Rational numbers are infinitely dense and so there is no such number. If for example, you considered F1 to be the fraction that was closet to 0 then what about half that number? F2 = (F1)/2 is closer to 0. And then what about F3 = (F2)/2? This could go on for ever.
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It is -1 or 1 because they are both integers
Fractions are infinitely dense and this means that between any two fractions there an infinite number of fractions. If any fraction, f, laid claims to being the nearest, there would be infinitely many fractions between 0 and f and so infinitely many fractions which were closer to 0. This means that f could not be the closest. The argument can be used again and again and so there cannot be a fraction closest to 0.
The integer/whole number closet to 0.1 is zero. If you can use decimal numbers, you will always be able to come up with a number closer to 0.1 than anyother number (exept 0.1 itself). For example 0.100000001 is very close to 0.1, but 0.10000000000000001 is even closer.
Yes, unless the number used for multiplication is 0. In that case you will have 0/0 which is not defined.
( 0 )/(any number) is.