Question

The following real numbers have decimal expansions, as given below. In each case, decide whether they are rational or not. If they are rational and of the form p, you say about the prime factors of q?
i.\[43.123456789\]
ii.\[0.120120012000120000...\]
iii.\[43.\overline {123456789} \]

Answer 1

Hint: Terminating decimal numbers are the numbers that contain a finite number of digits after the decimal point. A number 2.56 can be a terminating decimal if it is represented as 2.5600000000….. where 0 is terminating.
Any rational number \[\dfrac{p}{q}\] can be a terminating or repeating decimal if the factors of the decimal are in the form\[{2^m} \times {5^n}\], where m and n are non-negative integers.
If the factorial of the denominator is not in the form of \[{2^m} \times {5^n}\], then the number is non-terminating, and a number is an irrational number.
In this question, write the given decimal numbers in \[\dfrac{p}{q}\]form and then check the factors of the denominator q.


Complete step by step solution:
(i) \[43.123456789\]
\[x = 43.123456789\]is the rational number because digit, after decimal are terminating hence the number, can be written in\[\dfrac{p}{q}\]form,
Hence write the number in\[\dfrac{p}{q}\]the form
\[
  x = 43.123456789 \\
   = \dfrac{{43123456789}}{{1000000000}} \\
   = \dfrac{{43123456789}}{{{{\left( {10} \right)}^9}}} \\
   = \dfrac{{43123456789}}{{{{\left( {2 \times 5} \right)}^9}}} \\
   = \dfrac{{43123456789}}{{{{\left( {2 \times 5} \right)}^9}}} \\
   = \dfrac{{43123456789}}{{{2^9} \times {5^9}}} \\
 \]
Here the denominator q is in the form of\[{2^m} \times {5^n} = {2^9} \times {5^9}\], where m=9 and n=9
Hence we can say the number \[43.123456789\]is the terminating number.
(ii) \[0.120120012000120000...\]
\[x = 0.120120012000120000...\]the number is a non-terminating and non-repeating number; hence the number is an irrational number since the number of digits after the decimal is non-repeating.
(iii) \[43.\overline {123456789} \]
For \[x = 43.\overline {123456789} \], the number is a non-terminating and repeating number; hence the number is a rational number
The number can be written as
\[x = 43.123456789123456789.....\]
Since the number is non-terminating and repeating hence the number is a rational number.


Note: Every fraction number can be either terminating or non-termination (or, repeating). A decimal number can either be terminating or terminating; if the number is terminating, then the number is rational and if the number is non-terminating, then check whether they are repeating or non-repeating since a repeating number is rational and non-repeating numbers are an irrational number.