is 89.777... a rational number?

Answer 1

Yes, $89.777...$ is a rational number. A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero.

In this case, $89.777...$ can be represented as $89.777\bar{7}$, where the $7$ repeats indefinitely. This can be converted into a fraction.

Let $x=89.77777...$.

To eliminate the repeating part, we can multiply by 10 (to shift the decimal place one position to the right):

$$10x=897.77777...$$

Now, we can subtract the original $x$ from this equation:

$$10x-x=897.77777...-89.77777...$$ $$9x=808$$

Now, solving for $x$:

$$x=\frac{808}{9}$$

Since both 808 and 9 are integers and the denominator is not zero, $89.777...$ is indeed a rational number.