2 ms·

Say you have one apple and want to divide it over zero people. It can't be done - give everyone one, fifty, a million, infinity apples: you always have one app

Say you have one apple and want to divide it over zero people. It can't be done - give everyone one, fifty, a million, infinity apples: you always have one apple left over.

jacobolus 4y ago
The concept of "division" or "ratio" has different interpretations depending on context. The Euclidean (from The Elements) concept of ratio can certainly admit a 1:0 ratio, which we can write as 1/0 or ∞ if we like, and which is equivalent to the ratio of x:0 for any non-zero number x. There’s no inherent reason why the ratio of "apples per person" should be allowed to be 0:1 but not 1:0. From first principles it’s just as problematic to divide 0 apples among x people as to divide x apples among 0 people. When we e.g. write the values of trigonometric functions, these should ordinarily be interpreted to be among the extended reals, so that tan(π/2) = 1/0 = ∞ (i.e. the ratio 1:0) is entirely reasonable.
- SideburnsOfDoom 4y ago
  Is that +∞ or -∞ though? The problem with division by zero is not that it's infinite, it is that it limit could be either +∞ or -∞ depending on which side of the limit you look at. Therefor, it is not well-defined.
  - jacobolus 4y ago
    In some contexts +∞ = –∞ (e.g. tan ½π = tan –½π; these both represent the orientation of a line at a right angle). In other contexts +∞ ≠ –∞ (e.g. log ±0 ≠ log ±∞; these quantities can be thought of as the two opposite ends of an infinite cylinder)
    - SideburnsOfDoom 4y ago
      Well, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?
      - jacobolus 4y ago
        For example, when you look at a world map under the Mercator projection and you want to know where the north and south poles should be on the map, the answer is +∞ and –∞, which are not the same.

2 ms·

Say you have one apple and want to divide it over zero people. It can't be done - give everyone one, fifty, a million, infinity apples: you always have one app

by Scarblac 4y ago

Say you have one apple and want to divide it over zero people. It can't be done - give everyone one, fifty, a million, infinity apples: you always have one apple left over.

jacobolus 4y ago
The concept of "division" or "ratio" has different interpretations depending on context. The Euclidean (from The Elements) concept of ratio can certainly admit a 1:0 ratio, which we can write as 1/0 or ∞ if we like, and which is equivalent to the ratio of x:0 for any non-zero number x. There’s no inherent reason why the ratio of "apples per person" should be allowed to be 0:1 but not 1:0. From first principles it’s just as problematic to divide 0 apples among x people as to divide x apples among 0 people. When we e.g. write the values of trigonometric functions, these should ordinarily be interpreted to be among the extended reals, so that tan(π/2) = 1/0 = ∞ (i.e. the ratio 1:0) is entirely reasonable.
- SideburnsOfDoom 4y ago
  Is that +∞ or -∞ though? The problem with division by zero is not that it's infinite, it is that it limit could be either +∞ or -∞ depending on which side of the limit you look at. Therefor, it is not well-defined.
  - jacobolus 4y ago
    In some contexts +∞ = –∞ (e.g. tan ½π = tan –½π; these both represent the orientation of a line at a right angle). In other contexts +∞ ≠ –∞ (e.g. log ±0 ≠ log ±∞; these quantities can be thought of as the two opposite ends of an infinite cylinder)
    - SideburnsOfDoom 4y ago
      Well, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?
      - jacobolus 4y ago
        For example, when you look at a world map under the Mercator projection and you want to know where the north and south poles should be on the map, the answer is +∞ and –∞, which are not the same.