 # Rational Numbers on a Number line

### Arranging Rational Numbers on a Number line:

Numbers are the most basic elements of any Mathematical computations. There are different types of numbers used in various calculations. All the types of numbers are broadly classified into two types namely real numbers and imaginary numbers. Real numbers are the numbers that exist in reality. Imaginary numbers are the numbers assumed to exist to explain several Mathematical concepts efficiently. Real numbers are broadly classified into two types namely rational and irrational numbers. All these numbers can be represented on a number line. A number line is a straight line on which all the positive and negative real numbers are arranged towards the right and left of a fixed point called the reference point. The reference point on the number line is zero.

### How to Show Rational Numbers on a Number Line:

A rational number is a number that can be represented in the form of a fraction in which the value of the denominator is not equal to zero. If the denominator is zero, then the number is not a rational number. Rational numbers include whole numbers, positive and negative integers, decimal numbers, and fractions. A detailed description of steps to be followed for plotting the rational numbers on a number line for different types of numbers is elaborated in the subsequent sections.

### Plotting Rational Numbers on a Number Line (Integers):

Step 1:

Construct a number line with zero as a reference point, positive integers on the right of zero and negative integers on the left of zero.

Step 2:

The given number is represented at a point indicating that number towards the right if the number is positive and towards the left if the number is negative.

The blue spots in the above figure indicate -3, 2 and 4 as the representation of rational numbers on a number line.

### Plotting Rational Numbers on a Number Line (Number with a Decimal point):

Step 1:

If the number has a decimal point, it should be first converted into the form of a fraction.

Step 2:

If the fraction is improper, the steps for plotting improper fractions on the number line should be followed. If not, the steps to represent proper fractions should be followed.

### Plotting Rational Numbers on a Number Line (Improper or Mixed fraction):

Step 1:

If the given fraction is an improper fraction, it should be converted into a mixed fraction. The mixed fraction consists of a whole part and a proper fraction part. The whole part is generally a positive or negative integer.

Example: Suppose the given number is 65. It is converted into improper fraction as 115. The whole number part of this fraction is 1 and the proper fraction part is 15

Step 2:

Construct a number line with zero as a reference point, positive integers on the right of zero and negative integers on the left of zero.

Step 3:

Identify the point indicating the whole number part of the fraction on the number line. The distance between this point and the point indicating the immediately next whole number part is divided into a number of parts equal to the denominator in the fraction part of the mixed fraction.

Example: To indicate the fraction 115, the distance between the points indicating 1 and 2 is divided into 5 equal parts.

Step 4:

The part indicating the digit in the numerator of the fraction part in the mixed fraction indicates the given rational number.

The figure below indicates the example of representing 65 or 115on the number line.

### Plotting Rational numbers on a number line (Proper fraction):

Step 1:

Construct a number line with zero as a reference point, positive integers on the right of zero and negative integers on the left of zero.

Step 2:

Divide the distance between the points indicating 0 and 1 (-1 for negative rational numbers) into a number of parts equal to that of the denominator of the fraction.

Example: To represent -35 on a number line, the distance between 0 and 1 should be divided into 5 equal parts.

Step 3:

The part indicating the value in the numerator is the point indicating the given rational number.

Example: The figure below shows the representation of -35 on a number line.

### Fun Facts About Arranging Rational numbers on a number line:

• To understand the representation of rational numbers on a number line, the basic concept of types of fractions is mandatory. Proper fractions are the fractions with the value of numerator less than the value of denominator and the improper fractions are the fractions with the value of denominator less than that of the numerator.

• While arranging rational numbers on a number line, the rational numbers that are proper fractions lie between 0 and 1, and the rational numbers that are improper fractions are first converted to a mixed fraction and then represented on a number line.

1. What are Rational Numbers?

Rational numbers are the numbers that can be represented in the form of a fraction in which the numerator and denominators are integers and the denominator is not equal to zero. Also, the numerator and denominator should be coprime to each other. Mathematically, any number that can be represented as x/y where ‘x’ and ‘y’ are coprimes and y ≠ 0. Examples for rational numbers include prime numbers, composite numbers, odd numbers, even numbers, natural numbers, whole numbers, integers, fractions and decimals. Irrational numbers and imaginary numbers are not examples of rational numbers.

2. How to Represent Rational Numbers on a Number Line?

• The first step to be considered to show rational numbers on a number line is to check whether it is a proper, improper, or mixed fraction representation.

• If the given rational number is a proper fraction, then the distance between zero and one on the number line is divided into a number of parts equal to the denominator and the part indicating the value of the numerator is the point that gives the representation of rational numbers on a number line.

• If the given rational number is an improper fraction, it should be first converted into a mixed fraction. The rational number lies between the whole number represented in the mixed fraction and the immediate consecutive number. The distance between the whole number and the immediate consecutive number is divided into a number of equal parts as indicated by the denominator. The part indicating the number in the numerator is the point indicating the given number.

• If the number is a decimal, it is first converted into a fraction, and then the steps for plotting rational numbers on a number line when it is a fraction are followed.