Representation of Real Numbers on Number Line: A number line is a straight line that extends indefinitely on both ends. It is used to represent both positive and negative integers in equal intervals. Real numbers are set of integers, whole numbers, natural numbers, rational and irrational numbers.

A number line can be used to represent real numbers for comparing and ordering the numbers. This article will study how to represent the real number on a number line in detail.

Definition of Real Numbers

Real numbers are a combination of rational numbers \(Q\) and irrational numbers \(Q’.\) A set of real numbers is indicated by \(R.\) All numbers such as whole numbers, natural numbers, integers, decimals, rational and irrational numbers belong to a set of real numbers. Thus, we can write \(R=Q+Q’.\)

The following figure represents the real numbers:

Real Number Line

In a number line, the origin is \(0,\) and the number on the right side of zero are positive integers and negative integers on the left side. Every real number can be represented on a number line by a unique point. A number line representing the real number is simply called a number line. 

The figure below represents both the negative and positive numbers on a number line.

Steps to Represent Real Numbers on a Number Line

The following steps are followed to represent the real numbers in a number line:

1. Step 1: Draw a horizontal straight line with arrows on both ends and mark a point naming \(0\) anywhere. The number zero is called the origin.
2. Step 2: Locate real numbers at an equal interval on both sides of the origin and label them with a definite scale.
3. Step 3: Positive integers are on the right side of the origin, and the negative integers are on the left side of the origin. 
4. Step-4: Whole numbers, natural numbers, and integers can be easily marked by locating the particular point on the number line. In the case of large numbers, the scale of the number line may be adjusted accordingly. If the given point is a large number, let’s say \(120.\) We can choose \(1\) unit as \(20,\) so to mark \(120,\) we have to take \(6\) steps towards the right side of the origin.
5. Step-5: Rational and irrational numbers must be first converted to decimal equivalents and then marked on the number line. Sometimes, it is not possible to mark a number on the exact point. In such a case, we use the process of successive magnification. Visualizing the numbers on the number line using the magnifying glass is called the process of successive magnification.

Example: Mark the real numbers \(\frac{-7}{2}, \frac{-3}{2}, \frac{1}{2}\) and \(3\) on a number line.

Integers on Number Line

Integers consist of positive and negative integers. Integers increase from left to right on the number line. \(0\) lies at the centre of positive and negative integers. Positive integers are \(1, 2, 3, …\) while negative integers are \(-1, -2, -3, ….\)

The number line below represents the integers.

Rational Numbers on Number Line

Rational numbers are the number in the form of \(\frac{p}{q} .\) . We divide the two adjacent numbers into n numbers of parts to represent the rational numbers.

For example: To represent \(\frac{a}{3}\) we divide each equal interval into three parts and represent as \(\frac{1}{3}, \frac{2}{3}, \frac{3}{3} \ldots\) so on. Below is the number line that represents the required rational numbers.

Decimals on Number Line

To represent the decimal numbers, we divide the two consecutive integers into \(10\) parts for one decimal point or \(100\) parts for two decimal points on so on.

For example: Represent \( -0.7\) and \(0.4\) on the number line.
Since \(0.4\) has the whole number \(0,\) so it will lie between \(0\) and \(1\), and \(-0.7\) will lie between \(0\) and \(-1.\)
The below number line represents the required decimals.

Irrational Numbers on Number Line

Irrational numbers are real numbers that cannot be represented as a simple fraction. They cannot be expressed in the form of a ratio.
Let us learn how irrational numbers are located on a number line.

For example: Take \(\sqrt{2}\) which is an irrational number

1. Draw a number line with an origin, positive and negative integers.
2. Name origin as \(A,\) and \(1\) as \(B.\)
3. Using a compass, measure the distance between \(0\) and \(1.\) Draw a perpendicular at \(1\) of the same length as between \(0\) and \(1.\)
4. Name the end of the perpendicular line as \(C.\) Join \(AC.\)
5. Distance between \(AC\) is \(\sqrt{2}\) by Pythagoras theorem.\(\left(A C=\sqrt{1^{2}+1^{2}}=\sqrt{2}\right)\)
6. Using compass and radius \(AC,\) draw an arc to cut the number line at \( D.\)
7. \(D\) is the point representing an irrational number \(\sqrt{2}\)

Similarly, other irrational numbers like \(\sqrt{3}, \sqrt{5}, \ldots\) can be represented on a number line.

