Question

Determine if 0, point, 0, 5, 4, 2, 2, 7, 4, 0, 6, 1, 1, 0, 1, 7, 7, dot dot dot0.054227406110177... is rational or irrational and give a reason for your answer.

Answer 1

To determine if the number given is rational or irrational, we need to analyze its decimal representation. The number you provided appears to have a pattern or sequence followed by "dot dot dot," which implies that the decimal continues infinitely.

However, it is difficult to ascertain from your description whether the decimal is terminating or repeating. If it is claimed that the decimal representation is "0.054227406110177...", the expression suggests that it does not repeat and continues infinitely without a clearly defined pattern at the end.

A number is defined as rational if it can be expressed as a fraction of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Rational numbers either have a terminating decimal expansion or a repeating decimal expansion.

Conversely, an irrational number cannot be expressed as such a fraction and has a non-terminating, non-repeating decimal expansion.

From the decimal sequence "0.054227406110177...", since there is no indication of a repeating pattern and given that it appears to be non-terminating, we can conclude:

The number represented by "0.054227406110177..." is an **irrational number**.