Sure, let's determine which of these numbers is rational.
First, recall the definition of a rational number. A rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{a}{b}\)[/tex] of two integers, where [tex]\(a\)[/tex] (the numerator) and [tex]\(b\)[/tex] (the denominator) are integers, and [tex]\(b \neq 0\)[/tex].
Now, let's analyze each number provided:
1. [tex]\( -3 \frac{1}{2} \)[/tex]
This is a mixed number. A mixed number can be expressed as an improper fraction.
[tex]\[
-3 \frac{1}{2} = -\left(3 + \frac{1}{2}\right) = -\left(\frac{6}{2} + \frac{1}{2}\right) = -\frac{7}{2}
\][/tex]
Since [tex]\(-\frac{7}{2}\)[/tex] is a fraction of two integers, it is a rational number.
2. [tex]\( \sqrt{5} \)[/tex]
The square root of 5 is an irrational number because it cannot be expressed as a simple fraction. It is a non-repeating, non-terminating decimal. Thus, [tex]\( \sqrt{5} \)[/tex] is not rational.
3. [tex]\( \pi \)[/tex]
Pi ([tex]\(\pi\)[/tex]) is a well-known irrational number. It cannot be expressed as a fraction of two integers and its decimal representation is non-terminating and non-repeating. Therefore, [tex]\(\pi\)[/tex] is not a rational number.
4. [tex]\( 9.23157 \ldots \)[/tex]
The number [tex]\( 9.23157 \ldots \)[/tex] appears with an ellipsis, indicating it is a non-terminating decimal. Without any repeating pattern, it cannot be expressed as a fraction of two integers, making it an irrational number.
To summarize, the only rational number among the given options is:
[tex]\[
-3 \frac{1}{2}
\][/tex]
