Why the Sum or Product of Two Rational Numbers is Always a Rational Number: Understanding Rational Number Operations

sum or product of two rational numbers is

The sum or product of two rational numbers is also a rational number

The sum or product of two rational numbers is also a rational number.

To understand why this is true, let’s start by defining rational numbers. Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. In other words, a rational number can be written in the form a/b, where a and b are integers (b ≠ 0).

Now, let’s consider the sum of two rational numbers, say p/q and r/s, where p, q, r, and s are integers. To find the sum, we can first find a common denominator for q and s, and then add the numerators:

(p/q) + (r/s) = (ps + qr)/(qs)

Since ps + qr and qs are integers (since the sum of two integers is an integer), the sum of p/q and r/s is also a rational number, specifically (ps + qr)/(qs).

Now let’s consider the product of two rational numbers, p/q and r/s:

(p/q) * (r/s) = (pr)/(qs)

Again, pr and qs are integers (since the product of two integers is an integer), so the product of p/q and r/s is also a rational number, specifically (pr)/(qs).

Therefore, whether you add or multiply two rational numbers, the result will always be a rational number.

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