Rational Numbers

Introduction to Rational Numbers

What is a rational number?  The number of pages in a book, the fingers on your hand or the number of students in your classroom. These numbers are rational numbers. Let us study in detail about these numbers. Let’s find out more.

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Introduction to Rational Numbers

We already know about some types of numbers. The numbers that we are familiar with are natural numbers, whole numbers, and integers. Natural numbers are the ones that begin with 1 and goes on endlessly up to plus infinity. If we include 0 in these sets of numbers, then these numbers become whole numbers.

Now in these sets, if we also include the negative numbers, then we call it as integers. So all the numbers that we see collectively on the number line are called integers. To represent rationals, we divide the distance between two consecutive part into ‘n’ number of parts.

RationalFor example, to represent \( \frac{a}{3} \) divide the distance between 0 and 1, 1 and 2 and so on in three equal parts and that parts are represented as \( \frac{1}{3} \), \( \frac{2}{3} \), \( \frac{3}{3} \) and so on….

A rational number is a number that can be written in the form of a numerator upon a denominator. Here the denominator should not be equal to 0. The numerator and the denominator will be integers. A rational number is of the form

\( \frac{p}{q} \)

where p = numerator, q= denominator, p and q are integers and q ≠0. Examples, \( \frac{3}{5} \), \( \frac{-3}{10} \), \( \frac{11}{-15} \). Here we can see that all the numerators and denominators are integers and even the denominators should be non-zero.

Positive and Negative Rational Numbers

Any rational number can be called as the positive rational number if both the numerator and denominator have like signs. Similarly, a number which has either the numerator negative or the denominator negative is called the negative rational number.

Identify the Numbers

  • \( \frac{2}{7} \), here 2 is an integer, 7 is an integer so yes it is a rational number.
  • \( \frac{0}{0} \), here there is 0 in the denominator too. So it is not a rational number.
  • -9, here -9 can be written \( \frac{-9}{1} \). So it is a rational number.
  • 0, 0 is a  rational number.

Properties of Rational Number

  • A rational number remains unchanged when a non zero integer m is multiplied to both numerator and denominator.

\( \frac{p × m }{q ×m } \)

Suppose we take the number \( \frac{2}{5} \) and multiply both numerator and denominator by 3 then, \( \frac{2 × 3 }{5 ×3 } \) the result that we get is \( \frac{6}{15} \). Now this \( \frac{6}{15} \) is the standard form. If we express it in its simplest form we get it as \( \frac{2}{5} \).

  • A rational number remains unchanged when a non zero same integer m is divided to both numerator and denominator.

\( \frac{p ÷ m }{q ÷ m } \)

Suppose we take the number \( \frac{6}{15} \) and divide both numerator and denominator by 3 then, \( \frac{6 ÷ 3 }{15 ÷ 3 } \) the result that we get is \( \frac{2}{5} \)

Standard Form of Rational Number

Now \( \frac{24}{36} \) is a rational number. But when this number is expressed in its simplest form, it is \( \frac{2}{3} \). A rational number is in its standard form if it has no common factors other than 1 between the numerator and denominator and the denominator is positive.

Solved Examples for You

Question: \( \frac{11}{4} \) is the rational number between?

  1. 1 and 2
  2. 2 and 3
  3. 3 and 4
  4. 11 and 12

Solution: B. \( \frac{11}{4} \) = 2 \( \frac{3}{4} \). So, \( \frac{11}{4} \) lies between 2 and 3.

Question- What is a rational number simple definition?

Answer- A rational number is a number that we can write as a fraction. All real numbers which are positive or negative are rational numbers. A number which is not rational is referred to as irrational. We make use of numbers in our daily lives which are rational. Thus, it also includes fractions and integers.

Question- What are the positive and negative rational numbers?

Answer- We can refer to any rational number as a positive rational number if the numerator and denominator both have like signs. Further, a negative rational number is where the numerator or denominator is negative.

Question- List down two properties of rational numbers.

Answer- The two properties of rational numbers are that they do not change when a non-zero integer m is multiplied to both numerator and denominator. Moreover, a rational number does not change when a non-zero same integer m is divided into numerator and denominator.

Question- Can a rational number be negative?

Answer- We consider a rational number when we can write it as one integer divided by another integer. Thus, rational numbers may be positive, negative and zero as well. When writing a negative rational number, we put the negative sign either out in front of the fraction or with the numerator.

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