired jersey of Euler's number 2.718281828459... , an important universal constant found everywhere from compound interest to the bell-curve distribution in probability and statistics. Avoid Its use elsewhere if possible, proceed with caution if not.

Next up are f, g, and h. While geometers and engineers use these to label spatial distances, and physicists do their own thing with them, numerical analysts like using them to represent functions - series of arithmetical operations converting one number into another. A function is like a machine you dump a number into, which then spits out another number. Instead of repeatedly writing "Add 3 to some number ‘a’ and multiply what you get by ‘a’ itself, then subtract 7", he might just define that string of calculations as "f(a)" (pronounced "f of a") and use that whenever he wants to perform those operations. It's useful shorthand, and can sometimes reveal unsuspected relationships.

On to i through n. These letters are first in line when one wants to emphasize that we are dealing with whole numbers, where fractional parts need not apply, such as in counting elements of an open-ended series of similar objects. The venerable Fortran computer language even reserved these characters for the integers. While it was easy enough to override that default assignment, it was tacitly assumed that such behavior was not only in questionable taste, but quite possibly heretical as well. A notable exception is the use of "i" to represent the square root of -1 (though electrical engineers perversely prefer the letter "j"). Without getting into it, this imaginary quantity is incredibly useful in formulating an alternative representation of vectors and their associated operations.

Poor "o". Maybe because it looks too much like a zero, the only mathematical use of this rotund little glyph I'm aware of is in information theory, where upper case "O" is used as a measure of a problem's tractability, and the number of arithmetic operations required to solve it. I don't know enough about that field to say much more about it.

Number theorists seem to like using p and q as integers, particularly when dealing with divisibility of numbers. They figure prominently in the proof that the square root of 2 cannot be exactly calculated as the ratio of any whole numbers. Practitioners of this arcane art seem to prefer distancing themselves from practical applications, and are not above disregarding conventions established by their more plodding colleagues in engineering.

Physicists have a fondness for r, s, and t, to which they routinely assign radii of circles, linear distance, and elapsed time, respectively. The letters u, v, and occasionally w, are often used as functions (remember them?) in the intermediate steps of problems involving calculus.

Most everyone is familiar with x, y, and z as iconic symbols of Unknown Quantities. "Let x+y=5, and x-y=1." These last three letters of the Roman alphabet have become pop stars, frequently seen scrawled on blackboards in movies about scientists or mathematicians. On rare occasions, those cinematic background scribblings have actual meaning.

But why stop there? As Dr. Seuss wrote, "On Beyond Zebra!" The Greek alphabet consists of a few dozen wonderful symbols that have come to acquire particular meanings. As a prefix, delta almost always means degree of change in some quantity – capital Δ for a discrete amount, lower case δ for an infinitesimal amount. In statistics, epsilon (ε) figures prominently as a residual error quantity, while mu and sigma (μ, σ) often stand for average and standard deviation. Alpha, beta, and gamma (α, β, γ) frequently denote given or known angles, while phi and theta (φ, θ) commonly represent unknown or generalized angles. And who doesn't know what π stands for? Physicists use lambda (λ) to indicate wavelength, rho (ρ) for density, and omega (ω) for angular velocity; engineers use capital omega (Ω) for electrical resistance. Capital psi (Ψ) appears a lot in quantum mechanics. I'm personally fond of using β for dimensionless ratios, and Δ for the determinant of a matrix. I’d use zeta and xi more (ζ, ξ), but they're too hard to draw.

As with the Roman "o", the Greek omicron (ο) doesn't get much respect - except lately among epidemiologists. And I sincerely hope that its notoriety is fleeting.

Just as an aside, irrelevant but for the time of the year, the "X" is Xmas is actually the Greek capital chi, the first letter of "Christ" as written in that language: Χρίστοσ

And there we have the ABCs of mathematics. I haven't seen Cyrillic characters used much, but maybe that's because typesetting sorts for that family of languages aren’t common in the West. Still, I think it'd be really boss to send my physics bro Jim a derivation with the Russian Ю (“-yu”) symbolizing the mass of a trebuchet's counterweight, or Ж (“-zh”) for the sag of a long plank supported by two sawhorses...

But since the Equation Editor only supports Roman & Greek, I guess Cyrillic – or Arabic, or Hindi, or Cantonese – will have to wait.

*Sigh”
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"Falsehood flies, the Truth comes limping after it." - Jonathan Swift, ca. 1710

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stilltrucking
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Re: Now I Know my ABCs

Post by stilltrucking » December 24th, 2021, 2:07 pm

I am always on the look out for "non-obvious relationships between"... time and money.:wink:
interesting read 8)

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sasha
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Re: Now I Know my ABCs

Post by sasha » December 24th, 2021, 5:27 pm

Thanks for taking the time to read & comment, much appreciated. Curiously, right after I uploaded this I watched the following YT video on one of my favorite channels! Seems my description of the O(h) function was a bit off, including its name - should be the O(micron) function. Likewise my understanding of its significance was a bit off. My mistakes.

https://www.youtube.com/watch?v=JXtfKMH ... umberphile
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"Falsehood flies, the Truth comes limping after it." - Jonathan Swift, ca. 1710

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stilltrucking
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Joined: October 24th, 2004, 12:29 pm
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Re: Now I Know my ABCs

Post by stilltrucking » December 24th, 2021, 10:39 pm

It's all Greek to me :wink:
Have you ever had a teacher who loved to teach the subject he was teaching. A professor McCall at JHU I loved that guy.

I took a course in Classic Greek literature from him. When he would struggle with a translation that would not quite fit, he would sigh in mock despar and say, "What an enemy we have in Greek."
ACTG
I am a number and letter freak
I am forever a stranger to myself
a bundle of vague sensory perceptions

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