3 ms·
It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction
by devit 6y ago
It's easy to see that this cannot be a valid definition for fraction addition because applying it along with the rule that a/b = (ak)/(bk) gives a contradiction:
3/7 = 2/4 (+) 1/3 = 1/2 (+) 1/3 = 2/5, absurd because 2/5 != 3/7
Once this is realized, it's easy to see that the correct formulation for the "table joining" operation used here is a weighted arithmetic mean, i.e. a(/)b (+) c(/)d = weighted arithmetic mean of a/b and c/d with relative weights b and d = ((a/b) * b + (c/d) * d) / (b + d) = (a + c)/(b + d).
On the other hand, fraction addition is clearly determined as a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd given the a/b = (ak)/(bk) and (a/b + c/b) = (a + c)/b axioms.
- karatestomp 6y ago2/4 + 1/3 would clearly be 5/12. You halve each to put them in terms of the two wholes you're combining (which you must be doing, since you're adding them—else where do those go?) then combine those. Same process that gives you 1/3 + 1/3 = 1/3. Worked out (conventional meaning of + for clarity; we'd need a new operator otherwise): 1/4 + 1/6 = 6/24 + 4/24 = 10/24 = 5/12