Using rational approximations what is an inequality statement comparing pi and square root eight

Answer 1

To compare \(\pi\) and \(\sqrt{8}\) using rational approximations, we can start by calculating the approximate values of both.

1. Calculate \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \] Since \(\sqrt{2} \approx 1.414\), \[ 2\sqrt{2} \approx 2 \cdot 1.414 = 2.828 \]

2. Approximate the value of \(\pi\): \(\pi\) is approximately \(3.14159\).

Now, we can compare the two:

• \(\pi \approx 3.14159\)
• \(\sqrt{8} \approx 2.828\)

From these approximations, it is clear that: \[ \pi > \sqrt{8} \]

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