Real Number Line: Definition, Properties and Representing Real Numbers on the Number Line

The real number line is a fundamental concept in mathematics that serves as a visual representation of real numbers, including all rational and irrational numbers. This line provides a way to understand and work with numbers in a continuous manner, where each point on the line corresponds to a unique real number. Real numbers like 1, 2, 3, to fractions like ½ and decimals like 0.75, and even numbers like the square root of 2 that can’t be written as a simple fraction, all real numbers have their place on this line. A strong knowledge of the real number line is important for various competitive exams like JEE, NEET, SSC, Banking, and many state-level exams. To better understand this basic idea of the real number line, let’s go into more detail.

What is a Real Number Line?

The real number line is a one-dimensional line where every point corresponds to a unique real number. This concept allows for a visual and geometric interpretation of real numbers, making it easier to understand their relationships and properties. The real number line extends infinitely in both directions, typically represented with arrows, and includes all rational numbers (fractions and integers) and irrational numbers (non-repeating, non-terminating decimals).

<——I——I——-I——-I——I——I——-I——-I——I——>

    -3     -2      -1       0      1       2       3       4       5

Where,   -π ≈ -3.14 π ≈ 3.14 √2 ≈ 1.41

1. Integer on the Number Line:

• The number 3 is represented as a point three units to the right of zero on the number line.
• The number −2 is represented as a point two units to the left of zero.

2. Fraction on the Number Line:

• The number ½ ​ is represented as a point halfway between 0 and 1.

3. Irrational Number on the Number Line:

• The number √2, which is approximately 1.414, is represented as a point slightly more than 1.4 units to the right of Zero.
• The number π, which is approximately 3.14159, is represented as a point slightly more than 3.14 units to the right of Zero.

Also Read: Rational And Irrational Numbers: Differences, Examples

Steps to Represent Real Numbers on the Number Line

Here we have listed down the steps involved in representing real numbers on a number line, along with visual aids:

Step 1. Draw a Horizontal Line:

• Start by drawing a straight horizontal line. This line will represent all real numbers.

Step 2. Mark the Origin:

• Choose a point near the center of the line and label it as 0. This point is called the origin.

Step 3. Mark Positive Numbers:

• To the right of the origin, mark points at equal intervals and label them with positive integers (1, 2, 3, etc.). Each interval represents one unit.

Step 4. Mark Negative Numbers:

• To the left of the origin, mark points at the same intervals and label them with negative integers (-1, -2, -3, etc.).

Step 5. Include Fractions and Decimals:

• Between the integers, you can mark fractions and decimals. For example, halfway between 0 and 1, mark ½ ​ or 0.5.

Step 6. Add Irrational Numbers:

• Place irrational numbers approximately where they belong. For example, √2 (approximately 1.414) will be a bit to the right of 1, and π (approximately 3.14159) will be a bit to the right of 3.

  <——I——I——-I——-I——I——I——-I——-I——I——>

 -3     -2      -1        0       1       2       3       4       5

Where,   -π ≈ -3.14, π ≈ 3.14, √2 ≈ 1.41

Step 1: Draw the horizontal line.

Step 2: Mark and label the origin as 0.

Step 3: Mark and label positive integers to the right of 0.

Step 4: Mark and label negative integers to the left of 0.

Step 5: Mark fractions like ½ ​ between 0 and 1.

Step 6: Place irrational numbers like √2 and π at their approximate locations.

Properties of Real Number Line

Here are some of the important properties of the real number line.

1. Continuity:The real number line is continuous, meaning there are no gaps. Every point on the line represents a real number, and between any two points, there are infinitely many other points.
2. Order:Numbers on the real number line are ordered from left to right. Numbers to the right are greater, and numbers to the left are smaller. For example, 2 is to the right of 1, 2>1.
3. Origin:The point labeled 0 is called the origin. It divides the number line into positive numbers (to the right) and negative numbers (to the left).
4. Distance:The distance between any two points on the number line represents the absolute value of the difference between the two numbers. For example, the distance between 3 and −2 is ∣3−(−2)∣=∣3+2∣=5.
5. Density:The real number line is dense, meaning between any two real numbers, there is always another real number. For example, between 1 and 2, there is 1.5, and between 1 and 1.5, there is 1.25, and so on.
6. Symmetry:The number line is symmetric around the origin. For every positive number, there is a corresponding negative number at the same distance from 0. For example, 2 and −2 are equidistant from the origin but in opposite directions.

Real Number Line Solved Examples

Here are five solved examples of Real Number Line to help you understand how it’s shown and to get you practicing:

Example 1: Representing Integers on a Number Line

Problem: Represent the integers -3, 0, and 4 on a number line.

Solution:

• Draw a horizontal line.
• Mark a point as 0, the origin.
• Move 3 units to the left of 0 and mark it as -3.
• Move 4 units to the right of 0 and mark it as 4.

Example 2: Representing Fractions on a Number Line

Problem: Represent the fractions 1/2 and -3/4 on a number line.

Solution:

• Divide the line segment between 0 and 1 into two equal parts. The midpoint represents 1/2.
• Divide the line segment between -1 and 0 into four equal parts. The third point to the left of 0 represents -3/4.

Example 3: Representing Decimals on a Number Line

Problem: Represent the decimals 0.3 and -1.2 on a number line.

Solution:

• Divide the line segment between 0 and 1 into ten equal parts. The third point to the right of 0 represents 0.3.
• Divide the line segment between -1 and 0 into ten equal parts. The second point to the left of -1 represents -1.2.

Example 4: Representing Irrational Numbers on a Number Line (Approximation)

Problem: Approximate the position of √2 on a number line.

Solution:

• Since √2 is between 1 and 2, mark 1 and 2 on the number line.
• Divide the segment between 1 and 2 into ten equal parts.
• √2 is closer to 1.4, so approximate its position accordingly.

Also Read: Define Line in Maths: 9 Types and Examples

Example 5: Ordering Numbers on a Number Line

Problem: Order the numbers -2, 3/4, 0, -1.5, and 2.5 from least to greatest.

Solution:

• Plot the numbers on a number line.
• The order from left to right is the order from least to greatest.
• The order is: -1.5, -2, 0, 3/4, 2.5.

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