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Suppose x and y are two rational numbers. Therefore x = p/q and y = r/s where p, q, r and s are integers and q and s are not zero.

Then x - y = p/q - r/s = ps/qs - qr/qs = (ps - qr)/qs

By the closure of the set of integers under multiplication, ps, qr and qs are all integers,

by the closure of the set of integers under subtraction, (ps - qr) is an integer,

and by the multiplicative properties of 0, qs is non zero.

Therefore (ps - qr)/qs satisfies the requirements of a rational number.

Q: Why is the difference between two rational numbers always a rational number?

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The question cannot be answered because it is nonsensical. The difference between two rational numbers is very very rarely a whole number.

Yes. The rational numbers are a closed set with respect to subtraction.

When you consider how many rational numbers there are, the difference between any two of them is hardly ever an integer. Examples: 5 - 4/5 = 41/5 5/6 - 2/3 = 1/6 3.274 - 1.368 = 1.906 All of the nine numbers in these examples are rational numbers.

No. sqrt(3) - sqrt(2) is irrational.

They are always rational numbers.

Related questions

no

Yes, it is.

No.

The question cannot be answered because it is nonsensical. The difference between two rational numbers is very very rarely a whole number.

Yes. The rational numbers are a closed set with respect to subtraction.

Yes, that's true.

When you consider how many rational numbers there are, the difference between any two of them is hardly ever an integer. Examples: 5 - 4/5 = 41/5 5/6 - 2/3 = 1/6 3.274 - 1.368 = 1.906 All of the nine numbers in these examples are rational numbers.

No. sqrt(3) - sqrt(2) is irrational.

No, it is not true.

No. 5 and 2 are real numbers. Their difference, 3, is a rational number.

They will always be rational numbers.

They are always rational numbers.