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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.
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Why is the difference of two rational numbers always a whole number?
The question cannot be answered because it is nonsensical. The difference between two rational numbers is very very rarely a whole number.
Is it true that The difference of two rational numbers always a rational number?
Yes. The rational numbers are a closed set with respect to subtraction.
Is the difference between two rational numbers always an integer?
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.
Would the difference between two irrational numbers is always going to be rational?
No. sqrt(3) - sqrt(2) is irrational.
Are terminating decimals always nether or sometimes a rational numbers?
They are always rational numbers.
