Real Numbers (original) (raw)

Last Updated : 12 May, 2026

Real numbers are the set of numbers that can represent a quantity along a continuous number line. They include both rational and irrational numbers and can be positive, negative, or zero.

Real number sets

**Key Points

• Real numbers can be added, subtracted, multiplied, and divided (except by zero).
• They are used to measure continuous quantities such as length, time, temperature, and distance.
• Every point on the number line corresponds to a real number.

Types of Real Numbers

Real numbers are divided into two main types:

Hierarchy of real numbers based on their subsets

**1. **Rational Numbers (ℚ): Numbers that can be expressed as a fraction (p/q) where both the numerator represented as p and the denominator represented as q are integers, and the denominator (q) is not zero. Rational numbers include integers, finite decimals, and repeating decimals (e.g., 1/2, -3, 0.75).

2. **Irrational Numbers: Numbers that cannot be expressed in the form of a simple fraction p/q where 'p' and 'q' are integers and the denominator 'q' is not equal to zero (q≠0) and have non-terminating, non-repeating decimal expansions. They cannot be represented as a fraction of two integers (e.g., √2, π).

Real numbers can be further divided into the following subsets:

Symbols

We use **R to represent a set of real numbers, and other types of numbers can be represented using the symbol discussed below.

**Real Numbers on a Number Line

When real numbers are placed on a number line, each value corresponds to a unique point on that line. The number line extends infinitely in both the positive and negative directions.

Marked real numbers

**How Real Numbers Appear on a Number Line

• Zero (0) is at the center.
• Positive real numbers lie to the right of 0.
• Negative real numbers lie to the left of 0.
• The line continues forever on both sides.

Solved Examples

**Example 1: Add √3 and √5

(√3 + √5)

Now answer is an irrational number.

**Example 2: Multiply √3 and √3.

√3 × √3 = 3

Now answer is a rational number.

So we can say that result of mathematical operations on irrational numbers can be rational or irrational.

**Example 3: Represent the following numbers on a number line: 23/5, 6, and -33/7.

**Question 4: Plot the following rational numbers on the number line: −50/9, 3/2, 13/4.

Practice Problems

**Question 1: Add √2 and √8.

**Question 2: Multiply √7 by √14.

**Question 3: Add 5 to √9.

**Question 4: Add 3/2 (a rational number) to √3 (an irrational number)

**Question 5: Classify the Following Numbers as Rational or Irrational:

• a) \sqrt{49}
• b) \pi
• c) 0.75
• d) \sqrt{2}

**Question 6: Find the LCM and HCF of 24 and 36.

**Question 7: Express 0.3333… (repeating) as a fraction.

**Question 8: Prove that \sqrt{3} is an irrational number.

**Question 9: Find the Decimal Representation of \frac{7}{8}. Is it terminating or non-terminating?

**Question 10: Find the HCF of 75 and 105 using the Euclidean algorithm.

**Question 11: Write the prime factorization of 420 and 252. Use it to find their LCM.

**Question 12: Is 0.1010010001… a rational or irrational number? Justify your answer.

**Related Articles:

• Properties of real numbers
• Operations on real numbers
• Decimal Expansion of real numbers