105 LearnersLast updated on July 4th, 2025
Representation of Real Numbers on Number Line
Real numbers include natural numbers, whole numbers, as well as rational and irrational numbers. On a straight number line, each integer is placed at equal intervals. The number line extends infinitely in both directions. To organize and compare numbers, we can use a number line. In this article, we will learn about the representation of real numbers on the number line in detail.
What are Real Numbers?
The set of rational numbers Q and the set of irrational numbers Q’ together make up the set of real numbers. It is denoted as R. Subsets of whole numbers, natural numbers, integers, and rational and irrational numbers are all considered real numbers. Therefore,
R = Q ∪ Q’
What is the Real Number Line?
On a number line, each number has a unique point known as a coordinate and has a distinct position. For instance, the real number 3 is positioned between 2 and 4. On a number line, two numbers cannot share the same position. The origin of a number line is at 0. On the right side of the origin are the positive numbers, while on the left side are the negative numbers.
The visual representation of a real number line is:
How to Represent Real Numbers on a Number Line?
Following the given steps will help us to easily indicate real numbers on a number line using graphs and coordinates.
Step 1: Draw a straight line, mark the origin at 0, and draw arrows on both sides of the origin point.
Step 2: Use a fixed scale for marking real numbers. Place real numbers on both sides of the origin at equal intervals.
Step 3: Mark positive numbers to the right of the origin, and negative numbers to the left.
Step 4: By identifying the correct positions of natural numbers, whole numbers, and integers, we can easily place them on a number line. If we have a large number, such as 100, then we can use a larger scale, for example, marking each unit 1 as unit 20.
Through this, we can reach 100 after taking 5 steps.
Step 5: First, convert the rational or irrational numbers to a decimal form, to mark them on a number line.
Now we can mark real numbers such as -7/2, -2, 0, 1/2, and 3 on a number line.
How to Represent the Ordering of Real Numbers on a Number Line?
We use the number line to compare and arrange real numbers. Symbols such as greater than (>), less than (<), and equal to (=) are used to compare numbers.
Step 1: On a number line, the larger numbers are placed to the right and the smaller numbers are positioned to the left.
Step 2: On the number line, negative numbers are always on the left side of the zero. The negative numbers closer to zero are considered greater. For example, -3 is greater than -13 because -3 is closer to the origin point than -13.
Take a look at the given image to understand the comparison of real numbers on a number line.
Here -3 > -13.
Absolute Value of a Real Number on a Number Line
The absolute value is the distance between a real number and the origin of the number line. It is denoted |x|, where x is the real number. Since distance is always positive, the absolute value will also be positive. For instance, 3 is a real number, then the absolute value will be |3| = 3. Take a look at the given image:
Here, the real number (3) is 3 units away from the origin. The distance between a negative number and the origin point is the same as its equivalent positive number. For example, |-3| = 3, here also -3 is 3 units far from the origin.
Common Mistakes and How to Avoid Them in Representation of Real Numbers on Number Line
When students represent real numbers on a number line, they sometimes make some mistakes that lead to incorrect comparisons and arrangement of numbers. Here are some common mistakes and helpful solutions to avoid errors when representing real numbers on a number line.
Mistake 1
Forgetting the Origin as a Reference
Students should use the origin (0) as a reference when they mark real numbers on the number line. If they ignore the origin, they will mark real numbers at inconsistent intervals. They should place positive numbers to the right, while negative numbers to the left side of the origin.
Mistake 2
Ignoring the Equal Distance Between Numbers
Always remember to keep an equal interval between numbers when drawing a number line. They can use a ruler or a fixed scale to keep an equal spacing between numbers. If they mistakenly mark 3 closer to 4, the number line will be confusing and lead to incorrect conclusions.
Mistake 3
Placing Negative Numbers on the Right Side of the Origin
Remember to place negative numbers to the left of the origin (0). Only positive numbers should be marked to the right of the number line. Keep in mind that greater negative numbers are closer to the origin. For example, -2 is greater than -4 because it is closer to the origin (0).
Mistake 4
Confusing Positive and Negative Decimals
Students should remember that positive decimals must be placed between positive numbers and negative decimals marked between negative numbers. For example, it is incorrect when kids mistakenly mark -0.5 between 0.5 and 1. It should be placed on the left side of the origin, between -0.6 and -1 since -0.5 is a negative number.
Mistake 5
Forgetting to Convert Fractions into Decimals
Keep in mind to convert fractions to decimal form when marking fractions on a number line. By dividing the numerator by the denominator, students can change the fractions to decimals. Place the decimal number carefully on the number line between the whole numbers. For example, the fraction 1/2 can be converted to 0.5 and carefully marked between 0 and 1 on the number line.
