Question

is the following number rational or irrational?
$-0.overline{32}$
choose 1 answer:
a rational
b irrational

Explanation:

Step1: Define rational numbers

A rational number can be written as $\frac{p}{q}$, where $p,q$ are integers, $q
eq0$.

Step2: Convert repeating decimal to fraction

Let $x = -0.\overline{32} = -0.323232...$
Multiply by 100: $100x = -32.323232...$
Subtract $x$ from $100x$:
$100x - x = -32.323232... - (-0.323232...)$
$99x = -32$
Solve for $x$: $x = -\frac{32}{99}$

Step3: Verify rational form

$-\frac{32}{99}$ fits $\frac{p}{q}$ with $p=-32$, $q=99$ (integers, $q
eq0$).

Answer:

A. Rational