is 0.123124125…. rational or irrational

Answer 1

The number 0.123124125... is irrational.

To prove this, you would need to assume it is rational, which means it can be expressed as a fraction of two integers. Let's assume it can be written as p/q, where p and q are integers and q ≠ 0.

Since the decimal part of the number is non-repeating (0.123124125...), multiplying it by 1000 would give us:

1000 * 0.123124125... = 123.124125...

Now we have:

1000 * 0.123124125... = 123.124125...
- 0.123124125...
_________________________
= 123

From this, we can see that the difference between the initial number (0.123124125...) and its multiplication by 1000 (123.124125...) is 123. This means that we can rewrite the initial number as a difference between two integers:

0.123124125... = 123.124125... - 123

Since 123.124125... and 123 are both integers, we have expressed the initial number as a difference between two integers, which contradicts our initial assumption that the number can be written as a fraction of two integers. Therefore, 0.123124125... is irrational.