Question

Classify the following number as rational and irrational. \[\sqrt {1.44} \].

Answer 1

Hint: Here, we will find the number whether rational or irrational. First we will find the square root of the given number and then we will use the definition to find whether the number is rational or irrational. A rational number is defined as a number which can be expressed in the form of fractions. An irrational number is defined as a number which cannot be expressed in the form of fractions.

Complete step-by-step answer:
We are given a number \[\sqrt {1.44} \]to find whether it is a rational or irrational number.
We will convert the decimal number into fraction.
\[\sqrt {1.44} = \sqrt {\dfrac{{144}}{{100}}} \] ……………….\[\left( 1 \right)\]
We will use factorization method, to find the square root of a number.
\[\begin{array}{l}2\left| \!{\underline {\,
  {144} \,}} \right. \\2\left| \!{\underline {\,
  {72} \,}} \right. \\2\left| \!{\underline {\,
  {36} \,}} \right. \\2\left| \!{\underline {\,
  {18} \,}} \right. \\3\left| \!{\underline {\,
  9 \,}} \right. \\3\left| \!{\underline {\,
  3 \,}} \right. \end{array}\]
and
\[\begin{array}{l}5\left| \!{\underline {\,
  {100} \,}} \right. \\5\left| \!{\underline {\,
  {20} \,}} \right. \\2\left| \!{\underline {\,
  4 \,}} \right. \\2\left| \!{\underline {\,
  2 \,}} \right. \end{array}\]
Now we can write 144 and 100 in form of the factors:
\[144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\]
\[100 = 5 \times 5 \times 2 \times 2\]
Now ,
\[\sqrt {\dfrac{{144}}{{100}}} = \sqrt {\dfrac{{2 \times 2 \times 2 \times 2 \times 3 \times 3}}{{5 \times 5 \times 2 \times 2}}} \]
\[ \Rightarrow \sqrt {\dfrac{{144}}{{100}}} = \dfrac{{2 \times 2 \times 3}}{{5 \times 2}}\]
Multiplying the terms, we get
\[ \Rightarrow \sqrt {\dfrac{{144}}{{100}}} = \dfrac{{12}}{{10}}\]
Simplifying the terms, we get
\[ \Rightarrow \sqrt {\dfrac{{144}}{{100}}} = \dfrac{6}{5}\]
Since the number can be expressed as \[\dfrac{p}{q}\], then the number \[\sqrt {1.44} \] is a rational number.
Therefore, the number \[\sqrt {1.44} \] is a rational number.

Note: First, we will count the number of digits after the decimal point. If there \[n\] is the number of digits after the decimal point, then we have to multiply and divide \[{10^n}\] to remove the decimal and to convert the decimal into fraction. A rational number can also be defined as the ratio of two integers. An irrational number can also be defined which cannot be expressed as the ratio of two integers.
We might make a mistake by considering the given number as irrational because it is expressed as a square root. Before coming to a conclusion, we need to simplify the number and then if the number cannot be expressed in fraction then it is irrational.