To determine which of the given numbers is not a rational number, we first need to understand what qualifies a number as rational. A rational number is any 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].
Let's analyze each given number one by one:
1. Number: [tex]\( -5 \frac{4}{11} \)[/tex]
- This mixed number can be converted into an improper fraction:
[tex]\[
-5 \frac{4}{11} = -5 + \frac{4}{11}
\][/tex]
Convert [tex]\(-5\)[/tex] into a fraction:
[tex]\[
-5 = -\frac{55}{11}
\][/tex]
Now add the fractions:
[tex]\[
-\frac{55}{11} + \frac{4}{11} = -\frac{55 - 4}{11} = -\frac{51}{11}
\][/tex]
Since [tex]\(-\frac{51}{11}\)[/tex] is a fraction of two integers, it is a rational number.
2. Number: [tex]\( \sqrt{31} \)[/tex]
- The square root of 31 cannot be expressed as a fraction of two integers. It is a non-repeating and non-terminating decimal. Therefore, [tex]\(\sqrt{31}\)[/tex] is an irrational number.
3. Number: 7.608
- This is a terminating decimal number. Any terminating decimal can be written as a fraction of two integers:
[tex]\[
7.608 = \frac{7608}{1000}
\][/tex]
Simplify the fraction if necessary, but since it is in the form [tex]\( \frac{p}{q} \)[/tex] (where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers), 7.608 is a rational number.
4. Number: [tex]\( 18.4 \overline{6} \)[/tex]
- This is a repeating decimal. A repeating decimal can also be expressed as a fraction of two integers:
[tex]\[
18.4 \overline{6} = 18.466666... = 18 + 0.466666... = 18 + \frac{7}{15}
\][/tex]
Therefore, [tex]\( 18.4 \overline{6} \)[/tex] is a rational number as it can be converted to a fraction.
After evaluating, the only number that does not fit the definition of a rational number is [tex]\( \sqrt{31} \)[/tex].
Therefore, the number that is not a rational number is [tex]\( \sqrt{31} \)[/tex].
