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**Revisiting Irrationals, Rational And Their Decimal Expansions**

**Irrational numbers:**

All those numbers which can not be expressed as fractions i.e a number is an irrational number if it can not be written as $qp ,q=0$.

**In the view of decimal expansion:**

All those numbers whose decimal expansion is neither terminating nor repeating.

**Eg:**$2 ,3 ,35 ,10100100010000...,$etc.

**Proving irrationality of numbers:**

In order to prove the irrationality of numbers, we will use the method of contradiction and following results:

- The sum or difference of a rational and irrational number is an irrational number.
- The product and quotient of a non-zero rational number and an irrational number is an irrational number.
- If $p$ be a prime number and $a$ be a positive integer. If $p_{2}$ divides $a$, then $p$ divides $a$.

**Remarks:**

- For any prime number $p$,$p $ is an irrational number.

**Rational numbers:**

All those numbers which can be expressed as fractions i.e. a number is a rational number if it can be written as $qp ,q=0$.

**In the view of decimal expansion:**

All those numbers whose decimal expansion is either terminating or non-terminating repeating.

Eg:$12 ,100011 ,71113 ,13169 $,0.02,0.3412,0.121212, . . ., 0.010203, etc.

**Theorems to determine the nature of decimal expansions of rational numbers**

**Theorem 1.**

Let $x$ be a rational number whose decimal expansion terminates. Then,$x$ can be expressed in the form $qp $,where $p$ and $q$ are co-primes,and the prime factorization of $q$ is of the form $2_{m}5_{n}$, where $m,n$ are non-negative integers.

**Eg:**Let $x=0.242$

$⇒x=1000232 =500121 =2_{2}×5_{3}121 $

Thus,$q=2_{2}×5_{3}$, which is in the form of$2_{m}5_{n}$, where$m=2,n=3$.

**Theorem 2.**

Let $x=qp $ 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 $x$ has a decimal expansion which terminates.

**Eg:**Let $x=12539 $

$⇒x=5_{3}39 =5_{3}×2_{3}39×2_{3} =1000312 =0.312$.

Thus,$x$ has a terminating decimal expansion.

**Theorem 3.**

Let $x=qp $,where $p$ and $q$ are co-primes, be a rational number such that $q$ is not of the form $2_{m}5_{n}$, where $m$ and $n$ are non-negative integers. Then the decimal expansion of $x$ is non-terminating and repeating.

**Eg:**Let $x=37 $

$⇒x=2.333333...$

Thus,$x$ has decimal expansion which is non-terminating and repeating.