Question

How do you simplify rational numbers?

Answer 1

Hint: We will define rational numbers and from that definition, we will know some methods of simplifying rational numbers. We will clearly know about rational numbers with some examples, along with irrational numbers also. We will also define other kinds of numbers here.

Complete answer:
In mathematics, there are many kinds of numbers. They are: -
(1) Natural numbers \[\left( {1,2,3,4,5,6,........\infty } \right)\]
(2) Whole numbers \[\left( {0,1,2,3,4,5,6,........\infty } \right)\]
(3) Integers \[\left( { - \infty ,........ - 5, - 4, - 3, - 2, - 1,0,1,2,3,4,5,........\infty } \right)\]
(4) Rational numbers
(5) Irrational numbers
Let us know about rational numbers.
In mathematics, numbers which can be written in the form of \[\dfrac{p}{q}\] where \[q \ne 0\], are called Rational numbers. All the natural numbers, whole numbers and integers come under rational numbers, because, every natural number or whole number or integer can be written in \[\dfrac{p}{q}\] form.
Those numbers which cannot be written in the form of \[\dfrac{p}{q}\] are called irrational numbers.
Some examples of irrational numbers are \[\sqrt 3 ,\sqrt {9 + \sqrt 2 } , - 4.90248463.....\] and so on.
For example, take the number \[ - 7\]. It can be written as \[\dfrac{{ - 7}}{1}\].
The form \[\dfrac{p}{q}\] is called a “Fraction”.
And also, rational numbers can be non-recurring terminating decimal numbers or recurring non-terminating decimal numbers.
For example, take decimal number \[8.15\], which can be written as \[\dfrac{{815}}{{100}} = \dfrac{{163}}{{20}}\]
So, we have written the decimal number \[8.15\] in \[\dfrac{p}{q}\] form. So, it is a rational number.
Also take a number \[5.66666......\]
Let us simplify it and write in \[\dfrac{p}{q}\] form.
So, take \[x = 5.66666......\] -----(1)
Here, only the digit 6 is repeating. So, periodicity is 1. So, multiply the equation by 10.
\[ \Rightarrow 10x = 56.6666......\] -----(2)
 Subtract the equation (1) from equation (2)
\[ \Rightarrow 10x - x = (56.6666......) - (5.6666.......)\]
\[ \Rightarrow 9x = 51.0000..... = 51\]
So, on dividing this whole equation by 9, we will get,
\[ \Rightarrow x = \dfrac{{51}}{9}\]
So, thus we can simplify rational numbers.

Note: If in a recurring decimal, only one digit is repeating, then the periodicity is 1. If two digits are repeating, then periodicity is 2. And similarly, if three digits are repeating, then periodicity is 3.
So, periodicity is the number of digits repeating.
Suppose that, periodicity is \[n\], then multiply the whole equation by \[{10^n}\].