Determine if 0, point, 7, 6, 5, 3, 7, 6, 5, 3, 7, 6, 5, 3, 7, 6, 5, 3, point, point, .0.7653765376537653... is rational or irrational and give a reason for your answer.

Answer 1

To determine if the number \( 0.7653765376537653\ldots \) is rational or irrational, we need to analyze its decimal representation.

The decimal \( 0.7653765376537653\ldots \) appears to have a repeating pattern: it repeats the sequence "7653". This can be confirmed as follows:

1. Identify the repeating part: From the decimal notation, we see that the digits "7653" appear repeatedly.
2. Write the number as a fraction: Since the decimal representation repeats, we can express \( 0.7653765376537653\ldots \) as a fraction.

A decimal is considered rational if it can be expressed in the form \( \frac{a}{b} \) where \( a \) and \( b \) are integers (with \( b \neq 0 \)). A decimal representation is repeating or terminating when it is rational.

In this case, since the decimal \( 0.7653765376537653\ldots \) repeats, we conclude that it is a rational number.

Therefore, the number \( 0.7653765376537653\ldots \) is rational because it has a repeating decimal representation.