To determine which number supports the idea that rational numbers are dense, let's analyze each option and see if a rational number can be found within the specified ranges.
1) A natural number between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{3}\)[/tex]:
- [tex]\(\frac{\pi}{2}\)[/tex] is approximately 1.57 and [tex]\(\frac{\pi}{3}\)[/tex] is approximately 1.05.
- Natural numbers are the set \{1, 2, 3, ...\}.
- There are no natural numbers between these two values because the next natural number after 1.05 is 2, which is not within the interval.
2) An integer between -11 and -10:
- Integers are whole numbers that can be positive, negative, or zero.
- There are no integers between -11 and -10, because the closest integers to -11 are -10 and -12, but neither is between -11 and -10.
3) A whole number between 1 and 2:
- Whole numbers include 0 and all positive integers.
- There are no whole numbers that lie between 1 and 2. The closest whole numbers are 1 and 2, but neither is between these values.
4) A terminating decimal between -3.14 and -3.15:
- A terminating decimal is a decimal number that comes to an end.
- Since -3.14 and -3.15 are very close to each other, there cannot be a terminating decimal between them.
The concept of rational numbers being dense means that between any two real numbers, no matter how close they are, there exists a rational number. Thus, the correct number that supports the idea that rational numbers are dense is:
A rational number between 1 and 2, which, in this case, is 1.5.
So the final answer is:
1.5
