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Introduction to Rational numbers

Initially we only had the family of Natural Numbers. After the invention of the family changed into the family of Whole numbers. Then, after the invention of negative numbers the family became the family of Integers. So now the number line is as,
 

-$\infty$.……..-3,-2,-1,0,1,2,3,…….+$\infty$
 

Then, some numbers which were in form of $\frac{p}{q}$, where p and q are natural numbers also became the part of the family called Fractions. Some fractions are $\frac{4}{7}$, $\frac{2}{5}$ …etc. The family of fractions also contains the family of Rational Numbers in which all the numbers are expressed in form of $\frac{p}{q}$, where p and q are integers and q is non-zero.

Rational numbers can be both positive or negative. On the number line of rational numbers the Positive Rational Numbers are to the right of 0 and the Negative Rational Numbers are to the left of 0. A Standard Form of a Rational Number is that both numerator and denominator should not have any common factor except 1. Also the rational number should be a positive rational number. To convert a rational number to its standard form divide the numerator and denominator of the rational number by its highest common factor.

 

Note:

 

When we represent a number as a numerator/ denominator then it is called as a rational number where both numerator and denominator are integers. And we all know that rational numbers are finite and terminated numbers means when we calculate the decimal value of rational number, it gives finite and repeating digits. So, it’s not possible that denominator of rational numbers are 0 because it produces infinite value which is against the rational number property like if we have numerator 5 and denominator 0 then it produces 5/0 = ……∞

And we all know infinity value is not coming under rational number property. Now we take different-different examples which define that: 

Rational numbers cannot have 0 as denominator. 

Example 1: Prove that 1/0 is not a rational number?

Solution: When we calculate the decimal value of 1/0, then it produces 1/0 = ……∞

And we all know that rational number contains only finite and terminating values. So, 1/0 is not a rational number because it produces infinity value.

Example 2: Prove that 7/0 is not a rational number?

Solution: When we calculate the decimal value of 7/0, then it produces 7/0 = ……∞                                                                                                                

And we all know that rational number contains only finite and terminating values. So, 7/0 is not a rational number because it produces infinity value.

Example 3: Prove that 11/0 is not a rational number?

Solution: When we calculate the decimal value of 11/0, then it produces 11/0 = ……∞                                                                                                              

And we all know that rational number contains only finite and terminating values. So, 11/0 is not a rational number because it produces infinity value.

These are some examples which tell that rational numbers cannot accept denominator as a 0. So, we can say that Rational numbers cannot have 0 as denominator.

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