Answer :
Let's carefully analyze the given statements to determine which one is true when [tex]\( p \)[/tex] is an integer and [tex]\( q \)[/tex] is a nonzero integer.
### Statement A:
A rational number cannot be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is false because, by definition, a rational number is a number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
### Statement B:
All numbers can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is also false because not all numbers can be represented as fractions of two integers. Numbers that cannot be expressed in this way are known as irrational numbers (for example, [tex]\(\sqrt{2}\)[/tex] or [tex]\(\pi\)[/tex]).
### Statement C:
A rational number can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is true because, by definition, a rational number is any number that can be expressed as a quotient or fraction [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0\)[/tex]. This aligns perfectly with the definition of a rational number.
### Statement D:
An irrational number can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is false because, by definition, an irrational number cannot be written as a simple fraction [tex]\(\frac{p}{q}\)[/tex]. Irrational numbers have non-repeating, non-terminating decimal expansions and cannot be expressed as the ratio of two integers.
### Conclusion:
The true statement is:
C. A rational number can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
### Statement A:
A rational number cannot be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is false because, by definition, a rational number is a number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
### Statement B:
All numbers can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is also false because not all numbers can be represented as fractions of two integers. Numbers that cannot be expressed in this way are known as irrational numbers (for example, [tex]\(\sqrt{2}\)[/tex] or [tex]\(\pi\)[/tex]).
### Statement C:
A rational number can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is true because, by definition, a rational number is any number that can be expressed as a quotient or fraction [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0\)[/tex]. This aligns perfectly with the definition of a rational number.
### Statement D:
An irrational number can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
- This statement is false because, by definition, an irrational number cannot be written as a simple fraction [tex]\(\frac{p}{q}\)[/tex]. Irrational numbers have non-repeating, non-terminating decimal expansions and cannot be expressed as the ratio of two integers.
### Conclusion:
The true statement is:
C. A rational number can be written as a fraction [tex]\(\frac{p}{q}\)[/tex].
