3 ms·
The concept of "division" or "ratio" has different interpretations depending on context. The Euclidean (from The Elements) concept of ratio can certainly admit
by 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 agoIs 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 agoIn 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 agoWell, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?
- jacobolus 4y agoFor 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.