# Real Numbers – Revisiting Rational and Their Decimal Expansions

REVISITING RATIONAL NUMBERS AND THEIR DECIMAL EXPANSIONS :

We have already studied in the previous class that rational numbers have either a terminating decimal repeating decimal expansion. We have to consider a rational number as $\displaystyle \frac{p}{q}$ (where $\displaystyle q\ne 0$) as terminating or non-terminating repeating (or recurring) decimal expansion.

e.g.

(A) (i)    0.0527    = (ii)    26.12489     = (B) (i)    0.0875    = (ii)    23.3408    = Now, we have converted a real number whose decimal expansion terminates into a rational number of the form. $\displaystyle \frac{p}{q}$, where p and q are coprime, and prime factorization of denominator

(i.e., q) has only power of 2 or power of 5 or both. We should also understand that the denominator is a power of 10 and can be only prime factors 2 and 5.

Now, we get that this real number is a rational number of the form $\displaystyle \frac{p}{q}$, where the prime factorization of q is of the form 2n, 5m, and n, m are some non-negative integers.

Theorem : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form $\displaystyle \frac{p}{q}$, where p and q are coprime, and the prime factorization of q is of the form 2n5m, where n, m are non-negative integers.

Let $\displaystyle \frac{a}{b}$ be a rational number in the lowest form such that the prime factorization of b is of the form $\displaystyle {{2}^{n}}\times {{5}^{m}}$ . Where n and m are non-negative integers.

Example:

(i) $\displaystyle \frac{3}{8}=\frac{3}{{{2}^{3}}}=\frac{3\times {{5}^{3}}}{{{2}^{3}}\times {{5}^{3}}}=\frac{375}{{{10}^{3}}}=0.375$

(ii) $\displaystyle \frac{13}{125}=\frac{13}{{{5}^{3}}}=\frac{13\times {{2}^{3}}}{{{5}^{3}}\times {{2}^{3}}}=\frac{104}{{{10}^{3}}}=0.104$

(iii) $\displaystyle \frac{2139}{1250}=\frac{2139}{2\times {{5}^{4}}}=\frac{2139\times {{2}^{3}}}{{{2}^{4}}\times {{5}^{4}}}=\frac{2139\times 8}{{{\left( 2\times 5 \right)}^{4}}}=\frac{17112}{{{10}^{4}}}=1.7112$

Theorem : Let $\displaystyle x=\frac{p}{q}$ be a rational number, such that the prime factorization of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Example: $\displaystyle \frac{1}{7}$

In this case, denominator 7 is not of the form 2n5m. So, from above mentioned theorems, $\displaystyle \frac{1}{7}$ will not have a terminating decimal expansion. Hence, 0 (zero) will not be remainder and the remainders will start repeating after a certain stage. We have a block of digits, 142857, repeating in the quotient of $\displaystyle \frac{1}{7}$.

Hence, it is is true that any rational number is not converted by the above theorems. Theorem :Let $\displaystyle x=\frac{p}{q}$ be a rational number, such that the prime factorization of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is not-terminating repeating (recurring).

Hence, we can say that the decimal expansion of every rational number is either terminating or non-terminating repeating (recurring).