- #1
mirelo
- 19
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Is it zero divided by zero even a division? According to some, it is not, because division would have three requirements:
1) That any quotient must be a number.
2) That any quotient multiplied by a divisor must give us a dividend, which presupposes that any quotient is a number.
3) That no division can have more than one quotient.
My purpose here is to show the artificial nature of the third requirement, which has the sole purpose of invalidating the division of zero by zero, the first two requirements being the only true requirements of division (in fact, there is only one requirement, namely, that any quotient must be a number, since the concept of a quotient already presupposes a divisor and a dividend, which makes the second requirement redundant).
Those who deny the validity of the division of zero by zero allege that it is not only indeterminate -- meaning it has any quotient -- but also undefined -- meaning it has no quotient (I am not talking here about indeterminate forms, but simply about an indeterminate mathematical operation, meaning it has a multitude of equally valid outcomes). However, there is a division that is indeed undefined without being also indeterminate, which is the division of any other number (than zero) by zero, for which there is indeed no quotient, since no number multiplied by zero results in a nonzero number -- any number multiplied by zero results in zero, which is precisely what makes the division of zero by zero indeterminate in the first place. Therefore, once the division of any number by zero is reputed as undefined, and since the division of any other number than zero by zero is indeed undefined although without being indeterminate, we must add a fourth requirement to the list of division requirements, namely, that no division can have less than one quotient:
1) Any quotient must be a number.
2) Any quotient multiplied by a divisor must give us a dividend, which presupposes that any quotient is a number.
3) No division can have more than one quotient.
4) No division can have less than one quotient.
First of all, let us notice that the first two requirements apply to quotients, while the last two apply to divisions. In other words, the last two requirements presuppose a complete mathematical operation with a dividend, a divisor and -- already -- a quotient. However, at least when dealing with limits, we do not have such a situation: quotients of limits tend to some value, rather than being fixed once and for all whenever we simply formulate a division.
Of course we could say that for each possible value of a pair of numerator / denominator limits there is only one quotient, but then we are no longer talking about ratios of limits. If we accept the last two division requirements, then we are utterly forbidden of talking about such ratios -- their very mathematical concept becomes impossible: we will never be able to talk again about quotients tending to anything.
This is enough to show that the last two requirements are not only arbitrary, but also false:
1) The first two requirements refer to a different object than the last two, which refer to entire divisions -- while the first two (uncontroversial) requirements refer to quotients.
2) The last two requirements are incompatible with ratios involving limits.
This result is no surprise, as the last two requirements were specifically added to invalidate the division of any number by zero in general and the division of zero by zero in particular -- and without success, as they are incompatible with ratios involving limits.
1) That any quotient must be a number.
2) That any quotient multiplied by a divisor must give us a dividend, which presupposes that any quotient is a number.
3) That no division can have more than one quotient.
My purpose here is to show the artificial nature of the third requirement, which has the sole purpose of invalidating the division of zero by zero, the first two requirements being the only true requirements of division (in fact, there is only one requirement, namely, that any quotient must be a number, since the concept of a quotient already presupposes a divisor and a dividend, which makes the second requirement redundant).
Those who deny the validity of the division of zero by zero allege that it is not only indeterminate -- meaning it has any quotient -- but also undefined -- meaning it has no quotient (I am not talking here about indeterminate forms, but simply about an indeterminate mathematical operation, meaning it has a multitude of equally valid outcomes). However, there is a division that is indeed undefined without being also indeterminate, which is the division of any other number (than zero) by zero, for which there is indeed no quotient, since no number multiplied by zero results in a nonzero number -- any number multiplied by zero results in zero, which is precisely what makes the division of zero by zero indeterminate in the first place. Therefore, once the division of any number by zero is reputed as undefined, and since the division of any other number than zero by zero is indeed undefined although without being indeterminate, we must add a fourth requirement to the list of division requirements, namely, that no division can have less than one quotient:
1) Any quotient must be a number.
2) Any quotient multiplied by a divisor must give us a dividend, which presupposes that any quotient is a number.
3) No division can have more than one quotient.
4) No division can have less than one quotient.
First of all, let us notice that the first two requirements apply to quotients, while the last two apply to divisions. In other words, the last two requirements presuppose a complete mathematical operation with a dividend, a divisor and -- already -- a quotient. However, at least when dealing with limits, we do not have such a situation: quotients of limits tend to some value, rather than being fixed once and for all whenever we simply formulate a division.
Of course we could say that for each possible value of a pair of numerator / denominator limits there is only one quotient, but then we are no longer talking about ratios of limits. If we accept the last two division requirements, then we are utterly forbidden of talking about such ratios -- their very mathematical concept becomes impossible: we will never be able to talk again about quotients tending to anything.
This is enough to show that the last two requirements are not only arbitrary, but also false:
1) The first two requirements refer to a different object than the last two, which refer to entire divisions -- while the first two (uncontroversial) requirements refer to quotients.
2) The last two requirements are incompatible with ratios involving limits.
This result is no surprise, as the last two requirements were specifically added to invalidate the division of any number by zero in general and the division of zero by zero in particular -- and without success, as they are incompatible with ratios involving limits.
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