Dividing by 9, 99, 999....Date: 9/7/96 at 20:31:54 From: Joe Snyder Subject: Dividing by 9, 99, 999.... Dear Dr. Math, I was fooling around on my calculator. Somehow, after a while, I found a pattern in what I was keying in. This is what I found: x / 99... = .xx ... (the number of 9s is the same as the number of digits x has) Example No.1: 456 / 999 = .456456456... (456 repeat) Example No.2: 12,345,678 / 99,999,999 = .12345678 12345678 12345678... Example No.3: 9,999 / 9,999 = 1 and 0 / 9 = 0 (*cannot use all-9 numbers or 0s*) My question is: Is there any property that states this or is similar to this? What is it and why does it work that way? (This isn't really a property, but you know what I mean.) Any answers or replies will be greatly appreciated! Joe Snyder Date: 9/10/96 at 14:37:38 From: Doctor Ana Subject: Re: Dividing by 9, 99, 999.... My father, a math teacher, always talks about taking his calculator with him everywhere, even to bed, to play with numbers. I'm glad he's not the only one who likes math enough to do that! Here is one explanation for the property that you found. (Sorry, you weren't the first one to find it, but don't let that keep you from trying!) 456 456 456 --- = ---- + ------ 999 1000 999000 Prove that to yourself first of all. Then we can see that we have 456 456 --- = .456 + ------ 999 999000 456 456 456 ------ = ------- + --------- 999000 1000000 999000000 So, now we have 456 456 --- = .456 + .000456 + --------- 999 999000000 And the pattern keeps repeating forever. This same method will work for any denominator that contains all 9s, no matter how many of them there are. Have you looked at the pattern for the sevenths and the elevenths yet? They're almost as interesting as the ninths. Try 1/7, 2/7, 3/7 and 1/11, 2/11, 3/11, 4/11 on your calculator and figure out the pattern. -Doctor Ana, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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