To determine between which pair of numbers there exists an irrational number that supports the idea that irrational numbers are dense in real numbers, let's analyze each pair step-by-step:
1. Pair: 3.14 and [tex]\(\pi\)[/tex]
- [tex]\(3.14\)[/tex] is a rational approximation of [tex]\(\pi\)[/tex].
- [tex]\(\pi\)[/tex] (pi) is an irrational number.
- Therefore, there are definitely irrational numbers between [tex]\(3.14\)[/tex] and [tex]\(\pi\)[/tex]. However, this pair is not the most direct example since one is rational and the other is already irrational.
2. Pair: 3.33 and [tex]\(\frac{1}{3}\)[/tex]
- Both [tex]\(3.33\)[/tex] (a repeating decimal approximating [tex]\(\frac{1}{3}\)[/tex]) and [tex]\(\frac{1}{3}\)[/tex] are rational numbers.
- Rational numbers do not contain irrational numbers between them by definition.
- Therefore, this pair does not provide an example of an irrational number between them.
3. Pair: [tex]\(e^2\)[/tex] and [tex]\(\sqrt{54}\)[/tex]
- [tex]\( e^2 \)[/tex] (Euler's number squared) is an irrational number.
- [tex]\(\sqrt{54}\)[/tex] (square root of 54) is also an irrational number.
- Since irrationals are dense in the real numbers, there must be an irrational number between these two.
- Computing these values:
- [tex]\( e^2 \approx 7.389 \)[/tex]
- [tex]\(\sqrt{54} \approx 7.348 \)[/tex]
- A value between these two is [tex]\( \frac{e^2 + \sqrt{54}}{2} \approx 7.369 \)[/tex], which is still an irrational number.
4. Pair: [tex]\(\frac{\sqrt{64}}{2}\)[/tex] and [tex]\(\sqrt{16}\)[/tex]
- [tex]\(\frac{\sqrt{64}}{2} = \frac{8}{2} = 4\)[/tex], which is a rational number.
- [tex]\(\sqrt{16} = 4\)[/tex], which is also a rational number.
- Therefore, there cannot be any irrational numbers between these two rational numbers.
Considering the analysis above, the pair [tex]\( e^2 \)[/tex] and [tex]\( \sqrt{54} \)[/tex] demonstrates the idea that there are irrational numbers between any two given irrational numbers. Specifically:
- [tex]\( e^2 \approx 7.389 \)[/tex]
- [tex]\( \sqrt{54} \approx 7.348 \)[/tex]
- An example of an irrational number between them is [tex]\(\frac{e^2 + \sqrt{54}}{2} \approx 7.369\)[/tex].
Hence, the pair [tex]\( e^2 \)[/tex] and [tex]\( \sqrt{54} \)[/tex] supports the idea that irrational numbers are dense in the real numbers.
