To determine the inequality between [tex]\(\pi\)[/tex] and [tex]\(\sqrt{8}\)[/tex] using rational approximations, let's follow these steps:
1. Approximate the value of [tex]\(\pi\)[/tex]:
- [tex]\(\pi\)[/tex] is a well-known irrational number, and its approximate value is [tex]\(3.141592653589793\)[/tex].
2. Approximate the value of [tex]\(\sqrt{8}\)[/tex]:
- To find the approximate value of the square root of 8, we can consider the fact that [tex]\(\sqrt{8}\)[/tex] can be simplified as [tex]\(\sqrt{4 \times 2}\)[/tex], which is further simplified to [tex]\(2\sqrt{2}\)[/tex].
- The value of [tex]\(\sqrt{2}\)[/tex] is approximately [tex]\(1.4142135623730951\)[/tex], so multiplying this by 2 gives us:
[tex]\[
2 \times 1.4142135623730951 = 2.8284271247461903
\][/tex]
3. Compare the two approximations:
- We have [tex]\(\pi \approx 3.141592653589793\)[/tex]
- We have [tex]\(\sqrt{8} \approx 2.8284271247461903\)[/tex]
- Clearly, [tex]\(3.141592653589793\)[/tex] is greater than [tex]\(2.8284271247461903\)[/tex].
4. State the inequality:
Therefore, we can state the inequality as:
[tex]\[
\pi > \sqrt{8}
\][/tex]
The inequality is: [tex]\(\pi\)[/tex] [tex]\(>\)[/tex] [tex]\(\sqrt{8}\)[/tex]
