3 ms·

"5x less" would be "x - 5x" or -4x, which of course is nonsense as density can't be negative. "5x as sparse" works if you define sparsity as the inverse of dens

"5x less" would be "x - 5x" or -4x, which of course is nonsense as density can't be negative. "5x as sparse" works if you define sparsity as the inverse of density.

Dylan16807 6y ago
Don't be so weirdly picky. "times less" is division.
- nybble41 6y ago
  Which inconsistency do you prefer? • "90% less than x" does not equal "10% of x" • "90%" does not equal "0.9 times" • "0.9 times less" means multiplication and subtraction (x - 0.9x) but "1.1 times less" means division (x / 1.1) This is one of those cases where we know from the context what was meant but that isn't the same as what was said. Normalizing this abuse of language introduces unnecessary special cases and discontinuities for readers/listeners to deal with and creates ambiguity in situations where one of the two interpretations is not so obviously wrong as it is here. It's better to keep the rules simple: ratios, percentages and "times" always involve multiplication by the original value, and "more" and "less" refer to addition and subtraction. There should be a deterministic 1:1 correspondence between the phrasing and the formula.
  - Dylan16807 6y ago
    Does anyone actually say "zero point ___ times less"? I would say not to do that, and only use "times less" with numbers above 1. That's much easier than dealing with people who calculate percentages wrong or say ambiguous things about large increases.
    - nybble41 6y ago
      No, for ratios less than unity most people would use a shorter, more convenient fraction ("nine-tenths") or percentage ("ninety percent") which has exactly the same meaning as the decimal ratio ("zero point nine times"). For fractions the word "times" may be implied rather than explicit; for percentages it's always implied, whereas for decimal ratios it's necessary to distinguish ratios from absolute quantities. These forms are nonetheless all equivalent and substituting one for another should not change the meaning of the phrase. "X more than Y" means addition, even if X is expressed as a ratio ("three times more than Y" equals "300% more than Y" equals "Y plus 300% of Y" equals "Y plus three times Y"). Analogously, "X less than Y" means subtraction, not division, whether X is an absolute quantity, a percentage either less than or greater than 100%, a fraction, or a decimal ratio. This is human language, of course, so you can be as over-complicated, inconsistent, and ambiguous as you want and generally count on sufficiently smart listeners/readers to decode the resulting mess into something intelligible based on context. It's not good style, and it encourages innumeracy and presents math as being more complicated than it should be, but it will usually get the job done.

"5x less" would be "x - 5x" or -4x, which of course is nonsense as density can't be negative. "5x as sparse" works if you define sparsity as the inverse of density.

Dylan16807 6y ago

Don't be so weirdly picky. "times less" is division.

nybble41 6y ago
Which inconsistency do you prefer? • "90% less than x" does not equal "10% of x" • "90%" does not equal "0.9 times" • "0.9 times less" means multiplication and subtraction (x - 0.9x) but "1.1 times less" means division (x / 1.1) This is one of those cases where we know from the context what was meant but that isn't the same as what was said. Normalizing this abuse of language introduces unnecessary special cases and discontinuities for readers/listeners to deal with and creates ambiguity in situations where one of the two interpretations is not so obviously wrong as it is here. It's better to keep the rules simple: ratios, percentages and "times" always involve multiplication by the original value, and "more" and "less" refer to addition and subtraction. There should be a deterministic 1:1 correspondence between the phrasing and the formula.
- Dylan16807 6y ago
  Does anyone actually say "zero point ___ times less"? I would say not to do that, and only use "times less" with numbers above 1. That's much easier than dealing with people who calculate percentages wrong or say ambiguous things about large increases.
  - nybble41 6y ago
    No, for ratios less than unity most people would use a shorter, more convenient fraction ("nine-tenths") or percentage ("ninety percent") which has exactly the same meaning as the decimal ratio ("zero point nine times"). For fractions the word "times" may be implied rather than explicit; for percentages it's always implied, whereas for decimal ratios it's necessary to distinguish ratios from absolute quantities. These forms are nonetheless all equivalent and substituting one for another should not change the meaning of the phrase. "X more than Y" means addition, even if X is expressed as a ratio ("three times more than Y" equals "300% more than Y" equals "Y plus 300% of Y" equals "Y plus three times Y"). Analogously, "X less than Y" means subtraction, not division, whether X is an absolute quantity, a percentage either less than or greater than 100%, a fraction, or a decimal ratio. This is human language, of course, so you can be as over-complicated, inconsistent, and ambiguous as you want and generally count on sufficiently smart listeners/readers to decode the resulting mess into something intelligible based on context. It's not good style, and it encourages innumeracy and presents math as being more complicated than it should be, but it will usually get the job done.