Question

Using rational approximations, what is an inequality statement comparing π and √8? Enter <, >, or =.

Answer 1

To compare \( \pi \) and \( \sqrt{8} \), we can evaluate the numerical values of each.

First, we know that:

\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

We can use the approximate value of \( \sqrt{2} \). A common approximation is:

\[
\sqrt{2} \approx 1.414
\]

Thus, we can calculate \( 2\sqrt{2} \):

\[
\sqrt{8} \approx 2 \times 1.414 = 2.828
\]

Next, we use the known value of \( \pi \):

\[
\pi \approx 3.14159
\]

Now we can compare the two values:

\[
\sqrt{8} \approx 2.828
\]
\[
\pi \approx 3.14159
\]

Since \( 2.828 < 3.14159 \), we can conclude that:

\[
\sqrt{8} < \pi
\]

Therefore, the inequality statement comparing \( \pi \) and \( \sqrt{8} \) is:

\[
\sqrt{8} < \pi
\] or in the required form:

\[
\sqrt{8} < \pi
\]