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It follows from the closure of integers under addition and multiplication.

Any rational number can be expressed as a ratio of two integers, where the second is not zero. So two rational numbers may be expressed as p/q and r/s.

A common multiple of their denominators is qs. So the numbers could also have been expressed as ps/qs and qr/qs.

Both these have the same denominator so their sum is (ps + qr)/qs.

Now, because the set of integers is closed under multiplication, ps, qr and qs are integers and because the set of integers is closed under addition, ps + qr is an integer.

Thus the sum has been expressed as a ratio of two integers, ps + qr, and qs and so it is a rational number.

Q: Why is the multiplication of rational numbers always result in a rational number?

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Yes. Multiplication of integers, of rational numbers, of real numbers, and even of complex numbers, is both commutative and associative.

The product of two rational numbers is always a rational number.

The product of two rational numbers is always a rational number.

The way in which the binary functions, addition and multiplication, are defined on the set of rational numbers ensures that the set is closed under these two operations.

Dividing by a rational number (other than zero) is simply multiplication by its reciprocal.

Related questions

A rational number is not. But the set of ALL rational numbers is.

Rational numbers are closed under multiplication, because if you multiply any rational number you will get a pattern. Rational numbers also have a pattern or terminatge, which is good to keep in mind.

No. The set of rational numbers is closed under addition (and multiplication).

It the combination is multiplication and the rational number is 0, then the result is rational. Otherwise it is irrational.

The product of two rational numbers is always a rational number.

Yes. Multiplication of integers, of rational numbers, of real numbers, and even of complex numbers, is both commutative and associative.

The product of two rational numbers is always a rational number.

No, it cannot. The product of a rational and irrational is always irrational. And half a number is equivalent to multiplication by 0.5

No, and I can prove it: -- The product of two rational numbers is always a rational number. -- If the two numbers happen to be the same number, then it's the square root of their product. -- Remember ... the product of two rational numbers is always a rational number. -- So the square of a rational number is always a rational number. -- So the square root of an irrational number can't be a rational number (because its square would be rational etc.).

The way in which the binary functions, addition and multiplication, are defined on the set of rational numbers ensures that the set is closed under these two operations.

Dividing by a rational number (other than zero) is simply multiplication by its reciprocal.

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