Find solutions to your problems with the help of IDNLearn.com's knowledgeable users. Ask any question and get a thorough, accurate answer from our community of experienced professionals.
Sagot :
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.
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.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Your questions deserve reliable answers. Thanks for visiting IDNLearn.com, and see you again soon for more helpful information.
