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Re: What is zero divided by zero anyway?

by jynx (Priest)
on Oct 08, 2002 at 17:28 UTC ( #203698=note: print w/replies, xml ) Need Help??

in reply to What is zero divided by zero anyway?

Maybe this will help, maybe it will add to the confusion that this post has become. Who knows...

The easy explanation comes from back when i first thought of this as well. i asked a teacher. According to her, zero does not meet the requirements to be in the denominator of any fraction so it should just be avoided. That's the easy-peasy explanation, feel free to avoid the mathematics below if you want...

So, i've been thinking about this for a bit, and eventually i got back to the division algorithm. Let's first take the case of a/b, where a is non-zero and b is zero. According to the division algorithm, there are numbers q and r such that

a = b*q + r and 0 <= r <= b (where b in this case is zero).
But in the case of a/0, r will be a. So we have
a = b*q + r but not r = 0
(which is required in this case) which is against the definition. So we can't divide most things by zero because at a fundamental level it breaks the rules.

Zero over zero is a special case of the above, because r will fall into the appropriate range (it will be zero) but the fraction is still not defined. Why is that? Let's take a closer look...

Let us take the fraction 0/0. Let us attempt to obtain a value for this fraction. It is of the form a/a so it is clearly not reduced (any fraction with common factors in the numerator and denominator can be reduced, and in the case of a/a the fraction is usually reduced by whatever the number a is). We then take some number d in the integers to reduce this fraction by, but we must choose d carefully. When we're done we want the fraction to be in it's most reduced state. So we want to find d such that

d = gcd(0,0).
Note that no such number exists*. You can continue using larger and larger numbers, but you will never find one (infinity is not a number but a useful concept; it is unusable except in limits). Since we cannot find a number to reduce this fraction by it is irreducable, but it is not reduced. This is a contradiction, so i am forced to conclude that there is no value for this fraction.

This may or may not be the proof,
but hopefully it will shed a little light (if nothing else, it's convinced me :-)

*: gcd is a function that takes two integer numbers and returns the greatest common divisor of those numbers.

update: put in some formatting and refined some of the definitions. Hopefully this will be readable by non-mathematicians as well...

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