__Revisiting rational numbers and their decimal expansion__

**Revisiting Rational Numbers**

**Rational Numbers and their Decimal Expansion**

**The decimal expansion of every rational number is either terminating or non-terminating repeating.**

**Terminating Decimal Expansion**

**The number which terminates after a finite number of steps in the process of division is called terminating decimal expansion. E.g. 6.25, 1.14**

**Non-Terminating Decimal Expansion**

**The number which does not terminate in the process of division is called non-terminating decimal expansion.**

**There are two types of non-terminating decimal expansions**

**a) Non-terminating repeating Decimal Expansion**

**The number which does not terminate but repeats the particular number again and again in the process of division is said to be non-terminating repeating decimal. **

**E.g. ****0.333333**

**b) Non-terminating Non-repeating Decimal Expansion**

**The number which neither terminates nor repeats the particular number in the process of division is said to be non-terminating repeating decimal. E.g. 1.03303033**

**Theorem 1:**

**Let x be a rational number whose decimal expansion terminates.**

**Then x can be expressed in the form ** **, where p and q are coprime and the prime factorization of q is of the form 2 ^{m}5^{n }**

**, where m, n are non- negative integers.**

**Theorem 2:**

**Let x = be a rational number, such that the prime factorization of q is of ****the form 2**^{m}5^{n} , where m, n are non-negative integers, then ** has a ****terminating decimal.**

**Multiplying both numerator and denominator by **

**= = **

** = = = 0.2875**

**Theorem 3:**

**Let x =****be a rational number, such that the prime factorization of q is not of the form 2 ^{m}5**

^{n }

**, where m, n are non-negative integers, then x has**

**decimal expansion which is**

**non-terminating repeating.**

**Example:
Without actually performing the long division, state whether has a terminating decimal expansion or non-terminating recurring decimal expansion.**

**Since the factors of the denominator is not of the form 2 ^{m}5^{n }. **

**So, is non-terminating recurring decimal expansion.**

**Example:
The decimal expansion of the rational number will terminate after how many places of decimal.**

** **

**Now, multiplying both numerator and denominator by 5**

** = **

**The decimal expansion of the rational number will terminate after 4 places of decimal.**

**Example:If = , then find the value of m and n, where m and n are non- negative integers. Hence, write its decimal expansion without actual ****division.**

=

**m = 5 and n = 3**

**Now, can be expressed as **

**We have**,

**Multiplying both numerator and denominator by **

=

**Example:**

**What can you say about the prime factorization of the denominators of**

**the following rational:**

**i) 42.123456789**

**ii) **

**i) Since 42.123456789 has a terminating decimal expansion. So, its denominator is of the form 2 ^{m}5**

^{n }

**, where m and n are non-**

**negative integers.**

**ii) Since has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.**