2. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Answer: Any number that can be represented in p/q where q is greater than 0 is called a rational number. The real numbers cannot be expressed in p/q, where p and q are integers, and q ≠ 0 are known as irrational numbers.

q must satisfy the condition that if prime factors of q have only powers of 2 or power of 5 or both, then the rational numbers always have a terminating decimal expansion.

i.e. 2m × 5n, where m=1,2,3,⋯ or n=1,2,3

For Example

/2 = 0. 5, denominator q = 21

7/8 = 0. 875, denominator q =23

4/5 = 0. 8, denominator q = 51

So, we conclude that terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both. In the form of 2m × 5n, where n, m are natural numbers.