Question

Prove that if r and s are rational numbers, then r - s is a rational number.

Answer 1

To prove that if r and s are rational numbers, then r - s is a rational number, we can use the fact that the sum and difference of two rational numbers is always rational.

Step-by-step explanation:

In order to prove that if r and s are rational numbers, then r - s is a rational number, we can use the fact that the sum and difference of two rational numbers is always rational.

Let's assume that r = p/q and s = m/n, where p, q, m, and n are integers and q and n are not equal to zero.

Now, we can find r - s by subtracting the two fractions: r - s = (p/q) - (m/n) = (pn - mq) / (qn). Since pn - mq and qn are integers, r - s is also a rational number.

Answer 2

To prove that r - s is a rational number when r and s are rational numbers, we express them as a/b and c/d respectively, find a common denominator, and subtract to get (ad - bc)/bd, which is also rational.

Step-by-step explanation:

The question at hand requires proving that the difference of two rational numbers, r and s, is also a rational number. By definition, a rational number can be expressed as the quotient of two integers where the denominator is not zero. Let's assume r = a/b and s = c/d, where a, b, c, and d are all integers and b and d are not zero.

The difference of r and s is given by r - s = (a/b) - (c/d). To perform this subtraction, we need a common denominator, which would be bd. Therefore, r - s = (ad - bc) / bd. Since the numerator (ad - bc) and the denominator bd are also integers and the denominator isn't zero, the result is a rational number.

Our intuition regarding the addition and subtraction of fractions aligns with this formal proof, as we are familiar with finding a common denominator and then combining numerators to find the result.