There are an infinite number of rational numbers between 1 and 2.
This is because rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Between any two distinct rational numbers, you can always find another rational number. For example, the average of two rational numbers is always another rational number between them.
Let's illustrate this:
- We start with 1 and 2.
- The average of 1 and 2 is (1+2)/2 = 3/2, which is 1.5. So, 3/2 is between 1 and 2.
- Now, let's find a number between 1 and 3/2. The average is (1 + 3/2)/2 = (5/2)/2 = 5/4, which is 1.25. So, 5/4 is between 1 and 3/2 (and therefore also between 1 and 2).
- We can continue this process indefinitely. For example, the average of 3/2 and 2 is (3/2 + 2)/2 = (7/2)/2 = 7/4, which is 1.75.
Because this process can continue infinitely, there are infinite rational numbers between 1 and 2. Furthermore, any decimal that terminates (ends) or repeats is also a rational number. Think of numbers like 1.1, 1.01, 1.001, 1.9, 1.99, 1.999, and so on. All these numbers are rational and lie between 1 and 2.
