To determine which number among the given choices is irrational, we need to understand the definitions of rational and irrational numbers:
Rational Numbers: These are numbers that can be expressed as a fraction ([tex]\(\frac{a}{b}\)[/tex]) where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex]. Rational numbers include integers, finite decimals, and repeating decimals.
Irrational Numbers: These are numbers that cannot be expressed as a fraction of two integers. They are non-repeating and non-terminating decimals.
Let's analyze each choice one by one:
A. [tex]\(\sqrt{7}\)[/tex]
- The square root of 7 is not a perfect square and cannot be expressed as a fraction of two integers. It is a non-terminating and non-repeating decimal, which makes [tex]\(\sqrt{7}\)[/tex] an irrational number.
B. 0.8
- The number 0.8 is a finite decimal. It can be expressed as the fraction [tex]\(\frac{8}{10}\)[/tex] or [tex]\(\frac{4}{5}\)[/tex], which makes it a rational number.
C. [tex]\(0.333...\)[/tex]
- This number is a repeating decimal. It can be expressed as the fraction [tex]\(\frac{1}{3}\)[/tex], which makes it a rational number.
D. 0.020202...
- This number is a repeating decimal, specifically a repeating pair of "02". It can be expressed as a fraction: [tex]\(0.\overline{02} = \frac{2}{99}\)[/tex], which makes it a rational number.
Based on this analysis:
- [tex]\(\sqrt{7}\)[/tex] is irrational.
- 0.8, [tex]\(0.333...\)[/tex], and 0.020202... are all rational.
Therefore, the number that is irrational among the given choices is:
A. [tex]\(\sqrt{7}\)[/tex]
