Question
Short-Answer Questions:
Explain why $3.\overline{1416}$ is a rational number.
Explain why $3.\overline{1416}$ is a rational number.
✓
Answer
The number of the form $\frac{\text{p}}{\text{q}},$ where p and q are integers and $\text{q}\neq0$ are called rational numbers.
Let $\text{x}=3.\overline{1416}$
⇒ x = 3.141614161416... ...(i)
Since there are four repeating digits,
We multiply by 1000
⇒ 1000x = 31416.14161416... ...(ii)
Subtracting (i) from (ii), we get
999x = 31416
$\Rightarrow\text{x}=\frac{31416}{999}$
Which is of the form $\frac{\text{p}}{\text{q}}.$
So, $3.\overline{1416}$ is a rational number.
Let $\text{x}=3.\overline{1416}$
⇒ x = 3.141614161416... ...(i)
Since there are four repeating digits,
We multiply by 1000
⇒ 1000x = 31416.14161416... ...(ii)
Subtracting (i) from (ii), we get
999x = 31416
$\Rightarrow\text{x}=\frac{31416}{999}$
Which is of the form $\frac{\text{p}}{\text{q}}.$
So, $3.\overline{1416}$ is a rational number.
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