To determine if Abneke was correct, we need to check whether the numbers she suggested (1.66666... or 1.6) fall between \(\sqrt{2}\) and \(\sqrt{3}\).
First, we calculate the approximate values of \(\sqrt{2}\) and \(\sqrt{3}\):
\[ \sqrt{2} \approx 1.414 \] \[ \sqrt{3} \approx 1.732 \]
Next, we can look at the value of 1.66666... (which is also written as \( \frac{5}{3} \) or approximately 1.6667) and see if it lies between 1.414 and 1.732.
Since \(1.414 < 1.66666... < 1.732\), we find that:
- \(1.66666... > \sqrt{2}\)
- \(1.66666... < \sqrt{3}\)
Thus, \(1.66666...\) is indeed a rational number that lies between \(\sqrt{2}\) and \(\sqrt{3}\).
Therefore, Abneke was correct to suggest 1.66666..., as it is a rational number between \(\sqrt{2}\) and \(\sqrt{3}\). However, 1.6 is less than 1.66666..., but it also falls between \(\sqrt{2}\) and \(\sqrt{3}\) since:
\[ 1.414 < 1.6 < 1.732 \]
In conclusion, both numbers she suggested (1.66666... and 1.6) are rational numbers between \(\sqrt{2}\) and \(\sqrt{3}\).