Yes, the difference between two rational numbers is always a rational number. To understand why, let’s first define what rational numbers are. A rational number is any number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. For example, numbers like 1/2, -3/4, and 5 are all rational.
Now, if we take two rational numbers, say a/b and c/d, where a, b, c, and d are integers and b and d are not zero, we can find their difference:
Difference = (a/b) – (c/d)
To subtract these two fractions, we need a common denominator, which would be bd. This gives us:
Difference = (ad/bd) – (bc/bd) = (ad – bc) / bd
Here, ad – bc is an integer (since the set of integers is closed under subtraction), and bd is also a non-zero integer (since both b and d are non-zero). Therefore, the result of the difference is in the form of an integer over a non-zero integer, which means it is a rational number.
In conclusion, regardless of the specific rational numbers you choose, their difference will always yield another rational number.