To solve the problem of finding two irrational numbers between [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{60}\)[/tex], we can break it down into the following steps:
1. Calculate the numerical values of the boundaries:
- [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{60}\)[/tex] are our bounds.
- The approximate value of [tex]\(\sqrt{2}\)[/tex] is 1.4142135623730951.
- The approximate value of [tex]\(\sqrt{60}\)[/tex] is 7.745966692414834.
2. Identify two irrational numbers between these bounds:
- We need to find two irrational numbers that lie between 1.4142135623730951 and 7.745966692414834.
3. First Irrational Number:
- Consider [tex]\(\sqrt{3}\)[/tex].
- The approximate value of [tex]\(\sqrt{3}\)[/tex] is 1.7320508075688772.
- Check the boundary condition: 1.4142135623730951 < 1.7320508075688772 < 7.745966692414834. This is true since [tex]\(\sqrt{3}\)[/tex] lies between [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{60}\)[/tex].
4. Second Irrational Number:
- Consider [tex]\(\frac{\pi}{2}\)[/tex].
- The approximate value of [tex]\(\frac{\pi}{2}\)[/tex] is 1.5707963267948966.
- Check the boundary condition: 1.4142135623730951 < 1.5707963267948966 < 7.745966692414834. This is true since [tex]\(\frac{\pi}{2}\)[/tex] lies between [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{60}\)[/tex].
Based on these calculations:
- The first irrational number is [tex]\(\sqrt{3} \approx 1.7320508075688772\)[/tex].
- The second irrational number is [tex]\(\frac{\pi}{2} \approx 1.5707963267948966\)[/tex].
Thus, the two irrational numbers that lie between [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{60}\)[/tex] are:
- [tex]\(\sqrt{3} \approx 1.7320508075688772\)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.5707963267948966\)[/tex]
