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THE WORD “RATIONAL” is wondrous. One meaning is “having reason or understanding, balanced.” Another, in mathematics, describes the ratio of two numbers, a fraction. In my continuing fractional fun, here is a two-part exploration offering a glimpse at how mathematics leads to certainty, not simply rules of thumb. We’ll seek patterns of rationality, make conjectures about the patterns and then see how our conjectures can be firmed up with cold, solid logic. We might even cite some practical use for this rationality as well.
Today, we’re in pattern-seeking mode. Tomorrow, we’ll make conjectures and see where they lead.
Our little world of exploration is constituted of decimal representations of the rational numbers 1/2, 1/3, 1/4, …. Some of these decimals are familiar; others perhaps less so: 1/2 = 0.5; 1/3 = 0.3333.… (repeating forever); and 1/7 = 0.142857142857…… (with this block of digits repeating forever).
We’ll collect our exploratory data in the following table, most of its entries “left as an exercise for the reader,” as they used to say in textbooks before education reformers decided this might leave some at an emotional disadvantage.
These days, you have several choices: a handy cell phone’s calculator function, perhaps a more elaborate division function on a computer or traditional long division by hand. Be advised: Several representations may not exhibit their pattern unless the calculator has sufficient decimal places (my eight-place iPhone Calculator won’t cut it). You may enjoy resorting to traditional means, just for the practice.
Also, in filling out the table, identify the nature of each denominator: Is it a prime, 2, 3, 5, 7, 11, etc? Or is it a composite of primes, 6 = 2 x 3, 8 = 23, 12 = 22 x 3, etc?
Next, examine the decimal representation and identify its character. Some of these representations terminate: 1/2 = .5. Others repeat blocks of digits: 1/9 = .0909…, which can be denoted by a bar atop the repeating block.
Last, identify the lengths involved in each decimal representation: Does it terminate in one, two or more digits? Is its repeating cycle of length one or two or more? Does it have a digital string of some length, then the repetition?
I realize that many of you (indeed, all?) may well have more crucial things to do than fritter away time with this. For those understandably incurious sorts, I’ll present the completed table tomorrow, along with some conjectures and maybe even some good honest mathematics.
This may be one of the more honest things you’ll read on the Internet these days. ds
© Dennis Simanaitis, SimanaitisSays.com, 2017