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The value of the denominator cannot be zero.

Q: Why must you always be mindful of the final value of the denominator in a rational expression?

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Yes

A rational number is defined as a number that can be expressed as the ratio(fraction) of two integers.So if you see a fraction that you know is a rational number, and the denominator isn'tan integer, there's always something you can do to simplify it, without changing its value,so that the denominator and the numerator both areintegers.

5.01001000100001... is not a rational number. Rational numbers will always repeat when written in a digital form. Since it is not rational, it cannot be written as a fraction with integer numerator and denominator.

There is no fraction that is irrational. A rational number is defined as the division of two integers, with the denominator not being zero, so a division (or fraction) is always rational, not irrational.

Statement 1 is true but totally unnecessary. As integer is always a rational and you do not need to convert it to a fraction to determine whether or not it is rational. A negative fraction is can be rational or irrational. The fact that it is negative is irrelevant to its rationality. An integer number over a zero denominator is not defined and so cannot be rational or irrational or anything. It just isn't.

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Yes it is. That is the definition of rational numebrs.

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Get rid of the denominator.

A rational number is defined as a number that can be expressed as the ratio(fraction) of two integers.So if you see a fraction that you know is a rational number, and the denominator isn'tan integer, there's always something you can do to simplify it, without changing its value,so that the denominator and the numerator both areintegers.

5.01001000100001... is not a rational number. Rational numbers will always repeat when written in a digital form. Since it is not rational, it cannot be written as a fraction with integer numerator and denominator.

It is more likely that it will be irrational.

Always true. (Never forget that whole numbers are rational numbers too - use a denominator of 1 yielding an improper fraction of the form of all rational numbers namely a/b.)

There is no fraction that is irrational. A rational number is defined as the division of two integers, with the denominator not being zero, so a division (or fraction) is always rational, not irrational.

Statement 1 is true but totally unnecessary. As integer is always a rational and you do not need to convert it to a fraction to determine whether or not it is rational. A negative fraction is can be rational or irrational. The fact that it is negative is irrelevant to its rationality. An integer number over a zero denominator is not defined and so cannot be rational or irrational or anything. It just isn't.

Such a sum is always rational.

Yes. This is the same as asking for one rational number to be subtracted from another; to do this each rational number is made into an equivalent rational number so that the two rational numbers have the same denominator, and then the numerators are subtracted which gives a rational number which may possibly be simplified.