To determine which of the given numbers is irrational, we need to understand the definitions of rational and irrational numbers.
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]. Rational numbers include fractions, terminating decimals, and repeating decimals.
An irrational number is any number that cannot be expressed as a simple fraction or ratio of two integers. These numbers have non-terminating, non-repeating decimal expansions.
Let's analyze each of the given options one by one:
1. [tex]\(\pi\)[/tex]
- [tex]\(\pi\)[/tex] (pi) is a well-known mathematical constant. It represents the ratio of the circumference of a circle to its diameter. The decimal expansion of [tex]\(\pi\)[/tex] is non-terminating and non-repeating (approx. 3.141592653589793...). Therefore, [tex]\(\pi\)[/tex] is an irrational number.
2. [tex]\(0.1 \overline{6}\)[/tex] (0.1666...)
- The notation [tex]\( 0.1 \overline{6} \)[/tex] represents a repeating decimal, which continues indefinitely. Repeating decimals can always be expressed as a fraction. For example, [tex]\( 0.1 \overline{6} = \frac{1}{6} \)[/tex]. Therefore, [tex]\( 0.1 \overline{6} \)[/tex] is a rational number.
3. 2.53
- The number 2.53 is a terminating decimal. Any terminating decimal can be expressed as a fraction. For example, [tex]\( 2.53 = \frac{253}{100} \)[/tex]. Therefore, 2.53 is a rational number.
4. [tex]\(-\frac{1}{5}\)[/tex]
- The number [tex]\(-\frac{1}{5}\)[/tex] is already given in fractional form. Therefore, it is automatically a rational number.
Based on the above analysis, the only irrational number among the given options is:
[tex]\[ \pi \approx 3.141592653589793. \][/tex]