Here, the irrational number is \(\sqrt{2}\)

Comparison of Real Numbers on a Number Line

Real numbers can be compared on a number line. As we move from left to right, the value of numbers increases. In other words, numbers on the left side of zero are smaller numbers, while those on the right side of zero are larger numbers. Comparing negative real numbers is different. 

Example: \(-3\) is greater than \(-6\) as the negative numbers start from the left side of the origin. It becomes simpler once we use the number line for comparing.

Symbols like less than \((<),\) greater than \((>),\) and equal to \((=)\) are used to compare the real numbers. 

A number line is drawn below to represent the above example of comparing two real numbers.

Opposite Real Numbers on a Number Line

Opposite real numbers are numbers with the same value but opposite signs. In the number line, the opposite real numbers are at equal distances from the origin on the opposite side of the origin. 

Example: The opposite of \(7\) is \(-7.\) The below number line represents the example of the opposite real number.

Absolute Value of a Real Numbers on a Number Line

In mathematics, the absolute value or modulus of a real number \(x,\) denoted by \(|x|,\) is the non-negative value of \(x\) irrespective of its sign and indicates the distance from the origin. 

Example: \(|-4|=4\) and \(|4|=4.\) Look at the number line below for reference:

Solved Examples – Representation of Real Numbers on Number Line

Q.1. Represent \(-8\) on a number line.
Ans: Draw the number line with the origin, positive and negative integers
Now, locate \(-8\) on the number line

Q.2. Compare \(-9\) and \(2\) on a number line.
Ans: Locate \(-9\) and \(2\) on the number line.
We know that the value increases from left to right. In other words, the numbers on the right of the origin increase and the left of the origin decrease.

Thus, \(-9<2.\)

Q.3. Represent the opposite of  \(\frac{{ – 2}}{5}\)  on a number line
Ans: Opposite of any number is the number with the opposite sign. \(\frac{2}{5}\) is the opposite of \(\frac{{ – 2}}{5}\)
The following number line represents the opposite of  \(\frac{{ – 2}}{5}\)

Q.4. Graph the following set of real numbers on a number line \(\left\{ {0,\,0.3,\,0.6,\,0.9,\,1.2} \right\}\) 
Ans: The given set of real numbers are of decimal point one. So we divide one unit into \(10\) divisions and marks the decimal points as \(0.1, 0.2, 0.3,\) etc.

Q.5. List three integers greater than \(-10\) on a number line.
Ans: The value of numbers increases from left to the right. All the integers that lie on the right side of \(-10\) are greater than \(-10.\) Three integers \(-6, -4,\) and \(1\) are marked on the number line below are greater than \(-10.\)

Summary

In this article, we learnt the definition of real numbers and subsets of real numbers. We studied what a number line is and how to represent real numbers on a number line. We have explained the steps involved in representing the real numbers on a number line with examples. Also, there are solved examples that help you understand the concept clearly.

Frequently Asked Questions

Q.1. What is representing real numbers on the number line?
Ans:  Real numbers are a set of numbers that consist of whole numbers, natural numbers, rational and irrational numbers. Each real number has a unique point on the number line. And each point represents one and only one real number. We sometimes use the successive magnification method to represent real numbers on the number line. 

Q.2. Does the square root of a negative number belong to a real number?
Ans: No, the square root of a negative number does not belong to a real number.
For example, \(\sqrt{-2}, \sqrt{-3}\) are complex numbers that do not belong to the set of real numbers.

Q.3. If we subtract 1 from any number on the number line, on which side of the number, the new number will be? 
Ans: The new number will be on the left side of the original number. In the number line, numbers are in increasing order from left to right. The origin lies at the centre. The positive integers are on the right side, and the negative integers are on the left side of the origin.
When we have to compare two numbers on a number line,
numbers on right side \(>\) numbers on left side
If we subtract \(1\) from the number, then the new number will be smaller than the original number. Hence new number would be on the left side of the original number.

Q.4. Where does  \(\frac{2}{3}\)  lie on a number line?
Ans: On the real number line, it is between \(0\) and \(1.\) To be more precise, the distance between \(\frac{2}{3}\) and \(0\) is twice the distance between \(\frac{2}{3}\) and \(1.\)

Q.5. What does every point on a number line represent? 
Ans: Every point of a number line is assumed to correspond to a real number and every real number to a point. The integers are often shown as specially marked points evenly spaced on the line. It is often used to teach simple addition and subtraction, especially involving negative numbers.

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