Learn Decimal representation of rational numbers

Welcome to one of the most interesting concepts in CBSE Class 10 Mathematics: Real Numbers. By now, you know that rational numbers are those that can be expressed in the fractional form p/q, where p and q are integers and q is not equal to zero. But did you know that you can predict exactly how a rational number will behave when converted into a decimal, without even performing long division? By simply examining the denominator of the fraction in its simplest form, you can determine whether its decimal representation will eventually stop completely or keep repeating in a pattern forever.

The core logic behind this lies in the prime factorization of the denominator, q. Let x = p/q be a rational number where p and q are co-prime (meaning they share no common factors other than 1). The theorem states that if the prime factorization of q is strictly of the form 2ⁿ × 5^m (where n and m are non-negative integers), the rational number has a terminating decimal expansion. This happens because the base of our decimal system is 10, and the only prime factors of 10 are 2 and 5. On the other hand, if the prime factorization of q contains any prime number other than 2 or 5 (such as 3, 7, or 11), the number will have a non-terminating repeating (recurring) decimal expansion.

As illustrated in the infographic above, identifying the type of decimal is a straightforward two-step process. On the left side, we examine the fraction 13/20. The denominator 20 breaks down perfectly into 2² × 5¹. Because these factors are exclusively 2s and 5s, we can confidently state that it is a terminating decimal, ultimately resulting in exactly 0.65. Conversely, on the right side, the fraction 10/3 has a denominator of 3. Since 3 is neither a 2 nor a 5, the rule dictates it will be a non-terminating repeating decimal, creating the endless sequence 3.333... This concept is incredibly popular in CBSE Class 10 board exams. You will frequently face 1-mark and 2-mark questions requiring you to classify fractions without long division, or even calculate exactly how many decimal places a fraction will have before it terminates (which corresponds to the highest power of either 2 or 5 in the denominator's factorization).

Mastering the theorems of Real Numbers lays a strong foundation for your entire Class 10 Mathematics journey, but it is normal to occasionally feel stuck or overwhelmed by complex math proofs. If you are finding these concepts tricky or simply want to boost your board exam preparation, UrbanPro is here to help! Explore thousands of highly qualified and experienced CBSE Class 10 Mathematics tutors on UrbanPro. Whether you prefer the convenience of interactive online tuition or the personalized attention of local offline classes, UrbanPro connects you with top-rated educators who can make math engaging, simple, and scoring.