The number [tex]\(\sqrt{3}\)[/tex] is not a rational number because it cannot be expressed as a ratio of two integers [tex]\( \frac{a}{b} \)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]. By definition, a rational number can be written in the form [tex]\(\frac{a}{b}\)[/tex], and its decimal representation either terminates or repeats after a fixed number of digits.
However, the decimal expansion of [tex]\(\sqrt{3}\)[/tex] is approximately 1.732050807568877..., and it goes on forever without repeating. This non-repeating, non-terminating nature of the decimal expansion is a key characteristic of irrational numbers.
Since [tex]\(\sqrt{3}\)[/tex] cannot be accurately expressed as a ratio of two integers and its decimal form does not repeat, it classifies as an irrational number.
Thus, the statement "The number [tex]\(\sqrt{3}\)[/tex] goes on forever with no repeating pattern; therefore, it is rational." is
B. False.
