Answer :
To address the question of which number would support the idea that rational numbers are dense, let's first understand what it means for rational numbers to be dense:
### Explanation of Density of Rational Numbers
The density of rational numbers in the real number system implies that between any two distinct real numbers, there exists at least one rational number. In fact, there are infinitely many rational numbers between any two real numbers, making the rationals densely packed within the real number system.
### Evaluating Each Option
Now, let's go through each option provided:
1. A natural number between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{3}\)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex] and [tex]\(\frac{\pi}{3} \approx 1.0472\)[/tex]. There is no natural number (which are 1, 2, 3, ...) that lies between these two values. So this option does not demonstrate the density of rational numbers.
2. An integer between -11 and -10
- The integers are ...,-13,-12,-11,-10,... Between -11 and -10, there are no integers. Thus, this option also does not demonstrate the density of rational numbers.
3. A whole number between 1 and 2
- Whole numbers include non-negative integers (0, 1, 2, 3, ...). There is no whole number between 1 and 2. Therefore, this does not demonstrate the density of rational numbers either.
4. A terminating decimal between -3.14 and -3.15
- Terminating decimals are a subset of rational numbers. However, note that -3.14 and -3.15 themselves might create some confusion because they suggest a very small difference range using a typical decimal representation. To expressly show rational density, it would require more clear-cut divisions.
### Conclusion
To prove that rational numbers are dense, we particularly need to highlight the fact that there are rational numbers between any two real numbers. None of these options, however, provides a straightforward demonstration of this although the terminating decimal option closely relates to rational numbers but might not be clear-cut between -3.14 and -3.15.
Therefore, the best illustration of the density of rational numbers might not be immediately clear from these options, but in general, more specific ranges or descriptive examples are simpler to use in demonstrating the concept effectively in mathematical terms.
### Selection
None of the provided choices perfectly demonstrates the density property straightforwardly. The statement given supports and aims to clarify the common understanding of rational density, but precise examples within ranges are needed for definitiveness.
### Explanation of Density of Rational Numbers
The density of rational numbers in the real number system implies that between any two distinct real numbers, there exists at least one rational number. In fact, there are infinitely many rational numbers between any two real numbers, making the rationals densely packed within the real number system.
### Evaluating Each Option
Now, let's go through each option provided:
1. A natural number between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{3}\)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex] and [tex]\(\frac{\pi}{3} \approx 1.0472\)[/tex]. There is no natural number (which are 1, 2, 3, ...) that lies between these two values. So this option does not demonstrate the density of rational numbers.
2. An integer between -11 and -10
- The integers are ...,-13,-12,-11,-10,... Between -11 and -10, there are no integers. Thus, this option also does not demonstrate the density of rational numbers.
3. A whole number between 1 and 2
- Whole numbers include non-negative integers (0, 1, 2, 3, ...). There is no whole number between 1 and 2. Therefore, this does not demonstrate the density of rational numbers either.
4. A terminating decimal between -3.14 and -3.15
- Terminating decimals are a subset of rational numbers. However, note that -3.14 and -3.15 themselves might create some confusion because they suggest a very small difference range using a typical decimal representation. To expressly show rational density, it would require more clear-cut divisions.
### Conclusion
To prove that rational numbers are dense, we particularly need to highlight the fact that there are rational numbers between any two real numbers. None of these options, however, provides a straightforward demonstration of this although the terminating decimal option closely relates to rational numbers but might not be clear-cut between -3.14 and -3.15.
Therefore, the best illustration of the density of rational numbers might not be immediately clear from these options, but in general, more specific ranges or descriptive examples are simpler to use in demonstrating the concept effectively in mathematical terms.
### Selection
None of the provided choices perfectly demonstrates the density property straightforwardly. The statement given supports and aims to clarify the common understanding of rational density, but precise examples within ranges are needed for definitiveness.
