Now,let us express them in p ÷ q form.

1. 0.375 =375 ÷1000 =375 ÷10³


2. 1.04 =104 ÷100 =104 ÷10²


3. 0.0875 =875 ÷10000 =875 ÷10⁴


4. 12.5 = 125 ÷10 = 125 ÷10¹

We see that all terminating decimals taken by us can be expressed inp/q form whose denominators are powers of 10. Let us know factorize the numerator and denominator and then express then in the simplest form :

Now:

1. 0.375 =375 ÷10³ =(3×5³⁾÷(2³×5³⁾ =3 ÷2³ = 3 ÷8


2. 1.04 =104 ÷10² =(2³×13 ÷2²×5²) = 26 ÷5²=26 ÷25


3. 0.0875 = 875 ÷ 10⁴ = (5³×7 ÷2⁴×5⁴) = 7 ÷(2⁴×5) = 7 ÷ 80


4. 12.5 = 125 ÷10 = 5³ ÷(2 ×5) = 25 ÷2

Have you observed any pattern in the denominators of the above numbers? It appears that when the decimal is expressed in its simplest rational form then p and q are co-prime and the denominator (i.e., q) has only powers of 2, or powers of 5, or both. This is because 2 and 5 are the only prime factors of powers of 10.

From the above examples, you have seen that any rational number that terminates in its decimal form can be expressed in a rational form whose denominator is a power of 2 or 5 or both. So, when we write such a rational number, in p/q form, the prime factorization of q will be in 2ⁿ5^m, where n,m are some non-negative integers.

We can write our result formally :