Answer :
To determine which of the given numbers is not a rational number, we first need to understand what a rational number is. A rational number can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers [tex]\(p\)[/tex] and [tex]\(q\)[/tex], where [tex]\(q\)[/tex] is not zero.
Let's analyze each number step by step.
### 1. [tex]\(-5 + \frac{4}{11}\)[/tex]
[tex]\(-5 + \frac{4}{11}\)[/tex] is a sum of an integer and a fraction.
- [tex]\(-5\)[/tex] is an integer.
- [tex]\(\frac{4}{11}\)[/tex] is a fraction where the numerator is 4 and the denominator is 11.
Adding these together, you get [tex]\(\frac{-55}{11} + \frac{4}{11} = \frac{-55 + 4}{11} = \frac{-51}{11}\)[/tex], which is a fraction of two integers. Therefore, this number is rational.
### 2. [tex]\(\sqrt{31}\)[/tex]
The square root of 31 is an irrational number because 31 is not a perfect square. It cannot be expressed as a fraction of two integers. So [tex]\(\sqrt{31}\)[/tex] is irrational.
### 3. 7.608
The number 7.608 is a terminating decimal, which means it can be written as a fraction.
[tex]\[ 7.608 = \frac{7608}{1000} \][/tex]
Since both 7608 and 1000 are integers, this fraction can be simplified, but it will still be a quotient of two integers. Therefore, 7.608 is rational.
### 4. [tex]\(18.4\overline{6}\)[/tex]
[tex]\(18.4\overline{6}\)[/tex] represents a repeating decimal, which can be expressed as a fraction.
Let [tex]\(x = 18.4666666\cdots\)[/tex].
Multiplying both sides by 10, we get:
[tex]\[ 10x = 184.6666666\cdots \][/tex]
Subtracting the original equation from this:
[tex]\[ 10x - x = 184.6666666\cdots - 18.4666666\cdots \][/tex]
[tex]\[ 9x = 166.2 \][/tex]
[tex]\[ x = \frac{166.2}{9} \][/tex]
Since this fraction represents our repeating decimal, [tex]\(18.4\overline{6}\)[/tex] is a rational number.
### Conclusion
The irrational number in the list is:
[tex]\[ \boxed{\sqrt{31}} \][/tex]
Let's analyze each number step by step.
### 1. [tex]\(-5 + \frac{4}{11}\)[/tex]
[tex]\(-5 + \frac{4}{11}\)[/tex] is a sum of an integer and a fraction.
- [tex]\(-5\)[/tex] is an integer.
- [tex]\(\frac{4}{11}\)[/tex] is a fraction where the numerator is 4 and the denominator is 11.
Adding these together, you get [tex]\(\frac{-55}{11} + \frac{4}{11} = \frac{-55 + 4}{11} = \frac{-51}{11}\)[/tex], which is a fraction of two integers. Therefore, this number is rational.
### 2. [tex]\(\sqrt{31}\)[/tex]
The square root of 31 is an irrational number because 31 is not a perfect square. It cannot be expressed as a fraction of two integers. So [tex]\(\sqrt{31}\)[/tex] is irrational.
### 3. 7.608
The number 7.608 is a terminating decimal, which means it can be written as a fraction.
[tex]\[ 7.608 = \frac{7608}{1000} \][/tex]
Since both 7608 and 1000 are integers, this fraction can be simplified, but it will still be a quotient of two integers. Therefore, 7.608 is rational.
### 4. [tex]\(18.4\overline{6}\)[/tex]
[tex]\(18.4\overline{6}\)[/tex] represents a repeating decimal, which can be expressed as a fraction.
Let [tex]\(x = 18.4666666\cdots\)[/tex].
Multiplying both sides by 10, we get:
[tex]\[ 10x = 184.6666666\cdots \][/tex]
Subtracting the original equation from this:
[tex]\[ 10x - x = 184.6666666\cdots - 18.4666666\cdots \][/tex]
[tex]\[ 9x = 166.2 \][/tex]
[tex]\[ x = \frac{166.2}{9} \][/tex]
Since this fraction represents our repeating decimal, [tex]\(18.4\overline{6}\)[/tex] is a rational number.
### Conclusion
The irrational number in the list is:
[tex]\[ \boxed{\sqrt{31}} \][/tex]
