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Maybe the major source of confusion is that our decimal representation for whole numbers is supposed to be unique. Then when we extend it to rationals and reals
by steerablesafe 6y ago
Maybe the major source of confusion is that our decimal representation for whole numbers is supposed to be unique. Then when we extend it to rationals and reals this property fails at rationals in the form of a/10^n.
Arguably the sign symbol ruins it for whole numbers as well, as +0 and -0 could be equally valid representations of the number 0. We just conventionally don't allow -0 as a representation. There are other number representations that don't have this problem.
- virgilp 6y agoRight - I also find it easier to say that really, "1" is just a different/shorthand notation for 0.(9) It's not "two different, but equal numbers" - it's two different notations for the same number. Like how you can write same number in different ways in different bases - this is just writing the same number, in the "infinite number of decimals" vs "natural" way.
- steerablesafe 6y agoArguably 1 is just as an infinite number of decimals as 0.(9) . 1 is just short for 1.(0) .
- onion2k 6y agoI think the source of confusion is that people can't cope with recurring numbers. When someone says .999...=1 the listener assumes that the 0.999... stops at some point, and if that happens it'll always be below 1 because they can imagine adding another 9. Essentially, people actually ignore the "..." because that's the hard part.
- baddox 6y agoI don’t know, I don’t recall students having much trouble accepting that 1 = 1.0 = 1.000.