Answer :
To determine which number is rational among the given options, let's analyze each one step-by-step:
A. [tex]\( 0.83587643 \ldots \)[/tex]
- This number appears to be a non-repeating, non-terminating decimal. Numbers of this form are typically irrational because they cannot be expressed as a fraction of two integers.
B. [tex]\( \sqrt{7} \)[/tex]
- The square root of a non-perfect square is known to be irrational. Since 7 is not a perfect square, [tex]\( \sqrt{7} \)[/tex] cannot be expressed as a fraction of two integers, hence it is irrational.
C. [tex]\( \pi \)[/tex]
- The number [tex]\( \pi \)[/tex] (pi) is a well-known irrational number. It is a non-repeating, non-terminating decimal and cannot be expressed as a fraction of two integers.
D. [tex]\( 0.333 \ldots \)[/tex]
- This number is a repeating decimal. Repeating decimals can be expressed as a fraction of two integers. For instance, [tex]\( 0.333 \ldots \)[/tex] can be written as [tex]\( \frac{1}{3} \)[/tex]. Therefore, it is a rational number.
Based on this analysis, the rational number among the given options is:
D. [tex]\( 0.333 \ldots \)[/tex]
A. [tex]\( 0.83587643 \ldots \)[/tex]
- This number appears to be a non-repeating, non-terminating decimal. Numbers of this form are typically irrational because they cannot be expressed as a fraction of two integers.
B. [tex]\( \sqrt{7} \)[/tex]
- The square root of a non-perfect square is known to be irrational. Since 7 is not a perfect square, [tex]\( \sqrt{7} \)[/tex] cannot be expressed as a fraction of two integers, hence it is irrational.
C. [tex]\( \pi \)[/tex]
- The number [tex]\( \pi \)[/tex] (pi) is a well-known irrational number. It is a non-repeating, non-terminating decimal and cannot be expressed as a fraction of two integers.
D. [tex]\( 0.333 \ldots \)[/tex]
- This number is a repeating decimal. Repeating decimals can be expressed as a fraction of two integers. For instance, [tex]\( 0.333 \ldots \)[/tex] can be written as [tex]\( \frac{1}{3} \)[/tex]. Therefore, it is a rational number.
Based on this analysis, the rational number among the given options is:
D. [tex]\( 0.333 \ldots \)[/tex]
