Answer :
To determine which number among the given choices is irrational, let's examine each one step-by-step.
### Definition of Rational and Irrational Numbers
1. Rational Numbers: These 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]\(q \neq 0\)[/tex]. Examples include 0.5, -3, [tex]\(\frac{3}{4}\)[/tex], etc.
2. Irrational Numbers: These cannot be expressed as a fraction of two integers. Their decimal expansions are non-repeating and non-terminating. Examples include [tex]\(\pi\)[/tex], [tex]\(e\)[/tex], and [tex]\(\sqrt{2}\)[/tex].
### Analyzing Each Choice
#### Choice A: [tex]\(\sqrt{7}\)[/tex]
- The square root of 7 is approximately 2.6457513110645907.
- Since 7 is not a perfect square, [tex]\(\sqrt{7}\)[/tex] cannot be expressed as a fraction of two integers.
- Therefore, [tex]\(\sqrt{7}\)[/tex] is an irrational number.
#### Choice B: 0.8
- 0.8 is a terminating decimal.
- It can be expressed as the fraction [tex]\(\frac{4}{5}\)[/tex].
- Since it can be written as a fraction, 0.8 is a rational number.
#### Choice C: [tex]\(0.333\ldots\)[/tex] (repeating decimal)
- [tex]\(0.333\ldots\)[/tex] is a repeating decimal, which can be expressed as [tex]\(\frac{1}{3}\)[/tex].
- Since it can be written as a fraction, it is a rational number.
#### Choice D: [tex]\(0.020202\ldots\)[/tex] (repeating decimal)
- [tex]\(0.020202\ldots\)[/tex] is a repeating decimal, which can be written as [tex]\(\frac{2}{99}\)[/tex].
- Since it can be written as a fraction, it is a rational number.
### Conclusion
Based on the analysis above, the irrational number among the given choices is:
A. [tex]\(\sqrt{7}\)[/tex]
### Definition of Rational and Irrational Numbers
1. Rational Numbers: These 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]\(q \neq 0\)[/tex]. Examples include 0.5, -3, [tex]\(\frac{3}{4}\)[/tex], etc.
2. Irrational Numbers: These cannot be expressed as a fraction of two integers. Their decimal expansions are non-repeating and non-terminating. Examples include [tex]\(\pi\)[/tex], [tex]\(e\)[/tex], and [tex]\(\sqrt{2}\)[/tex].
### Analyzing Each Choice
#### Choice A: [tex]\(\sqrt{7}\)[/tex]
- The square root of 7 is approximately 2.6457513110645907.
- Since 7 is not a perfect square, [tex]\(\sqrt{7}\)[/tex] cannot be expressed as a fraction of two integers.
- Therefore, [tex]\(\sqrt{7}\)[/tex] is an irrational number.
#### Choice B: 0.8
- 0.8 is a terminating decimal.
- It can be expressed as the fraction [tex]\(\frac{4}{5}\)[/tex].
- Since it can be written as a fraction, 0.8 is a rational number.
#### Choice C: [tex]\(0.333\ldots\)[/tex] (repeating decimal)
- [tex]\(0.333\ldots\)[/tex] is a repeating decimal, which can be expressed as [tex]\(\frac{1}{3}\)[/tex].
- Since it can be written as a fraction, it is a rational number.
#### Choice D: [tex]\(0.020202\ldots\)[/tex] (repeating decimal)
- [tex]\(0.020202\ldots\)[/tex] is a repeating decimal, which can be written as [tex]\(\frac{2}{99}\)[/tex].
- Since it can be written as a fraction, it is a rational number.
### Conclusion
Based on the analysis above, the irrational number among the given choices is:
A. [tex]\(\sqrt{7}\)[/tex]
