Answer :
To determine which number among the options is irrational, we need to first understand what irrational numbers are. An irrational number is a number that cannot be expressed as a fraction of two integers. It has a non-repeating, non-terminating decimal representation.
Let's examine each option:
### Option A: [tex]$\pi$[/tex]
- [tex]$\pi$[/tex] (Pi) is a well-known mathematical constant that represents the ratio of a circle's circumference to its diameter.
- The decimal representation of [tex]$\pi$[/tex] is approximately 3.1415926535..., and it continues infinitely without repeating.
- [tex]$\pi$[/tex] cannot be expressed as a fraction of two integers, hence it is an irrational number.
### Option B: 0.7
- 0.7 is a finite decimal.
- It can be expressed as the fraction [tex]\(\frac{7}{10}\)[/tex], which is a ratio of two integers.
- Therefore, 0.7 is a rational number.
### Option C: 0.277277277...
- The number 0.277277277... is a repeating decimal.
- This can be expressed as the fraction [tex]\(\frac{277}{999}\)[/tex] (or in another form after simplifying, but it is clearly a ratio of two integers).
- Thus, 0.277277277... is a rational number.
### Option D: [tex]$0.333...$[/tex]
- The number 0.333... is also a repeating decimal.
- This can be expressed as the fraction [tex]\(\frac{1}{3}\)[/tex].
- Therefore, 0.333... is a rational number.
From this analysis, we can conclude that the number [tex]$\pi$[/tex] in option A is the only one that is irrational. Hence, the answer to the question "Which number is irrational?" is:
A. [tex]\(\pi\)[/tex]
Let's examine each option:
### Option A: [tex]$\pi$[/tex]
- [tex]$\pi$[/tex] (Pi) is a well-known mathematical constant that represents the ratio of a circle's circumference to its diameter.
- The decimal representation of [tex]$\pi$[/tex] is approximately 3.1415926535..., and it continues infinitely without repeating.
- [tex]$\pi$[/tex] cannot be expressed as a fraction of two integers, hence it is an irrational number.
### Option B: 0.7
- 0.7 is a finite decimal.
- It can be expressed as the fraction [tex]\(\frac{7}{10}\)[/tex], which is a ratio of two integers.
- Therefore, 0.7 is a rational number.
### Option C: 0.277277277...
- The number 0.277277277... is a repeating decimal.
- This can be expressed as the fraction [tex]\(\frac{277}{999}\)[/tex] (or in another form after simplifying, but it is clearly a ratio of two integers).
- Thus, 0.277277277... is a rational number.
### Option D: [tex]$0.333...$[/tex]
- The number 0.333... is also a repeating decimal.
- This can be expressed as the fraction [tex]\(\frac{1}{3}\)[/tex].
- Therefore, 0.333... is a rational number.
From this analysis, we can conclude that the number [tex]$\pi$[/tex] in option A is the only one that is irrational. Hence, the answer to the question "Which number is irrational?" is:
A. [tex]\(\pi\)[/tex]
