Dividing by zero

[CleanShavenJesus]

The "what is zero" thread has a video explaining it

http://atheistforums.org/thread-20325.html

Yeah, watch that video the really cool guy posted.

[FifthElement]

In algebra (where division is defined as opposite of multiplication) division with 0 is indeed undefined but the limit of 1/n as n approaches zero is infinity.

Weird, isn't it ?

[LostLocke]

I'll try to explain it with salami sandwiches (simply because I am eating one now)

Let's say that I have four sandwiches and I want to share it between four people.....

This whole example was great.
Nice and easy to understand using real world examples.

The one flaw in this set up was that, at the end, instead of trying to divide 4 sandwiches among 0 people, your example should have ended trying to divide 0 sandwiches among 4 people, because we all know that while the 4 mathematicians are discussing the 'divide by zero' concept, some idiot will come along and eat all the sandwiches...

[Faith No More]

Of course a more rote demonstration is to show that the answer to 20/5 is 4 and the test of that is that 4X5 = 20. Likewise 20/2 = 10 and it passes the test: 2X10 = 20. But if 20/0 = 0, then 0X0 would have to = 20. To say that the answer is infinity abuses the idea of infinity by treating it as a specific number, and anyway no very large number multiplied by zero will give you 20 (or 1 or any other number you care to divide by zero.)

While the salami sandwich is a nice, easy to digest example, I prefer whateverist's here, because it demonstrates the mathematical impossibility of dividing by zero equalling zero.

[pineapplebunnybounce]

In algebra (where division is defined as opposite of multiplication) division with 0 is indeed undefined but the limit of 1/n as n approaches zero is infinity.

Weird, isn't it ?

that is because dividing anything with a decimal (which can be written as 1/x) is actually multiplying it with the denominator of that fraction (x). so 1/n as n approaches 0 gives you 1 divided by n=1/x where x is a very big number so that 1/x is almost 0. So you have 1/n divided by 1/x = 1 * x = x (approaches infinity).

It's like saying if a quarter of a person gets 1 apple, how many apples does a whole person get? so just 1 divided by 1/4, which is 1 x 4 = 4. So if 1/1000 of a person gets 1 apple, a whole person would get 1000 apples. The smaller the fraction the more apples the whole person gets. And the smaller the fraction the more it approaches 0.

Anyway, the sandwich analogy is great. I use it to explain chemistry a lot.

[bennyboy]

If I divide any number by zero, I will get a kind of procedural infinity. But if I reverse the process, and multiply infinity by zero, there are a couple ways to look at it:
-Zero times anything is zero. Period. Therefore, math is broken, because you're supposed to be able to reverse by inverse, so to speak.
-Somehow, multiplying infinity, and only infinity, times zero magically allows me to arrive at ANY value X. A little juggling, and you will discover that 1=2=3=n.

You could look at it philosophically, and say that the eternal dance between zero and infinity is the motive for all of creation, and that paradox is itself the fundamental building block of existence. Or you could just walk away.

[Simsim]

Most of what can be said has already been said. But I would like to add another thing.

There are cases when a denominator approaches zero but the total fraction doesn't approach infinity. This when the numerator can be algebraically analyzed into two parts one of them corresponds to the denominator.

Example:

(x^2 + x) / x , when x approaches zero the total fraction doesn't go to infinity. Because you can put the fraction in the following formula:

x*(x+1)/ x

By omitting x in the numerator with the x in the denominator the final result will be x+1 . Thus, when x approaches zero the polynomial (x+1) approaches 1 not infinity. Consequently, the fraction (x^2+x)/ x approaches 1 also when x approaches zero.

[bennyboy]

Simsim that's interesting, but I think it's still ambiguous. If the reduction is valid, you should be able to prove it by plugging your answer into the original formula-- which is obviously impossible.

[Simsim]

Simsim that's interesting, but I think it's still ambiguous. If the reduction is valid, you should be able to prove it by plugging your answer into the original formula-- which is obviously impossible.

Sorry, Bennyboy, this is the first time I log to the forum since you had posted your comment.

I am not sure I understand your question well but I hope this is the required answer:

There are two cases of dividing by zero:

1- The first is when numerator doesn't equal zero, i.e: a/0 , where "a" doesn't equal zero.
In this case the division operation has no meaning. You can't find a number which is when multiplied by zero gives another number other than zero. This is a hopeless case. i.e, there is no number "b" when multiplied by "0" gives the number "a" (if "a" doesn't equal zero)

2- The second is when the numerator equals zero like the denominator. Generally the result of dividing zero by zero is undefined. Why do we use the expression undefined?? Because there is an infinite number of numbers when multiplied by zero give zero. So every number can be a result of the division operation 0/0

In Limits mathematicians deal with these cases when a operation such dividing by zero is undefined. In stead of dividing by zero directly they observe the function's value when the denominator approaches zero and then they find that there is a definite value the function approaches when the denominator approaches zero. For instance in the example which I posted previously:

x*(x+1)/x

when you make x equal these values:
1, 0.9, 0.7, 0.5, 0.3, 0.1, 0.05, 0.01, 0.001, 0.0000000001

you find that the fraction (or function) equals these values respectively:

2, 1.9, 1.7, 1.5, 1.3, 1.1, 1.05, 1.01, 1.001, 1.0000000001

i.e when x approaches zero the total fraction approaches 1.

Since making such calculations every time is difficult the easy way is to reduce the fraction into a simpler formula. The reduction process should include the omission of the "zero" in the denominator (x in our case).

So: x*(x+1)/x = (x+1), then when x approaches zero (x+1) approaches 1

Q.E.D

Regards.

[Rational AKD]

rather than using limits, you can use simple algebra to show why you can't divide by zero.
first, take expression like 1/0 and make it into an equation.
1/0=x
this equation would be equivalent to 1=0x
0x equals zero according to the multiplicative property of zero
so you get 1=0 which is simply not the case.

but what about 0/0?
this would be equivalent to 0=0x
which then gets you 0=0
so in the equation it works, so can it be done? well, not really. the problem is we have to solve for x, so then we have to answer what is x? well, x can literally be anything because anything times zero equals zero. so you pretty much get 0/0= all real numbers, which simply can't be the case for a division operation which is why it is instead expressed as undefined.