Sure! Let's determine which numbers in the given set are irrational.
We start with the given set of numbers:
[tex]\[
\left\{-4, 0, \frac{4}{5}, 0.44444, \sqrt{5}, \pi\right\}
\][/tex]
To classify these numbers, we need to recall the definition of irrational numbers. An irrational number is a number that cannot be expressed as a ratio of two integers, meaning it cannot be written as [tex]\( \frac{a}{b} \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers and [tex]\( b \neq 0 \)[/tex]. Additionally, irrational numbers have non-terminating and non-repeating decimal representations.
Let’s analyze each number in the set:
1. [tex]\(-4\)[/tex]: This is a whole number and can be written as [tex]\(-4/1\)[/tex]. Hence, [tex]\(-4\)[/tex] is a rational number.
2. [tex]\(0\)[/tex]: This is also a whole number and can be written as [tex]\(0/1\)[/tex]. Hence, [tex]\(0\)[/tex] is a rational number.
3. [tex]\(\frac{4}{5}\)[/tex]: This is already in the form of [tex]\( \frac{a}{b} \)[/tex] (with [tex]\( a = 4 \)[/tex] and [tex]\( b = 5 \)[/tex]), thus it is a rational number.
4. [tex]\(0.44444\)[/tex]: This is a terminating decimal number which can be expressed as a fraction. Thus, it is a rational number.
5. [tex]\(\sqrt{5}\)[/tex]: The square root of 5 is a non-terminating, non-repeating decimal and cannot be expressed as a ratio of two integers. Thus, [tex]\(\sqrt{5}\)[/tex] is an irrational number.
6. [tex]\(\pi\)[/tex]: Pi ([tex]\(\pi\)[/tex]) is a well-known irrational number with a non-terminating, non-repeating decimal expansion. Hence, [tex]\(\pi\)[/tex] is an irrational number.
Therefore, the irrational numbers in the given set are:
[tex]\[
\sqrt{5} \quad \text{and} \quad \pi
\][/tex]
So, the irrational numbers from the set [tex]\(\left\{-4,0, \frac{4}{5}, 0.44444, \sqrt{5}, \pi\right\}\)[/tex] are:
[tex]\[
\{2.23606797749979, 3.14159\}
\][/tex]
