Sure, let's determine which of the given numbers are rational.
What is a Rational Number?
A rational number is any number that can be written in the form of a 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 both terminating decimals and repeating decimals.
1. Number A: [tex]\(0.333 \ldots\)[/tex]
- The number [tex]\(0.333 \ldots\)[/tex] is a repeating decimal. Repeating decimals can always be expressed as fractions. For example, [tex]\(0.333 \ldots = \frac{1}{3}\)[/tex].
- Therefore, [tex]\(0.333 \ldots\)[/tex] is a rational number.
2. Number B: [tex]\(\sqrt{7}\)[/tex]
- The number [tex]\(\sqrt{7}\)[/tex] is the square root of a non-perfect square, which makes it an irrational number. Irrational numbers cannot be expressed as fractions of integers.
- Therefore, [tex]\(\sqrt{7}\)[/tex] is not a rational number, it is irrational.
3. Number C: 0.8
- The number 0.8 is a terminating decimal. Terminating decimals can always be written as fractions. For instance, [tex]\(0.8 = \frac{8}{10} = \frac{4}{5}\)[/tex].
- Therefore, 0.8 is a rational number.
4. Number D: [tex]\(\pi\)[/tex]
- The number [tex]\(\pi\)[/tex] (pi) is a well-known irrational number. Its decimal representation is non-terminating and non-repeating. It cannot be written as a fraction of integers.
- Therefore, [tex]\(\pi\)[/tex] is not a rational number, it is irrational.
Summary:
- [tex]\(0.333 \ldots\)[/tex] is rational.
- [tex]\(\sqrt{7}\)[/tex] is irrational.
- [tex]\(0.8\)[/tex] is rational.
- [tex]\(\pi\)[/tex] is irrational.
So the answer is:
- [tex]\(A\)[/tex] is rational.
- [tex]\(B\)[/tex] is not rational.
- [tex]\(C\)[/tex] is rational.
- [tex]\(D\)[/tex] is not rational.
