QuestionClassify the number $-0 . \overline{25}$ as rational or irrational and provide your reasoning.

Studdy Solution

STEP 1

Assumptions1. The number we are examining is $-0 . \overline{25}$ , where the overline indicates that the25 is repeating. . A rational number can be expressed as a fraction $a/b$ , where $a$ and $b$ are integers and $b$ is not zero.
3. An irrational number cannot be expressed as a simple fraction.

STEP 2

We observe that $-0 . \overline{25}$ is a repeating decimal, which means it can be expressed as a fraction.

STEP 3

We can express $-0 . \overline{25}$ as a fraction by using the method for converting repeating decimals to fractions. Let's denote $-0 . \overline{25}$ as $x$ .
$x = -0 . \overline{25}$

STEP 4

Multiply $x$ by100 to shift the decimal point two places to the right. This gives us a new equation.
$100x = -25 . \overline{25}$

STEP 5

Subtract the original equation from the new one to eliminate the repeating part.
$100x - x = -25 . \overline{25} - (-0 . \overline{25})$

STEP 6

implify the left side of the equation.
$99x = -25 . \overline{25} +0 . \overline{25}$

STEP 7

implify the right side of the equation.
$99x = -25$

STEP 8

olve for $x$ by dividing both sides of the equation by99.
$x = -25 /99$

STEP 9

We see that $- . \overline{25}$ can be expressed as the fraction $-25 /99$ , where both $-25$ and $99$ are integers.

STEP 10

Therefore, $-0 . \overline{25}$ is a rational number, because it can be expressed as a fraction where the numerator and the denominator are integers and the denominator is not zero.
The number $-0 . \overline{25}$ is rational.

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