Question

Which numbers are​ rational? The​ dots, . . . ​, indicate that the pattern continues. I. 9.9999999 . . . II. 9.234 III. 9.90990999 . . . Which numbers are​ rational?

Answer 1

All the numbers provided—9.9999999..., 9.234, and 9.90990999...—are rational because they either have a repeating pattern or terminate, allowing them to be expressed as the quotient of two integers.

Step-by-step explanation:

The question asks which of the given numbers are rational. Rational numbers can be written as the quotient of two integers, and they either terminate or have a repeating pattern after the decimal point.

• I. 9.9999999 . . . is rational because it represents the infinite decimal expansion of the number 10. The repeating pattern of 9s can be written as a fraction: 10/1.
• II. 9.234 is also a rational number because it is a terminating decimal that can be written as a fraction: 9234/1000.
• III. 9.90990999 . . . might seem tricky, but it's also rational as it has a repeating pattern. The portion '909' repeats indefinitely, which means it can be written in the form of a fraction, although the exact fraction would require a more detailed calculation to determine.

In conclusion, all three numbers given in the question are rational.

Answer 2

II. 9.234 it ends rather than continuous numbers