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Yes, the set of rational numbers is closed under addition.

Q: Can the sum of two rational numbers always be written as a fraction?

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Yes. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.

Yes. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.

Yes. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.

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.

Fractions where both the numerator and divisor are rational numbers are always rational numbers.

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They are not. Sometimes they are irrational. Irrational numbers cannot be expressed as a fraction.

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.

Fractions where both the numerator and divisor are rational numbers are always rational numbers.

Rational numbers can always be expressed as fractions.

If the number can be expressed as a ratio of two integer (the second not zero) then the number is rational. However, it is not always a simple matter to prove that if you cannot find such a representation, then the number is not rational: it is possible that you have not looked hard enough!

Yes, as long as the two nonzero numbers are themselves rational. (Since a rational number is any number that can be expressed as the quotient of two rational numbers, or any number that can be written as a fraction using only rational numbers.) If one of the nonzero numbers is not rational, the quotient will most likely 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 are are three types of decimals: terminating, repeating and non-terminating/non-repeating. The first two are rational, the third is not.