Simple, yet counter-intuitive. Surreal, but nice. (Okay, that didn’t make sense, but I just wanted to quote Hugh Grant from Notting Hill :P). I don’t know if it is good enough to write a blog post on it, but I find it quite interesting, considering my limited intellect and limited understanding of things in general.
Okay, the deal is: is the repeating decimal 0.9999…. less than or equal to 1? I would have thought “less than” at first, but then I thought why would that even be a question/problem had it been this direct. So I reflected upon it a little more carefully, and as it turns out with some scribbling on the paper, it is equal to 1. In fact, the methodology required here, is taught to us in secondary classes itself.
Let x = 0.99999….
so that 10x = 9.99999…. (Multiplying both sides with 10)
=> 10x – x = 9.9999999…… – 0.9999999……
=> 9x = 9
=> x = 1
That’s this completes the proof to verify that indeed, the equality holds, and the repeating decimal 0.999… is equal to 1. This statement has been accepted by mathematicians as far as written history dates, yet some students find it not so easy to “digest” even though they can’t find any error in the proof yet, they can’t accept it with grace.
Something always lingers on, and they can’t surely say what and why, but they just can’t consider it “natural”. Nonetheless, it is established and has been proved by many methods that indeed, 0.999.. = 1.