To find an irrational number between [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{2}{2}\)[/tex] and verify the bounds, let's follow these steps:
1. Identify the boundaries in decimal form:
- [tex]\(\frac{1}{2}\)[/tex] in decimal form is [tex]\(0.5\)[/tex].
- [tex]\(\frac{2}{2}\)[/tex] in decimal form is [tex]\(1.0\)[/tex].
2. Determine an irrational number between these boundaries:
- One commonly known irrational number is [tex]\(\sqrt{2}\)[/tex], which is approximately [tex]\(1.414213562\)[/tex].
- To confine it within the boundaries of [tex]\(0.5\)[/tex] and [tex]\(1.0\)[/tex], we divide it by 2.
3. Calculate [tex]\(\frac{\sqrt{2}}{2}\)[/tex]:
- [tex]\(\frac{\sqrt{2}}{2} \approx \frac{1.414213562}{2} \approx 0.7071067811865476\)[/tex].
Therefore, an irrational number between [tex]\(0.5\)[/tex] and [tex]\(1.0\)[/tex] is approximately [tex]\(0.7071067811865476\)[/tex].
