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.
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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.