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Can you say more? If "0 is not an allowable value for b", then it seems to me that (a/b)*b=a isn't true for all values. Specifically, it's false when b=0. IIUC

by beala 2y ago

Can you say more? If "0 is not an allowable value for b", then it seems to me that (a/b)*b=a isn't true for all values. Specifically, it's false when b=0. IIUC, codeflo is arguing that the division operation defined in the article isn't "actual division" because (a/b)*b=a isn't true for all values. But I can't think of a definition of division that satisfies that criteria.

jraph 2y ago
When we say "is not an allowable value", we are speaking about the domain [1]: all the values for which the function is defined. When we say "for all values", we implicitly mean for all values of the domain. The parallel in programming would be the contract : you provide a function that works on a given set of values. Or the type: the function would "crash" if you passed a value not of the type of its parameter, but it is admitted it won't be done. (In the remaining I'm referring to 1/x instead of a/b to simplify things a bit) Another way of saying it is that the function is undefined for 0. (Or on {0}). Then the property is true for all values (on which the function is defined, but saying it is redundant, the function can't be called outside its domain, it is an error to try to do this). The domain is often left out / implicit, but it is always part of the definition of a function. 0 is not in the domain, so it's not to be considered at all when studying the function (except maybe when studying limits, but the function will still not be called with it). [1] https://en.m.wikipedia.org/wiki/Domain_of_a_function https://en.m.wikipedia.org/wiki/Domain_of_a_function
zzo38computer 2y ago
If "0 is not an allowable value for b", then (a/b)*b=a is not defined when b=0, so it is neither true nor false, since you had previously agreed that b=0 is not allowed (regardless of what "/" and "*" are meaning in this context).
eru 2y ago
We don't allow division of apples and oranges, either. So why is excluding 0 weird, but excluding ice cream as an argument is not?

by beala 2y ago

jraph 2y ago

When we say "is not an allowable value", we are speaking about the domain [1]: all the values for which the function is defined. When we say "for all values", we implicitly mean for all values of the domain. The parallel in programming would be the contract : you provide a function that works on a given set of values. Or the type: the function would "crash" if you passed a value not of the type of its parameter, but it is admitted it won't be done. (In the remaining I'm referring to 1/x instead of a/b to simplify things a bit) Another way of saying it is that the function is undefined for 0. (Or on {0}). Then the property is true for all values (on which the function is defined, but saying it is redundant, the function can't be called outside its domain, it is an error to try to do this). The domain is often left out / implicit, but it is always part of the definition of a function. 0 is not in the domain, so it's not to be considered at all when studying the function (except maybe when studying limits, but the function will still not be called with it). [1] https://en.m.wikipedia.org/wiki/Domain_of_a_function https://en.m.wikipedia.org/wiki/Domain_of_a_function

zzo38computer 2y ago

If "0 is not an allowable value for b", then (a/b)*b=a is not defined when b=0, so it is neither true nor false, since you had previously agreed that b=0 is not allowed (regardless of what "/" and "*" are meaning in this context).

eru 2y ago

We don't allow division of apples and oranges, either. So why is excluding 0 weird, but excluding ice cream as an argument is not?