Question

1. john says that, because the decimal number $0.overline{347}$ never terminates, it cannot be written as a fraction. is john correct? explain.

Explanation:

Step1: Assign variable to repeating decimal

Let $x = 0.\overline{347}$, so $x = 0.347347347...$

Step2: Multiply to shift decimal

Multiply $x$ by $1000$ (3 repeating digits):
$1000x = 347.347347347...$

Step3: Subtract to eliminate repeating part

Subtract original $x$ from $1000x$:
$1000x - x = 347.347347... - 0.347347...$
$999x = 347$

Step4: Solve for x (fraction form)

Isolate $x$:
$x = \frac{347}{999}$

Answer:

John is not correct. The repeating decimal $0.\overline{347}$ can be written as the fraction $\frac{347}{999}$, so non-terminating repeating decimals are rational numbers that can be expressed as fractions.