Real-life Applications of Representation of Real Numbers on Number Line
The real number line plays an important role in various real-life situations by helping to compare and arrange numbers in order. The real-life applications of the real number line are as follows:
- Weather forecasters and geologists use the real number line to represent the temperatures and their variations. For example, the temperature in Los Angeles is 17℃, and in New York it is -7℃. By using a number line, they can easily compare temperatures and find the temperature differences between these two cities.
- In sports, scores are labeled on a number line, the positive scores of a team are marked on the right side, and the negative scores are on the left. For example, in a football match, team A scored + 3, which means they achieved 3 goals, whereas team B lost 2 points, bringing their score to -2.
- The savings and debts of an account holder can be represented using a number line. It helps the bank to understand the person’s balance and loans accurately. For example, if a person has $300 in his account and withdraws $500, now he has a balance of -$200.
- When cooking, we can use a real number line to mark the quantity of items used for a recipe. For instance, to bake a cake we need 2/5 cups of flour and 3/4 cups of milk. These fractions can be easily marked on a number line which helps visualize and measure ingredients easily.
Solved Examples of Representation of Real Numbers on Number Line
Problem 1
Represent -3.5 on the number line.
Explanation
To begin, draw a straight line and mark 0 as its origin.
Then find -3 on the number line.
Thus, -3.5 will fall exactly in the middle of -3 and -4.
Divide the space between -3 and -4 into 10 equal parts and mark the given number on the number line.
Problem 2
Find and plot a rational number between 1/10 and 3/8.
Explanation
We can convert 1/10 and 3/8 to a common denominator.
The common denominator of the unlike fractions 1/10 and 3/8 is 40.
1/10 = 1 × 4 / 10 × 4 = 4/40
Thus, 1/10 = 4/40.
Next, we can convert 3/8 to a denominator of 40.
3/8 = 3 × 5 / 8 × 5 = 15/40
Thus, 3/8 = 15/40.
Now, find the midpoint:
(4/40 + 15/40) ÷ 2 = 19/40 ÷ 2 = 19/80.
So, 19/80 is a rational number between 1/10 and 3/8, and plot the number on the number line.
Problem 3
Represent 5/6 on the number line.
Explanation
To begin, we can convert the fraction into its decimal form.
5 ÷ 6 = 0.8333...
0.8333… is a non-terminating decimal, so we can round it to 0.83, which is an approximate value.
This means, 5/6 is slightly greater than 0.8 and lower than 0.85.
Now we can draw a number line and mark 0 and 1.
To represent tenths, divide the space between 0 and 1 into 10 equal parts (0.1, 0.2,... 09, 1.0).
5/6 can be marked between 0.8 and 0.9.
Plot the points between 0.8 and 0.9 on the number line.
Again divide the space between 0.8 and 0.9 into 10 equal parts to represent hundredths (0.01, 0.02,...)
Now count 3 steps after 0.8 because:
0.83 = 0.80 + 0.03.
On the number line, mark the point as 5/6. In the image, the black dot stands for the 5/6.
Problem 4
Find and plot a rational number between -5/6 and -1/3.
Explanation
First, we need to convert -5/6 and -1/3 to like fractions.
For that, we must find a common denominator. 12 is the common denominator of 6 and 3 (because it is the least common denominator).
-5/6 =(-5 × 2) / (6 × 2) = -10/12
-1/3 = (-1 × 4) / (3 × 4) = -4/12
Now find the midpoint of -10/12 and -4/12:
(-10/12 + -4/12) ÷ 2 = -14/12 ÷ 2 = -7/12
Next, mark the spot -7/12 on the number line.
Therefore, -7/12 is a rational number between -5/6 and -1/3.
Problem 5
Represent 4.5 on the number line.
Explanation
Draw a number line and mark 4 and 5.
Divide the space between 4 and 5 into 10 equal parts.
Spot 4.5 between 4 and 5.
Mark 4.5 on the number line.
FAQs of Representation of Real Numbers on Number Line
1. Define real numbers.
2.Can we represent a fraction on a number line?
3.Can we represent a decimal number on a number line?
4.How can we represent negative and positive numbers on a number line?
5.Can we represent irrational numbers on a number line?
6.How can children in United States use numbers in everyday life to understand Representation of Real Numbers on Number Line?
7.What are some fun ways kids in United States can practice Representation of Real Numbers on Number Line with numbers?
8.What role do numbers and Representation of Real Numbers on Number Line play in helping children in United States develop problem-solving skills?
9.How can families in United States create number-rich environments to improve Representation of Real Numbers on Number Line skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
