8/2/2019 Before a Few Days Ago

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October 15, 2011

MY PHILOSOPHICAL ANALYSIS OF 0/0.

Before a few days ago, in an evening, I was just roaming arbitrarily around different websites in the

internet. Then suddenly I discovered myself on a page about number properties at MANHATTAN

GAMT FORUMS. Being mainly concerned about the properties of zero on the page Im describing, a

member of this forum, named thinkblue, had pointed out four properties of and raised a question on

zero as its below:

1. 0 is neither positive nor negative.

2. 0 is even.

3. 0 is a multiple for every integer. (*)

4. 0 is not a prime.

5. Is 0 a factor of every number? I think not but just checking. (*)

I believe, the points from 1 up to 4 are indisputable. But what made this discussion carry on was

point 5. According to Guest (a member of this forum who attended the discussion), though zero is a

multiple of every number, it is NOT a FACTOR of any number except zero. But Emily Sledge, an

instructor of ManhattanGMAT, argued - Zero is not a factor of anything; and thus, you cannot say

that 0/0 = 0 just because zero is the numerator. The denominator of 0 makes 0/0 undefined.

Well, this is, in short, all about the discussion in that page. Now as we know, point 3 is free of

doubts, I find it more logical to propose 0 must be a factor of 0. Lets look at the number line below:

The set of all possible numbers

-Infinity -4 -3 -2 -1 0 1 2 3 4 Infinity

Y X

This number line clearly shows that 0 is an integer and an element of the set of all possible numbersin between -Infinity up to Infinity. And as 0 is the multiple of every integer is true, it would also be

true that all the integers including 0 are factors of 0 only in the multiplicative form with 0 as its in

0x12 = 0. In this point I widely disagree with the instructor of ManhattanGMAT, Mrs. Emily Sledge,

though it puts me on the point of extreme uncertainty to cast doubts on such a person who scored

99th percentile on the GMAT, with an all-time high score of 790.

It often appears problematic to define 0/0. Some call it infinitive, others call it undetermined and the

others call it undefined. 0/0 is infinitive - is intuitively incorrect. The result of a division is infinitive

when and only when it doesnt terminate. For example 1/3 = 0.3333

Something is undetermined when you cant say which one would logically suit the best to a situation

at a given period of time. For example, if X says that he doesn t like Haque Chocolate Digestive

8/2/2019 Before a Few Days Ago

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October 15, 2011

MY PHILOSOPHICAL ANALYSIS OF 0/0.

Biscuits, you just cant logically determine whether X likes (or doesnt like) Globe Chocolate Digestive

Biscuits from the given statement. His liking or not liking GCDB is undetermined.

Now lets check what 0/0 is undetermined or undefined?

At the first glance, the equation that 0/0=0 is quite all right, because the divisor 0 x the quotient 0equals the dividend 0 and there remains 0 as remainder. Besides, one may argue: if 12/3=4 is

mathematically fine, then mathematicians should also be malleable to adjust with 0/0=0. In that

case, Id rather say 0/0=5 and 0/0=-7 and 0/0=(ANY REAL NUMBERS: positive, negative, rational,

irrational, integer) all seem to be logically correct, for the divisor 0 x the quotient 5 or 7 or any

number equals the dividend 0 and there remains 0 as remainder.

But to