Question

0.123124125…. rational or irrational

Answer 1

To determine whether the number 0.123124125... is rational or irrational, we need to analyze its decimal representation.

The number appears to have a repeating pattern, as the digits 123 are repeated. This suggests that it could be a rational number.

To confirm this, we can create an equation using the given number:

Let x = 0.123124125...
Then, 1000x = 123.124125...

Subtracting x from both sides,
1000x - x = 123.124125... - 0.123124125...
999x = 123.001001...

Dividing both sides by 999,
x = 123.001001... / 999

Now, we can simplify the fraction 123.001001... / 999:

Let y = 123.001001...
Then, 1000y = 123001.001001...

Subtracting y from both sides,
1000y - y = 123001.001001... - 123.001001...
999y = 122878

Dividing both sides by 999,
y = 122878 / 999 = 123.001001...

This shows that y is equal to 123.001001..., which is the same as x. Therefore, the number 0.123124125... is rational.