Question

is 1.222... an irrational number

Answer 1

The number \( 1.222\ldots \), where the digit 2 repeats indefinitely, is a rational number. It can be expressed as the fraction \( \frac{11}{9} \).

To see why it's rational, we can convert the repeating decimal into a fraction:

Let \( x = 1.222\ldots \).

This can be rewritten as:
\[
x = 1.222222\ldots
\]

Next, we can multiply both sides by 10 to shift the decimal point one place to the right:
\[
10x = 12.222222\ldots
\]

Now we subtract the first equation from this new equation:
\[
10x - x = 12.222222\ldots - 1.222222\ldots
\]
\[
9x = 11
\]

Dividing both sides by 9 gives:
\[
x = \frac{11}{9}
\]

Since \( \frac{11}{9} \) is a ratio of two integers, \( 1.222\ldots \) is a rational number, not an irrational number.