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It is like the infamous 0.999999... = 1. That one uses sloppy notation (what is "..."?) to make students think and talk about math.

mcphage 1y 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 1y ago
  Yeah; 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 1y ago
    There'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 1y ago
  Thanks for the reminder of the trailing zeroes. 1.000000000 is also a big number.
jjmarr 1y ago
It's not sloppy notation. It's an unambiguous infinite series of the form sum_n=1^infinity 9/10^n that converges to 1. It's the same reason that 0.333... = 1/3. It's an infinite series that converges on 1/3. Students learn repeating decimals before they understand infinite series.
- astrobe_ 1y ago
  It is ambiguous, that's the trick. One can interpret "..." as '"any number of 9s", and in this interpretation the equation is false. But the teacher will eventually ask, "what about an infinite number of 9s"? Bam! brilliant introduction to series and convergence. The correct notations to express exactly what one means feature the infinity symbol [1] - but if you use them, well, of course the students will see the trap. [1] https://en.wikipedia.org/wiki/Infinity#Real_analysis https://en.wikipedia.org/wiki/Infinity#Real_analysis

mcphage 1y 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 1y ago
Yeah; 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 1y ago
  There'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 1y ago
Thanks for the reminder of the trailing zeroes. 1.000000000 is also a big number.

jjmarr 1y ago

It's not sloppy notation. It's an unambiguous infinite series of the form sum_n=1^infinity 9/10^n that converges to 1. It's the same reason that 0.333... = 1/3. It's an infinite series that converges on 1/3. Students learn repeating decimals before they understand infinite series.

astrobe_ 1y ago
It is ambiguous, that's the trick. One can interpret "..." as '"any number of 9s", and in this interpretation the equation is false. But the teacher will eventually ask, "what about an infinite number of 9s"? Bam! brilliant introduction to series and convergence. The correct notations to express exactly what one means feature the infinity symbol [1] - but if you use them, well, of course the students will see the trap. [1] https://en.wikipedia.org/wiki/Infinity#Real_analysis https://en.wikipedia.org/wiki/Infinity#Real_analysis