Let $a$ and $b$ be rational numbers. Is $a \times b$ rational or irrational? $\newline$ Choose $1$ answer: $\newline$ (A) Rational $\newline$ (B) Irrational $\newline$ (C) It can be either rational or irrational

Full solution

Q. Let

a

and

b

be rational numbers. Is

a \times b

rational or irrational?

\newline

Choose

1

answer:

\newline

(A) Rational

\newline

(B) Irrational

\newline

Understand Rational Numbers: Step $1$ : Understand the properties of rational numbers. Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include $\frac{1}{2}$ , $-3$ , and $4$ .
Consider Multiplication of Rational Numbers: Step $2$ : Consider the multiplication of two rational numbers, $a$ and $b$ . Let $a = \frac{p}{q}$ and $b = \frac{r}{s}$ , where $p, q, r,$ and $s$ are integers and $q, s \neq 0$ .
Multiply Rational Numbers: Step $3$ : Multiply $a$ and $b$ . $a \times b = \left(\frac{p}{q}\right) \times \left(\frac{r}{s}\right) = \frac{p \times r}{q \times s}$ Since $p$ , $r$ , $q$ , and $s$ are all integers, and the product of integers is an integer, $p \times r$ and $q \times s$ are integers.
Check Result: Step $4$ : Check if the result is a rational number. $\newline$ The result $(p \times r) / (q \times s)$ is in the form of an integer divided by an integer (where the denominator is not zero). This is the definition of a rational number.