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

Updated: 9/25/2023

Wiki User

8y ago

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.

Wiki User

7y ago

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Q: Why is the difference between two rational numbers always a rational number?
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no

Yes, it is.

No.

### 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.

### The difference of two rational numbers is always a rational number?

Yes, that's true.

### 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.

### Is is true the difference of two rational numbers is always negative?

No, it is not true.

### Is The difference of two real numbers always an irrational number?

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

### Will repeating decimals always or never be rational numbers?

They will always be rational numbers.

### Are terminating decimals always nether or sometimes a rational numbers?

They are always rational numbers.