Re:Imaginary numbers are real
by SideburnsOfDoom on Sunday July 24, 2022 @01:21PM
(#32215580)
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.
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)
Re:Imaginary numbers are real
by SideburnsOfDoom on Monday July 25, 2022 @04:30AM
(#32221856)
Well, in that case the parent must ask themselves: am I doing geometry on an infinite cylinder or not?
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.
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.