Question When multiplying two rational numbers, the product is always a rational number.

Studdy Solution

STEP 1

Assumptions1. We are dealing with rational numbers. Rational numbers are numbers that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. . We are considering the product of two rational numbers.

STEP 2

Let's consider two general rational numbers for our analysis. We can represent them as $a = \frac{p}{q}$ $b = \frac{r}{s}$ where p, q, r, and s are integers, and q and s are not equal to zero.

STEP 3

The product of these two rational numbers a and b would be $a \times b = \frac{p}{q} \times \frac{r}{s}$

STEP 4

By the rules of multiplication of fractions, we multiply the numerators together and the denominators together $a \times b = \frac{p \times r}{q \times s}$

STEP 5

The product of two integers (pr and qs) is always an integer. Therefore, the result is a fraction of two integers, with the denominator not equal to zero. This is the definition of a rational number.
Therefore, the product of two rational numbers is always a rational number. The correct answer is A. a rational number.

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