Using rational approximations, what is an inequality statement comparing \pi and √(8)?

Question

Enter \ \textless \, \ \textgreater \, or =.

The inequality is: \pi \ \square \ √(8)

Answer 1

To compare \(\pi\) and \(\sqrt{8}\), we can use rational approximations:

1. We know that \(\pi\) is approximately equal to \(3.14\).
2. To calculate \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}. \] The approximate value of \(\sqrt{2}\) is about \(1.414\). Therefore, \[ 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828. \]

Now we compare:

• \(\pi \approx 3.14\)
• \(\sqrt{8} \approx 2.828\)

We can see that: \[ \pi > \sqrt{8} \]

Thus, the inequality statement is: \[ \pi \ > \ \sqrt{8} \]

So, the answer is: \(\pi \ \square \ \sqrt{8}\) where \(\square\) is \(>\).