bob x
Well-Known Member
There is a number-system called the "projective space" which are useful in some contexts. They are called "projective" because the main motivation is considering what happens if you can distinguish direction, but not distance, as if you were a Cyclops looking at the world; or more precisely, a "Bicyclops" with an eye on the front and an eye on the back of the head, both projecting onto a single retina, unable to tell which side the object is on either. Mathematically, all points on any line drawn through the center are equated, and all that is distinguished are the slopes of the lines: 1/1 is the slope of a 45-degree-angle line (up 1 for every 1 over), 1/2 a shallower slope (up 1 but over 2), 2/1 a steeper (up 2, over 1), and then 0/1 (the zero) is pure horizontal, 1/0 ("infinity") the pure vertical.
The arithmetical rules are the same as for fractions, a/b the same as ac/bc where c is any non-zero multiplier, a/b times c/d equals ac/bd, a/b + c/b = (a+c)/b but if the denominators are not the same you need to force them the same, a/b + c/d = (ad+bc)/bd. This has some odd results: there is an extra point 0/0 (the "indeterminate") which is deleted from the projective space and sometimes written "?"; anything plus "?" or anything times "?" gives "?", and zero times infinity is also "?" (0/1 = 0/2 = 0/3 etc. times 1/0 = 2/0 = 3/0 etc. comes to 0/0); anything plus zero gives itself, and anything except "?" or another infinity plus infinity gives infinity; anything except "?" or infinity times zero gives zero, and anything except "?" or zero times infinity gives infinity. Most find these results easy enough to rationalize, but "infinity plus infinity gives ?" is a little harder to understand: the reason is that "plus infinity" 1/0 and "minus infinity" -1/0 are the same, and "infinity minus infinity" can be ANYTHING. Consider all the natural numbers (1, 2, 3, ...): how many? Infinity. Consider all the numbers past 17 (18, 19, 20, ...): how many? Infinity. Take all the numbers past 17 away from all the numbers, and what is left? 1 through 17. Thus, infinity minus infinity is 17. Or, consider all the even numbers (2, 4, 6, ...): how many? Infinity. Take all the even numbers away from all the numbers, and you are left with all the odd numbers: thus, infinity minus infinity is infinity. Or, take all the numbers away from all the numbers, and nothing is left: thus, infinity minus infinity can also be zero; hence, infinity minus infinity is "?". Galileo is the first to publish this. Riemann extended the discussion to 3D -> 2D projective space, where each triple (x,y,z) is identified with all nonzero multiples (xc,yc,zc) so that we have a set of directions, but no distance distinction, in 3D space.
The point is: India first recognized the importance of including zero in the numbers, but did not work out all the implications, because they are genuinely difficult. Brahmagupta was not wrong to consider "1/0" as possible kind of number, but was wrong to think that "0/0" behaves the same as zero; the way I describe above is the only way it can work. Arabs didn't figure this out either; Europeans did, although it is quite true to point out that Europeans would never have heard of the concept of "zero" in the first place if Arabs had not carried the word.
The arithmetical rules are the same as for fractions, a/b the same as ac/bc where c is any non-zero multiplier, a/b times c/d equals ac/bd, a/b + c/b = (a+c)/b but if the denominators are not the same you need to force them the same, a/b + c/d = (ad+bc)/bd. This has some odd results: there is an extra point 0/0 (the "indeterminate") which is deleted from the projective space and sometimes written "?"; anything plus "?" or anything times "?" gives "?", and zero times infinity is also "?" (0/1 = 0/2 = 0/3 etc. times 1/0 = 2/0 = 3/0 etc. comes to 0/0); anything plus zero gives itself, and anything except "?" or another infinity plus infinity gives infinity; anything except "?" or infinity times zero gives zero, and anything except "?" or zero times infinity gives infinity. Most find these results easy enough to rationalize, but "infinity plus infinity gives ?" is a little harder to understand: the reason is that "plus infinity" 1/0 and "minus infinity" -1/0 are the same, and "infinity minus infinity" can be ANYTHING. Consider all the natural numbers (1, 2, 3, ...): how many? Infinity. Consider all the numbers past 17 (18, 19, 20, ...): how many? Infinity. Take all the numbers past 17 away from all the numbers, and what is left? 1 through 17. Thus, infinity minus infinity is 17. Or, consider all the even numbers (2, 4, 6, ...): how many? Infinity. Take all the even numbers away from all the numbers, and you are left with all the odd numbers: thus, infinity minus infinity is infinity. Or, take all the numbers away from all the numbers, and nothing is left: thus, infinity minus infinity can also be zero; hence, infinity minus infinity is "?". Galileo is the first to publish this. Riemann extended the discussion to 3D -> 2D projective space, where each triple (x,y,z) is identified with all nonzero multiples (xc,yc,zc) so that we have a set of directions, but no distance distinction, in 3D space.
The point is: India first recognized the importance of including zero in the numbers, but did not work out all the implications, because they are genuinely difficult. Brahmagupta was not wrong to consider "1/0" as possible kind of number, but was wrong to think that "0/0" behaves the same as zero; the way I describe above is the only way it can work. Arabs didn't figure this out either; Europeans did, although it is quite true to point out that Europeans would never have heard of the concept of "zero" in the first place if Arabs had not carried the word.