Dividing Zero By Zero

guigus wrote:

Unfortunately, Joe from Chicago did not try to teach me anything that I didn't already knew (neither was he willing to learn anything).

So then, are you going to use your administrator privileges to block me from answering to your posts? ...

Your "reality" exists only in your imagination. For instance: what is your source for thinking I have administrator privileges on this site? Finally: it's not only Joe who refused to "learn" from your magical thinking; it's also I who asked you to please not address me any longer, I won't bother replying.

The division of zero by zero is any number, since:
1. Any number multiplied by zero results in zero.
2. Division is defined as the inverse of multiplication.

On L'Hopital's rule (really discovered by Bernoulli) you're only 3 centuries late - maybe you should start with pictures

Quote:

While both f(x) and g(x) approach infinity as x->infinity, the limit of the ratio is bounded inside the interval [1/e,e], while the limit of f^'(x)/g^'(x) approaches 0.

http://mathworld.wolfram.com/LHospitalsRule.html

Limits of the form



in which both ƒ(x) and g(x) approach 0 as x approaches 0, may equal any real or infinite value, or may not exist at all, depending on the particular functions ƒ and g (see l'Hôpital's rule for discussion and examples of limits of ratios). These and other similar facts show that the expression 0/0 cannot be well-defined as a limit.

The division of zero by zero is any number, since:
1. Any number multiplied by zero results in zero.
2. Division is defined as the inverse of multiplication.

On L'Hopital's rule (really discovered by Bernoulli) you're only 3 centuries late - maybe you should start with pictures

Quote:

While both f(x) and g(x) approach infinity as x->infinity, the limit of the ratio is bounded inside the interval [1/e,e], while the limit of f^'(x)/g^'(x) approaches 0.

http://mathworld.wolfram.com/LHospitalsRule.html

The division of zero by zero is any number, since:
1. Any number multiplied by zero results in zero.
2. Division is defined as the inverse of multiplication.

Additionally, the reason why ƒ(x)/g(x) as x approaches zero is occasionally nonexistent is precisely because in this case we have ƒ(x) and g(x) approaching zero instead of being zero itself: any nonzero value divided by zero is nonexistent (undefined), since no value different from zero multiplied by zero results in a nonzero value, so no wonder if we take zero to be some ƒ(x) or g(x) approaching it we occasionally get a nonexistent result, just to remind us that:

1. Zero is nothing rather than something.
2. If something can approach zero, then that something must not be just identical to it.

So despite zero as an infinitesimal divided by zero as an infinitesimal being undefined (possibly nonexistent), zero as itself divided by zero as itself is rather indeterminate (always existent by being any number).

Indeed, the notion that ƒ(x)/g(x) as x approaches zero can be taken to represent zero divided by zero builds upon the idea that a number can be both different from zero and identical to it (an infinitesimal), which is a patent contradiction -- a contradiction that resides in the very heart of calculus -- which made Newton at first avoid championing it explicitly.

Now considering that modern mathematicians accept a number that is at once different from and identical to zero, it shouldn't be that much of a problem to accept that any truth must be false and any falsehood must be the truth it falsifies. However, the problem here is that we are no longer in the realm of mathematics -- which means mathematics does not account for everything, hence is no ultimate truth.

The division of zero by zero is any number, since:
1. Any number multiplied by zero results in zero.
2. Division is defined as the inverse of multiplication.

you seem to have covered zero fairly thoroughly i might add

What you term "variability" is nonsensical. To see why, consider division of any real number except zero by zero - and read Cantor on infinities.

But zero does not fall into the divisibility rules of 3,6, or 9. But it does fall into 5, 10, and 2. So I would say 0 goes into 0, one time.

guigus wrote:

because if we multiply two by zero then we also recover zero

I know what you are trying to say here, but when multiplying you are taking the number x number of times e.x. 2 times 3 would be 3 groups of 2, or 2 groups of 3 which equals 6. But in 2 times zero, the reason why we recover zero is because when you have zero groups of 2 what do you have but zero? You have zero groups, no groups.

When you divide a number it results in a smaller number, why is it that when you divide a number into zero you get the same number?

I disagree that zero by itself is not a number. It has value depending on how it it used. What is ground zero? It is the point at which a nuclear devise is activated. Zero also has value in programming computers. It's a value depending on how many zeros are used. We also hear of "less than zero." That is a value.

were zero not a number, negative numbers would not be numbers as well.

Quote:

were zero not a number, negative numbers would not be numbers as well.

what would they be?