How to Represent Rational Numbers on a Number Line Step by Step with Solved Examples
A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions.
Arranging Rational Numbers on a Number line
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.
Rational Numbers Vs Irrational Numbers
Rational numbers are those numbers that can be expressed in p/q form that is represented in a fraction from where q is never equal to zero. On the other hand, irrational numbers can not be represented in a ratio format or p/q form. Rational numbers include finite and recurring decimals whereas irrational numbers include non-recurring and non-terminating decimals. Both the numerator and denominator of a rational number is a whole number or positive integer whereas an irrational number can never be expressed in that format.
Number Lines
Number Lines are nothing but a horizontal representation of numbers and their values on a line with equal intervals having certain values. Numbers with a particular sequence can only be expressed on a number line and this can be extended indefinitely on either side. Generally, zero is taken as a reference value and all the positive numbers or values are pointed to the right of the number line whereas all the negative values are marked to its left. It is comparatively easy to explain arithmetic operations like addition, subtraction, etc, on a number line.
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.
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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.
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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.
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.
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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.
Step 4:
The part indicating the digit in the numerator of the fraction part in the mixed fraction indicates the given rational number.
Plotting Rational Numbers on a Number Line (Proper fraction)
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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 -3/5 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 -3/5 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.
Conclusion
Rational Numbers are the foundation of mathematical equations. If it weren’t for these, we wouldn’t have linear or binary equations. They substantiate numbers. This article focuses on Rational numbers, their applications, and how to plot them. Go through it properly for a better understanding.
FAQs on Rational Numbers on a Number Line with Clear Explanation and Examples
1. What are rational numbers on a number line?
Rational numbers on a number line are numbers that can be written in the form p/q (where q ≠ 0) and can be represented as exact points between integers. A rational number includes fractions, integers, and terminating or repeating decimals.
- Examples: -3, 1/2, 0.75, -4/5
- They can lie to the left (negative) or right (positive) of 0.
- Every rational number has a fixed position on the number line.
2. How do you represent a rational number on a number line?
To represent a rational number on a number line, divide the interval between integers into equal parts based on the denominator and mark the required position.
- Step 1: Draw a number line and mark integers.
- Step 2: For a fraction like 3/4, divide the space between 0 and 1 into 4 equal parts.
- Step 3: Count 3 parts from 0 and mark the point.
3. How do you plot negative rational numbers on a number line?
Negative rational numbers are plotted to the left of 0 on the number line according to their value. For example, to plot -3/5:
- Divide the space between 0 and -1 into 5 equal parts.
- Move 3 parts to the left from 0.
- Mark the third division as -3/5.
4. How do you compare rational numbers on a number line?
To compare rational numbers on a number line, the number located further to the right is greater. For example:
- On the number line, 1/2 lies to the right of 1/4, so 1/2 > 1/4.
- -1/3 lies to the right of -2/3, so -1/3 > -2/3.
5. Can every rational number be represented on a number line?
Yes, every rational number can be represented as a unique point on the number line. Since a rational number is defined as p/q (q ≠ 0), it always corresponds to a specific distance from 0.
- Integers like 5
- Fractions like 7/8
- Decimals like 0.125
6. How do you find the distance between two rational numbers on a number line?
The distance between two rational numbers is the absolute value of their difference, given by |a − b|. For example:
- Distance between 3/4 and 1/4 = |3/4 − 1/4| = 1/2
- Distance between -2 and 3 = |-2 − 3| = 5
7. What is the difference between integers and rational numbers on a number line?
Integers are whole numbers without fractions, while rational numbers include integers and fractions placed on the number line.
- Integers: ..., -2, -1, 0, 1, 2 ...
- Rational numbers: All numbers of the form p/q, including fractions like 3/5.
8. How do you divide a number line to represent fractions?
To represent fractions, divide the space between two consecutive integers into equal parts equal to the denominator. For example, to represent 5/6:
- Focus between 0 and 1.
- Divide the segment into 6 equal parts.
- Count 5 parts from 0 and mark the point.
9. Why is 0 considered a rational number on a number line?
Zero is a rational number because it can be written as 0/1 (or 0/q where q ≠ 0). On the number line, 0 is the central reference point separating positive and negative numbers.
- It has an exact position.
- It satisfies the definition p/q.
10. Can rational numbers be ordered on a number line?
Yes, rational numbers can be arranged in increasing or decreasing order based on their positions on the number line. Numbers to the right are greater, and numbers to the left are smaller.
- Example (increasing order): -3/4, -1/2, 0, 1/4, 2/3
- Example (decreasing order): 2/3, 1/4, 0, -1/2, -3/4
