Last updated on July 4th, 2025
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.
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’
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:
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.
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.
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.
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.
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:
Represent -3.5 on the number line.
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.
Find and plot a rational number between 1/10 and 3/8.
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.
Represent 5/6 on the number line.
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.
Find and plot a rational number between -5/6 and -1/3.
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.
Represent 4.5 on the number line.
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.
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.
: She loves to read number jokes and games.