To determine the type of number that cannot be written as a fraction [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex], let's analyze the given options:
A. A decimal: Decimals can be divided into two categories:
- Terminating decimals: These can be represented as fractions, for example, [tex]\(0.5\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
- Repeating decimals: These can also be represented as fractions, for example, [tex]\(0.3333...\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
Since both kinds of decimals can be written as fractions, this option is not the correct answer.
B. An irrational number: By definition, an irrational number is a number that cannot be expressed as a simple fraction [tex]\(\frac{p}{q}\)[/tex]. Examples include [tex]\(\pi\)[/tex] and [tex]\(\sqrt{2}\)[/tex]. These numbers cannot be written as a fraction with integer numerator and denominator where the denominator is not zero.
C. A rational number: By definition, a rational number can be expressed as a fraction [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. Examples include numbers like [tex]\(2\)[/tex] (which can be written as [tex]\(\frac{2}{1}\)[/tex]) and [tex]\(\frac{4}{5}\)[/tex].
D. All numbers can be written in this way: This statement is incorrect since irrational numbers cannot be expressed as fractions.
Among the given options, the correct answer is:
B. An irrational number
Irrational numbers cannot be expressed in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
