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Math Parables 2: Welcome to the World of Hyper-Sur-Reality

In my last article, “Can Mathematics be Parables?” I considered the fantastical realm of “imaginary” numbers. Now, wander with me across a terrain of numbers even more dazzlingly head-spinning . . . and even more hazardous, perhaps, to encounter.

I’m sure you’ll remember that there is a whole category of numbers that is (gasp) “irrational”? I think that to be irrational in our current culture may be even worse than having too much imagination. Here in the 21st-century West, an educated person must be centered in factual objectivity (no fake facts allowed, no irrationality tolerated – never mind that we break this rule all the time through our relativistic idolatries).

To be irrational is to be positively outside the realm of what is reasonable. To be reasonable is what we consider “sane.” And if I’m not mistaken, what is outside the realm of sanity is crazy.

Yet, here we have a mathematical universe populated by “irrationals”: apparently nutty mathematical entities who know no limits. They just refuse to be pinned down. And the stuff they can do is near miraculous.

Think of the famous “golden ratio,” also known as φ (phi), an irrational number that permeates the creation like a charm from a cosmic genie, blessing all it touches with not only beauty but seemingly generative powers (picture an Abracadabra from Cinderella’s fairy godmother: Suddenly you have not only a gorgeous gown and a great hairdo, but pumpkins grow into coaches and mice into footmen).

Or, consider the ubiquitous number, at least in our experience with math education, π (pi). Who has managed to get through a basic education without meeting π? This fantastic entity is practically man’s best friend in the mathematical universe when it comes to computing anything that has to do with circles.

Now, take an even more amazing leap with me and consider the concept of ∞, which is the symbol indicating “infinite processes.” It’s used in extremely ingenious ways in scientific calculations. This brilliant maneuver (one could almost call it magical) simply uses the concept of infinity as a mathematical creature that can be included in equations! Some of these are extraordinarily important equations. Consider, for instance, the Gaussian Integral, which represents the normal distribution in statistics:

See those infinity symbols? This elegant notation with the swooping line (which looks strikingly like a magic wand, if you ask me), flanked by infinity symbols, reflects the entire scope of possibility from “negative infinity” to infinity! Wrap your head around that concept!

By the way, this equation also includes the base of the natural logarithms, “e”, another alluring creature in the wild and woolly world of math that we don’t have time to even scratch the surface of here; incidentally, it’s worth noting that it’s considered to be a fine example of mathematical beauty, so “e” isn’t just any old nymph, she’s an exceptionally gorgeous one. Being practically omnipresent is another of her characteristics. She’s one of the most important numbers in mathematics and she shows up everywhere that things grow or decay (which is pretty much in everything, right?), like population and economics.

Isn’t it remarkable that in our fact-based culture, obsessed with statistics and finances, we discover these amazing creatures whom we are hard pressed to categorically define? Incidentally, π and e are also two of the most famous so-called “transcendental” numbers, and transcendental in normal parlance generally means related to the nonphysical or spiritual realm. Shouldn’t it give us a good case of the heebie-jeebies to think of these powerful entities permeating the supposed concrete substrata of our society?

Now get this: Infinity is part of a system called “hyperreal” numbers. If that word “hyperreal” doesn’t strike you as eerily evocative of “supernatural” – and therefore above and probably barely accessible to our human comprehension – I don’t know what would.

Finally, ponder with me the fact that in mathematics we lump the “hyperreal” numbers with the “real” numbers to get . . . wait for it . . . Ta Da: The “surreal” number system! Anyone with language literacy can see that number literacy involves just as much mystery, transcendent conceptualization, and as many supernatural characters as any good parable or story worth its salt.

I invite you to join me next time to hike deeper into the realm of math as parable in The Fable of the Fearsome √2, as respectable and renowned an irrational number as one could possibly care to meet.



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