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I'm not sure the "..." is sloppy notation—it can be made rigid pretty easily. The surprise is that students' expectations that if two decimal expressions are
by mcphage 2y ago
I'm not sure the "..." is sloppy notation—it can be made rigid pretty easily. The surprise is that students' expectations that if two decimal expressions are distinct, that the real number they correspond to must be distinct also. (Even there, students have already gotten used to trailing zeros being irrelevant).
- dwohnitmok 2y agoYeah; students often conflate numerals with numbers and assume that if two numerals are distinct then the numbers they represent must also be distinct. It's not trivially true! You need to prove that distinct numerals lead to distinct numbers and as 0.999... vs 1.000... demonstrates it is not always true for various kinds of numerals and numbers! I think teasing apart numerals and numbers is a good first step on one's journey in mathematical logic.
- mcphage 2y agoThere's also another numeral system called "Bijective Numeration" (https://en.wikipedia.org/wiki/Bijective_numeration https://en.wikipedia.org/wiki/Bijective_numeration) which does is a bijection between numeral sequences and numbers (hence the name). It's pretty neat... it doesn't have a 0, so you can't get an infinite string of trailing 0s.
- cwmoore 2y agoThanks for the reminder of the trailing zeroes. 1.000000000 is also a big number.