) On Jul 2, 4:31 am, sulekhaswe...@gmail.com wrote: )> Hi, )> )> suppose a is any non zero natural number , )> then a x 5 = a/2 x 10 is n't it ? ) ) this statement is true: ) 'a' times 5 = ('a' divided by 2) times 10
What does ('a' divided by 2) mean for natural numbers ?
If I ask someone: "What is 5 divided by 2", then he could answer: "2, with a remainder of 1"
SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT
>>> suppose a is any non zero natural number , >>> then a x 5 = a/2 x 10 is n't it ? >> Only if a is even > No, it's true for odd a ...
Depends on whether or not it's intended that the intermediate results are also required to be natural numbers. -- Mark Brader | "But [he] had already established his own reputation Toronto | as someone who wrote poetry that mentioned the el." m...@vex.net | --Al Kriman
On Wed, 2 Jul 2008 15:56:31 +0000 (UTC), Willem <wil...@stack.nl> wrote:
>Nishant Shukla wrote: >) On Jul 2, 4:31 am, sulekhaswe...@gmail.com wrote: >)> Hi, >)> >)> suppose a is any non zero natural number , >)> then a x 5 = a/2 x 10 is n't it ? >) >) this statement is true: >) 'a' times 5 = ('a' divided by 2) times 10
>What does ('a' divided by 2) mean for natural numbers ?
>If I ask someone: "What is 5 divided by 2", then >he could answer: "2, with a remainder of 1"
Hmm, and what would he say if you asked "What is '2, with a remainder of 1' multiplied by 10?" :-)
>>>> suppose a is any non zero natural number , >>>> then a x 5 = a/2 x 10 is n't it ?
>>> Only if a is even
>> No, it's true for odd a ...
> Depends on whether or not it's intended that the intermediate results > are also required to be natural numbers.
And on whether we are allowed to rearrange the equation. In mathematics, it is normal to be granted such licence, provided it doesn't break any rules. (It is in quietly breaking such rules that paradoxists arrive at some of their curios.)
) And on whether we are allowed to rearrange the equation. In mathematics, it ) is normal to be granted such licence, provided it doesn't break any rules. ) (It is in quietly breaking such rules that paradoxists arrive at some of ) their curios.)
In the natural numbers, the division operator is such that you would break some rules when rearranging equations. I don't know offhand what they are called, but I'm sure there are those more knowledgable who do.
SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT
) Hmm, and what would he say if you asked "What is '2, with a remainder ) of 1' multiplied by 10?" :-)
Err, he would probably say that that doesn't make sense.
What is '5, with a remainder of 3' multiplied by 10 ?
SaSW, Willem -- Disclaimer: I am in no way responsible for any of the statements made in the above text. For all I know I might be drugged or something.. No I'm not paranoid. You all think I'm paranoid, don't you ! #EOT
> And on whether we are allowed to rearrange the equation. In mathematics, it > is normal to be granted such licence, provided it doesn't break any rules.
Yes, but that's precisely avoiding the question: Mark's point is that he is stating what might be an implicit rule in this case, which (if so) rearranging the equation would break.
The rationale behind rearranging and simplifying an equation is that you apply transformations which do not affect the answer - or lack of answer - given by the equation. That's the principle on which you decide what rearrangements are valid in the first place. _If_ (as Mark suggests is a possibility) we are constrained to work entirely within the natural numbers and thus dividing an odd number by two is forbidden, then we may not simplify "divide by two and then multiply by ten" into "multiply by five", precisely _because_ the latter would yield a result in cases where the former does not. You can't get around such a restriction by citing rearrangement, because by definition if the rearrangement gets you round a restriction then it wasn't permitted anyway.
It is certainly true that naively rearranging this particular equation can turn it into one which yields the same answer in all cases where the original _did_ yield an answer and which also produces answers with an internal consistency in some additional cases, and this sort of possibility has historically tended to inspire mathematicians to search for an expanded algebraic structure which contains the previous one and in which more rearrangements become valid and more equations soluble. Such processes have been responsible for extending the natural numbers into integers, rationals, reals, complex numbers and no end of rings, fields, groups and more exotic stuff still. But none of that excuses the breaking of the rules that bind you here and now in any given situation; they merely suggest that _after_ you've finished solving this particular problem and are no longer bound by its rules, then there's an interesting area in which you might like to do further research. -- Simon Tatham "A cynic is a person who smells flowers and <ana...@pobox.com> immediately looks around for a coffin."
>Richard Heathfield wrote: >) And on whether we are allowed to rearrange the equation. In mathematics, it >) is normal to be granted such licence, provided it doesn't break any rules. >) (It is in quietly breaking such rules that paradoxists arrive at some of >) their curios.)
>In the natural numbers, the division operator is such that you would break >some rules when rearranging equations. I don't know offhand what they are >called, but I'm sure there are those more knowledgable who do.
I think you're looking for the word "associative".
If a set is associative under multiplication, it means that (a * b) * c = a * (b * c) for all a, b and c in the set.
Thus: (a/2 * 2) * 5 = a/2 * (2 * 5)
The natural numbers are associative under multiplication.
So if "a/2" is allowed to have any meaning for odd naturals, and conforms to the expected behaviour (a/2 * 2) = a then we're OK.
