Friday, June 5, 2015

Number Sense – or, Is It Nonsense?

An over-arching sense of Order pervades Numbers and the arithmetic manipulation thereof. Whether just adding or subtracting, determining probability and statistics, or graphing quadratic equations in three-dimensional space – however simple or complex the task, however trivial or crucial the context – that which sits to the left of the “equal sign” must necessarily possess the same numerical value as that which sits to the right.

Approximation Not Allowed.

Please understand that this strict sense of balance was not conjured up by some old-fart Greek or Egyptian thousands of years ago. Numbers themselves probably began to evolve when some Troglodyte happened to notice that some things are, ya know, more plentiful than others. Critters large and small, who instinctively – or do they count? – recognize when they are out-numbered and flee, must have experienced a similar epiphany.

No, this orderliness does not exist by mandate – rather, it simply exists … is a component of the natural world, just like the survival instinct of the beast or the complex architectural physics that “just happened” in the formation of our many natural wonders. You don’t have to take somebody’s word for it – just open your eyes.

Numbers present the illusion of random chaos, while in actuality providing a system of pattern, progression and proportion. Packaged therein you’ll find prime numbers, perfect squares, powers of two, sooner or later even a dude named Pythagoras.

Prime numbers (whole numbers whose only factors are “1” and the number itself), all but one of which are odd, seem totally random. In actuality, though, the primes are the remnants of a very meticulous process by which “composite” numbers (i.e. those that are not prime but rather the “product” of two other whole numbers) reveal themselves. Start with a prime number like “2” – every second number that follows (i.e. all the rest of the even numbers) is automatically composite since “2” times a number other than “1” will yield that result. Starting from “3”, every third number becomes composite; every other landing spot (6, 12, 18, 24, etc.), being an even number, would already have been removed from “prime” consideration. Repeating the process for “5” and “7” will expose all the primes less than 121. (Can you figure out the significance of “121” here?)

With perfect squares (the result obtained when multiplying a whole number by itself), the pattern is more direct and apparent. The square of every number turns out to be the sum of the first “that many” odd numbers. 1 + 3 + 5 = 32 = 9. Just add seven to get the square of “4”. Works every time, though I’m really not sure why.

And the powers of two are on display any time the sports world, from the babies to the pros, utilizes a single-elimination bracket.

From “cancelling” common factors in a fraction to “Casting Out Nines” to the FOIL method, the patterns and associations inherent in our number system appear to be countless.

That might seem ironic, but the myriad combinations and permutations of numbers provide fodder aplenty for a mischievous ex-schoolteacher.

Say, for instance, the young store cashier rings … I mean, scans up my purchase at $30.42. I’m already aware that, with sales tax, the financial damage will be a little north of $30. (If the line’s long enough and distractions few enough, my mental calculation will be within a few cents.)

Other than a couple of single dollars and a little small change, only “twenties” remained in my pocket from the last bank run. Now, I’m determined to (a) get back a “ten” or two “fives” and (b) avoid further cluttering my pocket with more coins.

So, I hand over two “twenties,” a single, a dime, a nickel and two pennies. In return (along with a confused glance that might need reassurance), I get $10.75 …

… and sometimes a convert to the NBA – Number Believers Anonymous.