To demonstrate that rational numbers are dense, we need to find a rational number that lies between any two given numbers. Density of rational numbers means that between any two real numbers, there exists a rational number.
Let's consider the options provided and determine which one fits:
1. A natural number between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\frac{\pi}{3}\)[/tex]
- Natural numbers are the positive integers: 1, 2, 3, etc.
- The value of [tex]\(\frac{\pi}{2}\)[/tex] is approximately 1.57, and [tex]\(\frac{\pi}{3}\)[/tex] is approximately 1.05.
- There is no natural number between 1.05 and 1.57.
2. An integer between -11 and -10
- Integers are the whole numbers that can be positive, negative, or zero: -1, 0, 1, 2, etc.
- Between -11 and -10, the only integer is -10, which is not in between.
3. A whole number between 1 and 2
- Whole numbers are the non-negative integers: 0, 1, 2, etc.
- Between 1 and 2, there is no whole number.
4. A terminating decimal between -3.14 and -3.15
- Terminating decimals are decimals that have a finite number of digits after the decimal point.
- A number that lies between -3.14 and -3.15 can certainly be a rational, terminating decimal.
Considering these explanations, the best candidate to demonstrate the density of rational numbers is the option that asks for:
A terminating decimal between -3.14 and -3.15.
Now, let's find such a number:
1. We are looking for a decimal that is in between -3.14 and -3.15.
2. One such number could be -3.145.
Thus, a terminating decimal between -3.14 and -3.15 is [tex]\(-3.145\)[/tex], which supports the idea that rational numbers are dense.
